#  Combining Evidence Across Filtrations

Yo Joong Choe
Data Science Institute
University of Chicago
<EMAIL_ADDRESS>Part of the work was done while this author was at
Carnegie Mellon University. Aaditya Ramdas
Department of Statistics and Data Science
Machine Learning Department
Carnegie Mellon University
<EMAIL_ADDRESS>
(February 14, 2024)

###### Abstract

In anytime-valid sequential inference, it is known that any admissible
inference procedure must be based on test martingales and their composite
generalization, called _e-processes_ , which are nonnegative processes whose
expectation at any arbitrary stopping time is upper-bounded by one. An
e-process quantifies the accumulated evidence against a composite null
hypothesis over a sequence of outcomes. This paper studies methods for
_combining_ e-processes that are computed using different information sets,
i.e., filtrations, for a null hypothesis. Even though e-processes constructed
on the same filtration can be combined effortlessly (e.g., by averaging),
e-processes constructed on different filtrations cannot be combined as easily
because their validity in a coarser filtration does not translate to validity
in a finer filtration. We discuss three concrete examples of such e-processes
in the literature: exchangeability tests, independence tests, and tests for
evaluating and comparing forecasts with lags. Our main result establishes that
these e-processes can be _lifted_ into any finer filtration using _adjusters_
, which are functions that allow betting on the running maximum of the
accumulated wealth (thereby insuring against the loss of evidence). We also
develop randomized adjusters that can improve the power of the resulting
sequential inference procedure.

###### Contents

  1. 1 Introduction
    1. 1.1 Motivating example: Combining e-processes for testing exchangeability
    2. 1.2 Related work on combining e-values (and p-values)
    3. 1.3 Outline
  2. 2 Preliminaries
    1. 2.1 Anytime-valid inference: E-processes, p-processes, and sequential tests
    2. 2.2 Calibrating an e-process and a p-process
    3. 2.3 Adjusting the running maximum of an e-process
    4. 2.4 One-to-one correspondence between p-to-e calibrators and adjusters
  3. 3 Combining evidence across filtrations via lifting
    1. 3.1 $\mathfrak{p}$-lifting: How to combine an e-process with a p-process from a sub-filtration
    2. 3.2 $\mathfrak{e}$-lifting: How to combine an e-process with an e-process from a sub-filtration
  4. 4 Three illustrative use cases
    1. 4.1 Lifting and combining e-processes for testing exchangeability (revisited)
    2. 4.2 Combining e-processes across different sub-filtrations: the case of comparing forecasts with lags
    3. 4.3 Combining e-processes constructed using $k$-wise examples: the case of sequential independence testing
  5. 5 Randomized adjustments of e-processes across filtrations
    1. 5.1 The lift-then-randomize procedure
    2. 5.2 Lifting the randomized Ville’s inequality
    3. 5.3 Can a randomize-then-lift procedure also be $\mathbb{F}$-anytime-valid?
  6. 6 Discussion
    1. 6.1 A theoretical implication on when powerful e-processes exist for a testing problem
    2. 6.2 A practical implication on designing e-processes by coarsening the filtration
    3. 6.3 Other variations of the setup
    4. 6.4 A game-theoretic protocol for combining evidence: Cooperative skepticism
  7. A Details on e-processes for testing exchangeability
  8. B Additional example: Testing a scale-invariant Gaussian mean
  9. C Proofs
    1. C.1 Proof of Lemma 1
    2. C.2 Proof of Theorem 2
    3. C.3 Proof of Lemma 2

##  1 Introduction

Given a null hypothesis over a sequence of observations, how do we combine
pieces of evidence that are based on different information sets? This question
arises in hypothesis testing scenarios where multiple evidence processes for a
null are constructed w.r.t. different filtrations. In this paper, we
specifically consider the problem of combining evidence for composite null
hypotheses that are measured in terms of _e-processes_ (Ramdas et al.,, 2020),
which are sequential generalizations of _e-values_ (Vovk and Wang,, 2021;
Grünwald et al.,, 2023) or _betting scores_ (Shafer,, 2021). An e-value $E$
for a composite null hypothesis $\mathcal{P}$ is a nonnegative random variable
whose expectation is at most one under any null distribution, that is,
$\mathbb{E}_{P}[E]\leq 1$ for any $P\in\mathcal{P}$. E-values generalize
likelihood ratios to composite and nonparametric hypothesis tests, and any
p-value can be _calibrated_ into an e-value (Shafer et al.,, 2011). A key
benefit of using e-values, as opposed to using p-values, is that we can
seamlessly combine e-values across tests with arbitrary dependence structures
by averaging them.

Despite much attention on the ease of combining e-values, recent work on
designing e-processes for _sequential_ testing scenarios has shown that we
face a rather subtle yet substantial challenge when combining e-processes that
have different underlying filtrations (Grünwald et al.,, 2023; Ramdas et al.,,
2022; Henzi and Ziegel,, 2022; Choe and Ramdas,, 2023; Arnold et al.,, 2023;
Pérez-Ortiz et al.,, 2022). As we will see later, there are various types of
scenarios in which this challenge arises. Here, as a starting point, we
describe one such scenario, where one e-process is constructed on a
_coarsening_ of the filtration under which another e-process is computed,
using a concrete example.

###  1.1 Motivating example: Combining e-processes for testing exchangeability

######  Example 1 (Combining e-processes for testing exchangeability).

Let $\mathcal{P}^{\mathsf{exch}}$ describe the set of all distributions under
which the sequence of binary random variables $X_{1},X_{2},\dotsc$ are
exchangeable.111Formally, a sequence of random variables $(X_{t})_{t\geq 1}$
is _exchangeable_ if, for any $t\geq 1$ and any permutation $\sigma$ over
$\\{1,\dotsc,t\\}$, $(X_{1},\dotsc,X_{t})$ has the same distribution as
$(X_{\sigma(1)},\dotsc,X_{\sigma(t)})$. Since any independent and identically
distributed (i.i.d.) sequence is exchangeable, any (anytime-)valid test for
exchangeability is also an (anytime-)valid test for whether the sequence is
i.i.d. (also known as a test for _randomness_). Ramdas et al., (2022,
corollary 16) further show that a sequence is an e-process for testing the
i.i.d. assumption if and only if it is also an e-process for testing
exchangeability. Let $\mathfrak{e}=(\mathfrak{e}_{t})_{t\geq 0}$ be an
e-process for $\mathcal{P}^{\mathsf{exch}}$ w.r.t. the data filtration
$\mathcal{F}_{t}=\sigma(X_{1},\dotsc,X_{t})$, such as the _universal
inference_ (UI; Wasserman et al.,, 2020) e-process of Ramdas et al., (2022);
let $\mathfrak{e}^{\prime}=(\mathfrak{e}_{t}^{\prime})_{t\geq 0}$ be another
e-process for $\mathcal{P}^{\mathsf{exch}}$ w.r.t. a sub-filtration
$\mathbb{G}$ of $\mathbb{F}$, such as the _conformal test martingale_ of Vovk,
2021b . For the conformal test martingale, the sub-filtration $\mathbb{G}$
consists of conformal p-values, i.e.,
$\mathcal{G}_{t}=\sigma(p_{1},\dotsc,p_{t})$, where each conformal p-value
$p_{t}$ is a function of some nonconformity scores for $X_{1},\dotsc,X_{t}$.
(We defer the precise definitions of $\mathfrak{e}$ and
$\mathfrak{e}^{\prime}$ to Appendix A.)

The two e-processes are known to be useful in their separate ways. The UI
e-process yields the only known anytime-valid sequential test for
exchangeability that allows freely peeking at the data sequence
$X_{1},X_{2},\dotsc$ as it becomes available over time. This notion of
anytime-validity, w.r.t. the filtration $\mathbb{F}$, directly follows from
the definition of an e-process: at any $\mathbb{F}$-stopping time
$\tau^{\mathbb{F}}$, an e-process is expected to be upper-bounded by one under
$\mathcal{P}^{\mathsf{exch}}$, i.e.,
$\mathbb{E}_{P}[\mathfrak{e}_{\tau^{\mathbb{F}}}]\leq 1$ for all
$P\in\mathcal{P}^{\mathsf{exch}}$. The UI e-process is also powerful against
first-order Markovian alternatives. In contrast, the conformal test martingale
requires that the experimenter only peeks at the “conformal p-values”
$p_{1},\dotsc,p_{t}$, and _not_ the actual data, although it is powerful
against changepoint alternatives. In practice, we may want to combine the two
e-processes into one e-process that retains power against both alternatives.

To combine the two e-processes $\mathfrak{e}$ and $\mathfrak{e}^{\prime}$, it
is tempting to consider their mean $M=(M_{t})_{t\geq 0}$:

$M_{t}=\frac{1}{2}\left(\mathfrak{e}_{t}+\mathfrak{e}_{t}^{\prime}\right),\quad\forall
t\geq 0.$ (1)

Since each conformal p-value $p_{t}$ is $\mathcal{F}_{t}$-measurable, we know
that $M$ is a process adapted to $\mathbb{F}$. However, because
$\mathfrak{e}^{\prime}$ is valid at only $\mathbb{G}$-stopping times and _not_
$\mathbb{F}$-stopping times, $M$ is not an e-process w.r.t. $\mathbb{F}$. On
the other hand, while $M$ is valid at arbitrary $\mathbb{G}$-stopping times,
it is not an e-process w.r.t. $\mathbb{G}$ either because it is simply not
adapted to $\mathbb{G}$ ($\mathfrak{e}_{t}$ is not
$\mathcal{G}_{t}$-measurable in general).

In Figure 1, we empirically demonstrate how the validity of
$\mathfrak{e}^{\prime}$ at $\mathbb{G}$-stopping times does _not_ carry over
to $\mathbb{F}$-stopping times. For each run, we sample data from an i.i.d.
$\mathsf{Ber}(0.3)$, which belongs to $\mathcal{P}^{\mathsf{exch}}$, and
define the stopping time $\tau^{\mathbb{F}}$ as the first time we observe five
consecutive zeros, that is,

$\tau^{\mathbb{F}}=\inf\\{t\geq 5:X_{t-4}=X_{t-3}=X_{t-2}=X_{t-1}=X_{t}=0\\}.$
(2)

While $\tau^{\mathbb{F}}$ is a $\mathbb{F}$-stopping time, it is not a
$\mathbb{G}$-stopping time as it depends directly on the individual data
points and not just the conformal p-values. Then, for 100 repeated runs, we
plot both the UI e-process and the conformal test martingale, specifically of
the form $\mathfrak{e}_{t}^{\prime}=\prod_{i=1}^{t}(1+(p_{i}-1/2))$, up until
they each reach $\tau^{\mathbb{F}}$. Figure 1(a) shows that, unlike the UI
e-process that remains small ($\leq 1$) across repeated runs, many instances
of conformal test martingales reach values exceeding 1 at $\tau^{\mathbb{F}}$.
In Figure 1(b), we further plot the histogram and the estimated mean (vertical
line) of the stopped e-values ($\mathfrak{e}_{\tau^{\mathbb{F}}}$ and
$\mathfrak{e}_{\tau^{\mathbb{F}}}^{\prime}$) across 10,000 simulations. We
find that the mean of stopped conformal e-values is significantly larger than
$1$ ($1.33\pm 0.016$), while the mean of stopped UI e-values remains small
($0.25\pm 0.008$).

(a) E-processes and their stopping times

(b) Histograms of stopped e-values (vertical line: mean)

Figure 1: Test martingales and e-processes defined w.r.t. a coarsened
filtration $\mathbb{G}$ of $\mathbb{F}$ are not valid at $\mathbb{F}$-stopping
times. In 1(a), we plot the universal inference e-process (solid blue) and the
conformal test martingale (dashed orange) for testing exchangeability using
data generated from i.i.d. $\mathsf{Ber}(0.3)$. These e-processes are stopped
at an $\mathbb{F}$-stopping time $\tau^{\mathbb{F}}$: the first time five
consecutive zeros are observed in the data. Even though the data comes from a
null distribution, many stopped e-values are above $1$ for the conformal test
martingale. In 1(b), we plot the histogram of stopped e-values at
$\tau^{\mathbb{F}}$ over 10,000 repeated simulations. The estimated mean of
stopped e-values for the conformal test martingale (vertical dashed orange
line) is $1.33\pm 0.016$, implying that
$\mathbb{E}[\mathfrak{e}_{\tau^{\mathbb{F}}}]>1$.

Example 1 makes clear that an e-process w.r.t. a coarsened filtration
($\mathbb{G}$) need not be an e-process w.r.t. the original filtration
($\mathbb{F}$). This phenomenon poses both a technical and practical challenge
when combining sequentially obtained evidence across different
filtrations.222Technically, in the example above, the simple average of the
two e-processes may appear valid empirically, but this is only because the UI
e-process is relatively conservative. If we were to combine the conformal test
martingale with any other e-process that remains close to 1 under i.i.d.
$\mathsf{Ber}(0.3)$, the mean of the stopped simple average
$a_{\tau^{\mathbb{F}}}$ would exceed $1$. On the technical side, this puts an
emphasis on specifying an e-process’ underlying filtration, which is often
“swept under the rug.” On the practical side, this means that the experimenter
must be cautious about what information is included in the stopping rule,
since accessing extra information (either inadvertently or deliberately) can
invalidate the resulting test, whenever that information “leaks” into the
decision to stop the experiment. This can be impractical when the e-process is
only valid under a coarse filtration and the extra information is easily
accessible, such as when the coarse filtration does not include the data
sequence.

In Appendix B, we include another example due to Pérez-Ortiz et al., (2022)
that explicitly shows how an e-process for testing a scale-invariant Gaussian
mean is not valid w.r.t. the data filtration.

Motivated by these examples, we develop methods for combining e-processes (and
p-processes) defined across different filtrations, such that the combined
process is valid at any data-dependent stopping times. We then illustrate
their applications to various sequential testing scenarios. Central to our
approaches is the use of _adjusters_ and _(p-to-e) calibrators_ (Shafer et
al.,, 2011; Dawid et al., 2011b, ; Ramdas et al.,, 2020; Vovk and Wang,, 2021;
Ramdas et al.,, 2022), which allows us to _lift_ an e-process from a coarse
filtration into a finer one. This is somewhat surprising because these tools
were originally devised for different purposes, namely to be able to use the
running maximum of an e-process as valid evidence (for adjusters) and to
convert p-values to e-values (for p-to-e calibrators).

In addition to the practical motivation, our main result (of lifting
e-processes) has a standalone implication in enhancing our theoretical
understanding of e-processes. It establishes that, unlike in the case of test
martingales, there always exists a powerful e-process for a testing problem in
the data filtration whenever there exists one in a coarser filtration. We will
return to this point in Section 6.1.

###  1.2 Related work on combining e-values (and p-values)

Many recent papers highlighted how e-values, unlike p-values, can be combined
seamlessly, even when they are arbitrarily dependent (Vovk and Wang,, 2021;
Wang and Ramdas,, 2022; Grünwald et al.,, 2023). In particular, Vovk and Wang,
(2021) show that the arithmetic mean function, which straightforwardly yields
a valid combined e-value, dominates any other symmetric “e-merging” function.
This is useful when combining two or more different e-values for a hypothesis
$\mathcal{P}$ and each e-value is constructed using the same data sample
$X_{1},\dotsc,X_{n}$. In the sequential context, we will see that any
e-process w.r.t. a filtration $\mathbb{F}$ is an e-value at any
$\mathbb{F}$-stopping time; thus, any two e-processes w.r.t. the same
filtration $\mathbb{F}$ for a (shared) null hypothesis $\mathcal{P}$ can also
be combined seamlessly by averaging. The main challenge that we highlight in
this work is when the two e-processes are defined w.r.t. two _different_
filtrations, such that one (or both) e-processes must first be _lifted_ into a
common filtration before the averaging step.

Much of the literature also concerns cases where the e-values exhibit various
dependence structures that can be exploited. Generally, taking the product of
e-values is a (weakly) dominating e-merging strategy (Vovk and Wang,, 2021,
proposition 4.2) when the e-values either are independent or form a
supermartingale. Taking the product is a useful strategy in sequential
settings involving continuous monitoring (Howard et al.,, 2021; Waudby-Smith
and Ramdas,, 2023) and in meta-analysis (ter Schure and Grünwald,, 2022;
Grünwald et al.,, 2023). In this work, we focus on cases where the two
e-processes are constructed using (some version of) the same data sequence, so
the product strategy does not apply. Others study e-values obtained for
multiple hypotheses with different, including arbitrary, dependence structures
(Wang and Ramdas,, 2022; Xu et al.,, 2021, 2022; Ignatiadis et al.,, 2023; Ren
and Barber,, 2023; Chi et al.,, 2022). We do not generally consider such
multiple testing scenarios.

Finally, there are various strategies for combining p-values (see, e.g., Vovk
and Wang, 2020a, ). Yet, Vovk et al., (2022)’s recent results suggest that any
admissible combination function for p-values must go through e-values. Thus,
we primarily focus on e-processes, which are also the fundamental notion of
evidence in anytime-valid sequential inference (Ramdas et al.,, 2020, 2023).
Nevertheless, we discuss the combination of e-processes with p-processes,
which are the anytime-valid counterpart to p-values, as a preliminary step to
our main result; this is related to Ignatiadis et al., (2023)’s study on
combining (non-sequential) e-values and p-values for multiple testing. In this
work, we will discover that p-processes can be “lifted” freely across
filtrations (Theorem 3.1) and that p-processes appear in an intermediary step
when combining e-processes across filtrations. The underlying connection is
made via p-to-e calibrators, which we review in Section 2.2.

###  1.3 Outline

The rest of the paper is organized as follows. In Section 2, we review the key
objects of anytime-valid sequential inference (e-processes, p-processes, and
sequential tests) as well as calibration and adjustment among e-/p-processes.
In Section 3, we develop the main technical tools that we call _$\mathfrak{p}$
-lifting_ and _$\mathfrak{e}$ -lifting_ and prove their validity. In Section
4, we illustrate three concrete use cases of the $\mathfrak{e}$-lifting
procedure, including how our method works on the exchangeability e-processes
from Example 1. In Section 5, we further develop a randomized variant of the
$\mathfrak{e}$-lifting approach to improve the power of a lifted e-process
when it is used in a sequential test. In Section 6, we conclude with a
discussion of the theoretical and practical implications of our main results,
additional use cases, and a game-theoretic interpretation of
$\mathfrak{e}$-lifting.

##  2 Preliminaries

###  2.1 Anytime-valid inference: E-processes, p-processes, and sequential
tests

To set the stage, we formally define the relevant core tools for quantifying
evidence in anytime-valid sequential inference (Ramdas et al.,, 2023). Let
$(\Omega,\mathcal{F})$ be a measurable space, and let
$\mathbb{F}=(\mathcal{F}_{t})_{t\geq 0}$ and
$\mathbb{G}=(\mathcal{G}_{t})_{t\geq 0}$ be two filtrations on
$(\Omega,\mathcal{F})$, with
$\mathcal{F}_{0}=\mathcal{G}_{0}=\\{\emptyset,\Omega\\}$. Unless specified
otherwise, we focus on the discrete-time scale ($t=0,1,2,\dotsc$). One of the
main cases we consider is when $\mathcal{G}_{t}\subseteq\mathcal{F}_{t}$ for
each $t$, in which case $\mathbb{G}$ is called a _sub-filtration_ of
$\mathbb{F}$, denoted as $\mathbb{G}\subseteq\mathbb{F}$. Because $\mathbb{F}$
contains more information than $\mathbb{G}$, $\mathbb{F}$ is _finer_ than
$\mathbb{G}$ (or $\mathbb{G}$ is _coarser_ than $\mathbb{F}$).

A sequence of random variables $(X_{t})_{t\geq 0}$ is a _process_ w.r.t. a
filtration $\mathbb{F}$ if it is $\mathbb{F}$-_adapted_ , i.e., $X_{t}$ is
$\mathcal{F}_{t}$-measurable for each $t$. A random variable $\tau$ taking
values in $\mathbb{N}\cup\\{\infty\\}$ is a _$\mathbb{F}$ -stopping time_ if
$\\{\tau\leq t\\}\in\mathcal{F}_{t}$ for each $t$. In this paper, any stopping
time we consider is possibly infinite.

Given a family of probability distributions $\mathcal{P}$ and a filtration
$\mathbb{F}$, we define an _e-process_ (Ramdas et al.,, 2020, 2022) for
$\mathcal{P}$ as a nonnegative process $(\mathfrak{e}_{t})_{t\geq 0}$ that
starts at one, or $\mathfrak{e}_{0}=1$, and satisfies:

$\text{for any $\mathbb{F}$-stopping time $\tau$ and any
$P\in\mathcal{P}$,}\quad\mathbb{E}_{P}[\mathfrak{e}_{\tau}]\leq 1.$ (3)

E-processes strictly generalize _test (super)martingales_ (Shafer et al.,,
2011), which are nonnegative (super)martingales whose initial value is 1,
i.e., any nonnegative process $(M_{t})_{t\geq 0}$ w.r.t. $\mathbb{F}$ such
that $M_{0}=1$ and $\mathbb{E}_{P}[M_{t}\mid\mathcal{F}_{t-1}]=1$ (or
$\mathbb{E}_{P}[M_{t}\mid\mathcal{F}_{t-1}]\leq 1$ for test supermartingales)
for all $t\geq 1$ and $P\in\mathcal{P}$. By the optional stopping theorem for
supermartingales (e.g., Durrett,, 2019, theorem 4.8.4), every test
supermartingale for $\mathcal{P}$ is an e-process for $\mathcal{P}$. When
$\mathcal{P}$ is understood as a (composite) null hypothesis, an e-process for
$\mathcal{P}$ quantifies evidence _against_ $\mathcal{P}$, as it is not
expected to grow large at arbitrary stopping times _unless_ $\mathcal{P}$ does
not adequately describe the data. Note that the definition of an e-process
explicitly depends on the filtration $\mathbb{F}$, particularly through the
choice of the stopping time $\tau$.

A closely related notion is that of a p-process, which is the sequential
generalization of a p-value. We say that a $[0,1]$-valued process
$(\mathfrak{p}_{t})_{t\geq 0}$ is a _p-process_ for $\mathcal{P}$ if, for any
$\alpha\in(0,1)$,

$\text{for any $\mathbb{F}$-stopping time $\tau$ and any
$P\in\mathcal{P}$,}\quad P(\mathfrak{p}_{\tau}\leq\alpha)\leq\alpha.$ (4)

Further, given a significance level $\alpha\in(0,1)$, we say that a _level-
$\alpha$ sequential test for $\mathcal{P}$ w.r.t. $\mathbb{F}$_ is a
$\\{0,1\\}$-valued process $\phi=(\phi_{t})_{t\geq 0}$ such that,

$\text{for any $\mathbb{F}$-stopping time $\tau$ and any
$P\in\mathcal{P}$,}\quad P(\phi_{\tau}=1)\leq\alpha.$ (5)

A p-process straightforwardly yields the level-$\alpha$ sequential test
$\phi_{t}=\mathds{1}\left(\mathfrak{p}_{t}\leq\alpha\right)$ for each
$\alpha\in(0,1)$. The notions of validity at stopping times defined in (3),
(4), and (5) are collectively referred to as ($\mathbb{F}$-)_anytime-validity_
(Johari et al.,, 2022; Howard et al.,, 2021) or ($\mathbb{F}$-)_safety_
(Grünwald et al.,, 2023; Ramdas et al.,, 2022), as they generalize their
corresponding validity notions in fixed sample sizes.

An e-process can directly yield a sequential test via _Ville’s inequality_
(Ville,, 1939), which was originally shown for test supermartingales but has
since been extended to e-processes (Ramdas et al.,, 2020). Ville’s inequality
for e-processes states that, for an e-process $\mathfrak{e}$ for $\mathcal{P}$
w.r.t. $\mathbb{F}$, the probability under $\mathcal{P}$ of $\mathfrak{e}$
exceeding a large threshold at any $\mathbb{F}$-stopping time is small: for
any $\alpha\in(0,1)$,

$\text{for any $\mathbb{F}$-stopping time $\tau$ and for any
$P\in\mathcal{P}$,}\quad
P\left(\mathfrak{e}_{\tau}\geq\frac{1}{\alpha}\right)\leq\alpha.$ (6)

This implies that $\phi_{t}=\mathds{1}\left(\mathfrak{e}_{t}\geq
1/\alpha\right)$ is level-$\alpha$ sequential test for each $\alpha\in(0,1)$.

We say that a level-$\alpha$ sequential test is _powerful_ against
$\mathcal{Q}$ (a composite alternative) if $Q(\tau_{\alpha}<\infty)=1$ for any
$Q\in\mathcal{Q}\setminus\mathcal{P}$, where $\tau_{\alpha}=\inf\\{t\geq
1:\phi_{t}=1\\}$, and that an e-process $\mathfrak{e}$ or a p-process
$\mathfrak{p}$ is _powerful_ against $\mathcal{Q}$ at level $\alpha$ if its
corresponding sequential test, $\phi_{t}=\mathds{1}\left(\mathfrak{e}_{t}\geq
1/\alpha\right)$ or
$\phi_{t}=\mathds{1}\left(\mathfrak{p}_{t}\leq\alpha\right)$, is powerful
against $\mathcal{Q}$. Further, an e-process that is powerful against
$\mathcal{Q}$ at every level $\alpha\in(0,1)$ is referred to as _consistent_
or _power one_ ; when it is clear from the context, we will broadly refer to
such e-processes as powerful (without reference to $\alpha$). Consistent
e-processes have an equivalent definition that
$\limsup_{t\to\infty}\mathfrak{e}_{t}=\infty$, $Q$-a.s., for every
$Q\in\mathcal{Q}\setminus\mathcal{P}$ (Ramdas et al.,, 2022).

Our main goal in this paper is to study methods for combining e-/p-processes
that are defined on different filtrations, such that the combined process
retains anytime-validity in the largest filtration.

###  2.2 Calibrating an e-process and a p-process

Combining evidence that takes the forms of an e-process and a p-process
requires calibrating one to another. Prior work by Shafer et al., (2011); Vovk
and Wang, (2021) suggests that any e-process can be calibrated into a
p-process and vice versa. Hereafter, we allow e-processes and calibrators to
be defined on the extended nonnegative reals, $[0,\infty]$, and we refer the
reader to Wang and Ramdas, (2023, section 2.2) for the arithmetic and measure-
theoretic properties on the extended reals.

####  $\mathfrak{e}$-to-$\mathfrak{p}$ calibration.

First, any e-process $(\mathfrak{e}_{t})_{t\geq 0}$ can be calibrated into a
p-process via either

$\mathfrak{p}_{t}=\frac{1}{\mathfrak{e}_{t}}\wedge
1\quad\text{or}\quad\mathfrak{p}_{t}^{*}=\frac{1}{\mathfrak{e}_{t}^{*}}\wedge
1,$ (7)

where $\mathfrak{e}_{t}^{*}=\sup_{i\leq t}\mathfrak{e}_{i}$ is the running
maximum of the e-process and $\wedge$ denotes the minimum operator. Note that
$\mathfrak{p}_{t}^{*}\leq\mathfrak{p}_{t}$, meaning that taking the running
maximum can lead to more powerful p-processes (though the running maximum
process itself is _not_ an e-process—we return to this soon).

####  $\mathfrak{p}$-to-$\mathfrak{e}$ calibration.

Converting a p-process into an e-process requires using a p-to-e calibrator
(Shafer et al.,, 2011; Ramdas et al.,, 2020; Vovk and Wang,, 2021). A
nonincreasing function $\mathsf{C}:[0,1]\to[0,\infty]$ is a _p-to-e
calibrator_ if $\int_{0}^{1}\mathsf{C}(\mathfrak{p})d\mathfrak{p}\leq 1$, and
it is _admissible_ (i.e., no other p-to-e calibrator strictly dominates it) if
and only if it is right-continuous, $\mathsf{C}(0)=\infty$, and
$\int_{0}^{1}\mathsf{C}(\mathfrak{p})d\mathfrak{p}=1$ (Vovk and Wang,, 2021,
proposition 2.1). Since there is no benefit to using a non-admissible p-to-e
calibrator, we restrict our attention to admissible ones hereafter and treat
the latter condition as an alternative definition.

Examples of admissible p-to-e calibrators include the family of functions
$\mathsf{C}_{\kappa}(\mathfrak{p})=\kappa\mathfrak{p}^{\kappa-1}$ for
$\kappa\in(0,1)$, as well as the (hyperparameter-free) function

$\mathsf{C}(\mathfrak{p})=\frac{1-\mathfrak{p}+\mathfrak{p}\log\mathfrak{p}}{\mathfrak{p}(-\log\mathfrak{p})^{2}}.$
(8)

###  2.3 Adjusting the running maximum of an e-process

Unlike a p-process, an e-process is not valid when taking the running maximum
over time, and in fact, the running maximum of an e-process for $\mathcal{P}$
will typically diverge to infinity even under $\mathcal{P}$ (Ramdas et al.,,
2022). An adjuster can convert the running maximum process into an e-process,
thereby “insuring against the loss of evidence” when, e.g., the e-process
sharply decreases following a changepoint.

Formally, an (e-to-e) _adjuster_ (Dawid et al., 2011a, ; Dawid et al., 2011b,
; Shafer et al.,, 2011) is an increasing function
$\mathsf{A}:[1,\infty]\to[0,\infty]$ such that, for any e-process
$(\mathfrak{e}_{t})_{t\geq 0}$ for $\mathcal{P}$ and its running supremum,
$\mathfrak{e}_{t}^{*}=\sup_{i\leq t}\mathfrak{e}_{i}$, there exists a
dominating e-process $(\mathfrak{e}_{t}^{\prime})_{t\geq 0}$ for $\mathcal{P}$
that satisfies

$\mathsf{A}(\mathfrak{e}_{t}^{*})\leq\mathfrak{e}_{t}^{\prime},\quad\forall
t\geq 0.$ (9)

Since $(\mathfrak{e}_{t}^{\prime})_{t\geq 0}$ is an e-process for
$\mathcal{P}$, inequality (9) implies that
$(\mathsf{A}(\mathfrak{e}_{t}^{*}))_{t\geq 0}$ is also an e-process for
$\mathcal{P}$.

Dawid et al., 2011b (, theorem 1) show that an adjuster $\mathsf{A}$ is
admissible, i.e., no other adjuster strictly dominates it on $[1,\infty]$, _if
and only if_ it is right-continuous,
$\mathsf{A}(\infty)=\lim_{\mathfrak{e}\to\infty}\mathsf{A}(\mathfrak{e})=\infty$,
and

$\int_{1}^{\infty}\frac{\mathsf{A}(\mathfrak{e})}{\mathfrak{e}^{2}}d\mathfrak{e}=1.$
(10)

As before, we restrict our attention to only the admissible ones hereafter and
treat condition (10) as an alternative definition. One concrete example of an
admissible adjuster is the following function:

$\mathsf{A}(\mathfrak{e})=\frac{\mathfrak{e}^{2}\log
2}{(1+\mathfrak{e})(\log(1+\mathfrak{e}))^{2}}.$ (11)

See Dawid et al., 2011a ; Dawid et al., 2011b ; Shafer et al., (2011) for more
examples. (These papers use the terms _simple lookback adjuster_ , _capital
calibrator_ , and _martingale calibrator_ , respectively.) Ramdas et al.,
(2022, section 4.7) discuss the use of adjusters in the context of testing
exchangeability.

###  2.4 One-to-one correspondence between p-to-e calibrators and adjusters

Finally, there is a straightforward one-to-one correspondence between p-to-e
calibrators and adjusters via the change-of-variables formula,

$\mathsf{A}(\mathfrak{e})=\mathsf{C}\left(\frac{1}{\mathfrak{e}}\right),\quad\forall\mathfrak{e}\in[1,\infty]$
(12)

(denoting $1/\infty=0$). To see this, observe that for any p-to-e calibrator
$\mathsf{C}:[0,1]\to[0,\infty]$, such that it is right-continuous,
$\mathsf{C}(0)=\infty$, and
$\int_{0}^{1}\mathsf{C}(\mathfrak{p})d\mathfrak{p}=1$, we have

$\int_{1}^{\infty}\frac{\mathsf{A}(\mathfrak{e})}{\mathfrak{e}^{2}}d\mathfrak{e}=\int_{1}^{\infty}\frac{\mathsf{C}(1/\mathfrak{e})}{\mathfrak{e}^{2}}d\mathfrak{e}=\int_{0}^{1}\mathsf{C}(\mathfrak{p})d\mathfrak{p}=1,$
(13)

by setting $\mathfrak{p}=1/\mathfrak{e}$. That is, $\mathsf{A}$ defined
according to (12) is an adjuster. The analogous argument in the converse
direction further shows that any adjuster $\mathsf{A}$ can be mapped to a
p-to-e calibrator via
$\mathsf{C}(\mathfrak{p})=\mathsf{A}(1/\mathfrak{p}),\,\forall\mathfrak{p}\in[0,1]$
(denoting $1/0=\infty$).

##  3 Combining evidence across filtrations via lifting

In this section, we describe the two main results of this paper, referred to
as _$\mathfrak{p}$ -lifting_ and _$\mathfrak{e}$ -lifting_, that allow us to
combine an e-process with a p- or e-process from a coarse filtration.

###  3.1 $\mathfrak{p}$-lifting: How to combine an e-process with a p-process
from a sub-filtration

We first establish that any p-process can be “lifted”333This notion of
“lifting” is largely distinct from that of Getoor and Sharpe, (1972), who used
the term in the (unrelated) context of continuous-time, complex-valued
martingales. Their notion has since been called Hypothèse (Brémaud and Yor,,
1978), extension (Dubins et al.,, 1996), and immersion (Beghdadi-Sakrani and
Emery,, 2006), among others, as it has been adapted to the (also unrelated)
literature of Brownian innovations (e.g., Tsirelson,, 1998). Our terminology
only concerns the lifting of events and evidence processes across filtrations
defined in discrete time, with a focus on stopping times. Further, as noted in
Vovk, 2021a (, remark 1), it is a coincidence that the subject of Getoor and
Sharpe, (1972)’s study was termed “conformal martingales” (unrelated to Vovk,
2021a ; Vovk, 2021b ’s “conformal test martingales” that we discuss in Example
1). into a finer filtration without calibration. This somewhat surprising
result is a direct consequence of the “equivalence lemma” (Ramdas et al.,,
2020; Howard et al.,, 2021) between time-uniform validity, random time
validity, and anytime-validity of sequential tests. When it comes to
probability statements involving whether a p-process will erroneously reject
the null hypothesis at level $\alpha\in(0,1)$, the lemma says that the
following notions are equivalent:

  1. (i)

_Time-uniform validity_ : The probability of falsely rejecting the null _ever_
, over infinite time, is at most $\alpha$.

  2. (ii)

_Random time validity_ : The probability of falsely rejecting the null at any
_random_ times $T$, which may be chosen post-hoc, is at most $\alpha$.

  3. (iii)

_Stopping time validity_ , or _anytime-validity_ : The probability of falsely
rejecting the null using a p-process adapted to $\mathbb{F}$, at arbitrary
$\mathbb{F}$-_stopping_ times $\tau$ (possibly infinite), is at most $\alpha$.

The equivalence lemma holds for both point and composite nulls (Ramdas et
al.,, 2020, lemma 2).

In our case, notice that we can specifically use the equivalence between ii
and iii to move from a sub-filtration to the data filtration (or any finer
filtration) by avoiding the dependence on the filtration in the first place.
We summarize this result as the _lifting lemma_. In the following, we say that
a sequence of events $(\xi_{t})_{t\geq 0}$ is adapted to $\mathbb{G}$ if
$\xi_{t}\in\mathcal{G}_{t}$ for $t\geq 0$, and we denote
$\xi_{\infty}=\lim\sup_{t\to\infty}\xi_{t}=\cap_{t}\cup_{i\geq t}\xi_{i}$.

######  Lemma 1 (The lifting lemma).

Let $\mathbb{G}$ be a sub-filtration of $\mathbb{F}$. Let $\mathcal{P}$ be a
family of distributions, $(\xi_{t})_{t\geq 0}$ be a sequence of events adapted
to $\mathbb{G}$, and $\alpha\in(0,1)$. Then, for any $P\in\mathcal{P}$, the
following statements are equivalent:

  1. (a)

For any $\mathbb{G}$-stopping time $\tau$, $P(\xi_{\tau})\leq\alpha$.

  2. (b)

For any $\mathbb{F}$-stopping time $\tau$, $P(\xi_{\tau})\leq\alpha$.

We give a self-contained proof in Appendix C.1. Only the lifting direction, a
$\Rightarrow$ b, is nontrivial: it states that, if any fixed upper bound on
the probability of a $\mathbb{G}$-adapted event holds at all stopping times
w.r.t. $\mathbb{G}$, then the bound further holds at any stopping time w.r.t.
any finer filtration $\mathbb{F}\supseteq\mathbb{G}$. Note that the coarse
filtration $\mathbb{G}$ has to be the one to which the sequence of events is
adapted—if $(\xi_{t})_{t\geq 0}$ were adapted to $\mathbb{F}$ but not
$\mathbb{G}$, then the equivalence would no longer hold. We also remark that
the analogous claim involving upper bound statements about an _expectation_
(e.g., $\mathbb{E}_{P}[\mathfrak{e}_{\tau}]\leq 1$) is NOT true in general,
and this necessitates the use of an adjuster when lifting e-processes (as
opposed to p-processes) in Section 3.2.

The lifting lemma directly implies the following result.

######  Theorem 1 ($\mathfrak{p}$-lifting).

Let $\mathbb{G}$ be a sub-filtration of $\mathbb{F}$. If
$(\mathfrak{p}_{t})_{t\geq 0}$ is a p-process for $\mathcal{P}$ w.r.t.
$\mathbb{G}$, then $(\mathfrak{p}_{t})_{t\geq 0}$ is a p-process for
$\mathcal{P}$ w.r.t. $\mathbb{F}$.

That is, $\mathfrak{p}$-lifting is “free”: any p-process can be lifted without
calibration. Theorem 1 follows immediately by applying Lemma 1 to the sequence
of $\mathbb{G}$-adapted events, $\xi_{t}=\\{\mathfrak{p}_{t}\leq\alpha\\}$ for
each $t$.

We note in passing that, by complete analogy to Theorem 1, we can also
“freely” lift sequential tests and even confidence sequences (CS; Darling and
Robbins,, 1967; Howard et al.,, 2021) at arbitrary stopping times. We do not
focus on them in this work, however, as it is unclear how to combine multiple
sequential tests or CSs without involving their underlying e-processes (Ramdas
et al.,, 2020).

#### How to combine an e-process with a p-process from a sub-filtration.

The p-lifting result (Theorem 1), combined with any p-to-e calibrator, gives
an answer for how to combine an e-process with a p-process from a sub-
filtration. We can first lift the p-process into the finer filtration,
calibrate it into an e-process in the finer filtration, and then combine it
using the fact that any set of e-processes (in the same filtration) can be
combined into an e-process via averaging (Vovk and Wang,, 2021). This is
summarized in the following corollary.

######  Corollary 1 (Combining an e-process with a p-process from a sub-
filtration).

Let $\mathbb{G}$ be a sub-filtration of $\mathbb{F}$. If
$(\mathfrak{e}_{t})_{t\geq 0}$ is an e-process for $\mathcal{P}$ w.r.t.
$\mathbb{F}$ and $(\mathfrak{p}_{t})_{t\geq 0}$ is a p-process for
$\mathcal{P}$ w.r.t. $\mathbb{G}$, then for any p-to-e calibrator
$\mathsf{C}$, the sequence $(\bar{\mathfrak{e}}_{t}^{\mathsf{cal}})_{t\geq 0}$
defined by

$\bar{\mathfrak{e}}_{t}^{\mathsf{cal}}=\gamma\mathfrak{e}_{t}+(1-\gamma)\mathsf{C}(\mathfrak{p}_{t}),\quad\forall
t\geq 0,$ (14)

for any fixed $\gamma\in[0,1]$, is an e-process for $\mathcal{P}$ w.r.t.
$\mathbb{F}$.

We note that, while $\bar{\mathfrak{e}}_{t}^{\mathsf{cal}}$ is an e-process
for any fixed constant $\gamma\in[0,1]$, the default choice should be to use
the arithmetic mean via $\gamma=1/2$, which dominates any other symmetric
e-merging functions (Vovk and Wang,, 2021, proposition 3.1). The only
exception would be when there is a prior belief that one evidence is more
valuable than the other, in which case $\gamma$ must be specified _a priori_.

###  3.2 $\mathfrak{e}$-lifting: How to combine an e-process with an e-process
from a sub-filtration

Next, we consider the problem of combining two e-processes, one residing in a
sub-filtration of another’s filtration. Recall the motivating example from
earlier (Example 1), where we saw that simply taking the average of the two
e-processes does not yield an e-process in the larger filtration.

How can we avoid this issue and successfully “lift” the validity of
$\mathfrak{e}^{\prime}$ to the larger filtration $\mathbb{F}$? Surprisingly,
an answer comes directly from using an adjuster. Leveraging the one-to-one
correspondence between adjusters and p-to-e calibrators, we can now lift an
e-process to any finer filtration.

######  Theorem 2 ($\mathfrak{e}$-lifting).

Let $\mathbb{G}$ be a sub-filtration of $\mathbb{F}$, and let
$(\mathfrak{e}_{t})_{t\geq 0}$ be an e-process for $\mathcal{P}$ w.r.t.
$\mathbb{G}$. Then, for any adjuster $\mathsf{A}$, the sequence of random
variables
$\mathfrak{e}^{\mathsf{adj}}=(\mathfrak{e}_{t}^{\mathsf{adj}})_{t\geq 0}$
defined by

$\mathfrak{e}_{t}^{\mathsf{adj}}=\mathsf{A}\left(\mathfrak{e}_{t}^{*}\right)=\mathsf{A}\left(\sup_{i\leq
t}\mathfrak{e}_{i}\right),\quad\forall t\geq 0,$ (15)

is an e-process for $\mathcal{P}$ w.r.t. $\mathbb{F}$.

Theorem 2 establishes that $\mathfrak{e}_{\tau}^{\mathsf{adj}}$ is a valid
e-value for $\mathcal{P}$ at _any_ $\mathbb{F}$-stopping time $\tau$. Its
proof consists of three steps, each of which follows from a known or previous
result:

  1. (i)

$\mathfrak{p}_{t}^{*}=(1/\mathfrak{e}_{t}^{*})\wedge 1$ is a p-process for
$\mathcal{P}$ w.r.t. $\mathbb{G}$ (Section 2.2);

  2. (ii)

$\mathfrak{p}_{t}^{*}$ is also a p-process for $\mathcal{P}$ w.r.t.
$\mathbb{F}$, by the $\mathfrak{p}$-lifting result (Theorem 1); and

  3. (iii)

for each adjuster $\mathsf{A}$, there is a corresponding p-to-e calibrator
$\mathsf{C}$ (Section 2.4) such that
$\mathfrak{e}_{t}^{\mathsf{adj}}=\mathsf{A}(\mathfrak{e}_{t}^{*})\leq\mathsf{C}(\mathfrak{p}_{t}^{*})$,
proving that $\mathfrak{e}^{\mathsf{adj}}$ an e-process for $\mathcal{P}$
w.r.t. $\mathbb{F}$.

We include a complete proof in Appendix C.2.

Figure 2: E-processes $(\mathfrak{e}_{t}^{\prime})_{t\geq 0}$ constructed
w.r.t. a sub-filtration $\mathbb{G}=(\mathcal{G}_{t})_{t\geq 0}$ (orange) can
be $\mathfrak{e}$-lifted to any finer filtration
$\mathbb{F}=(\mathcal{F}_{t})_{t\geq 0}$ (blue), such that
$\mathbb{G}\subseteq\mathbb{F}$, using an adjuster $\mathsf{A}$ (Theorem 2;
blue arrow). _Unlike $(\mathfrak{e}_{t}^{\prime})_{t\geq 0}$, the lifted
e-process $(\mathsf{A}(\mathfrak{e}_{t}^{\prime*}))_{t\geq 0}$ is valid at an
arbitrary $\mathbb{F}$-stopping time (or $\mathbb{F}$-anytime-valid)._ The
lifted e-process can be combined with any other e-process
$(\mathfrak{e}_{t})_{t\geq 0}$ w.r.t. $\mathbb{F}$ via averaging (Corollary 2;
gray). In the figure, whether each random variable is anytime-valid w.r.t.
each filtration ($\tau$ being a stopping time in the filtration) is marked
with a green check ( ✓) or a red cross ( ✗).

#### How to combine an e-process with an e-process from a sub-filtration.

Analogous to $\mathfrak{p}$-lifting (Theorem 1), $\mathfrak{e}$-lifting
(Theorem 2) gives a recipe for how to combine an e-process with an e-process
from a sub-filtration.

######  Corollary 2 (Combining an e-process with an e-process from a sub-
filtration).

Let $\mathbb{G}$ be a sub-filtration of $\mathbb{F}$. If
$(\mathfrak{e}_{t})_{t\geq 0}$ is an e-process for $\mathcal{P}$ w.r.t.
$\mathbb{F}$ and $(\mathfrak{e}_{t}^{\prime})_{t\geq 0}$ is an e-process for
$\mathcal{P}$ w.r.t. $\mathbb{G}$, then for any adjuster $\mathsf{A}$, the
sequence $(\bar{\mathfrak{e}}_{t}^{\mathsf{adj}})_{t\geq 0}$ defined by

$\bar{\mathfrak{e}}_{t}^{\mathsf{adj}}=\gamma\mathfrak{e}_{t}+(1-\gamma)\mathsf{A}(\mathfrak{e}_{t}^{\prime*}),\quad\forall
t\geq 0,$ (16)

is an e-process for $\mathcal{P}$ w.r.t. $\mathbb{F}$, where
$\mathfrak{e}_{t}^{\prime*}=\sup_{i\leq t}\mathfrak{e}_{i}^{\prime}$.

Corollary 2 establishes that we can first lift the e-process from the sub-
filtration and then average the lifted e-process with the other e-process.
Note the asymmetry in (16): we leave the e-process $\mathfrak{e}$ w.r.t.
$\mathbb{F}$ as-is, while we take the running supremum of the e-process
$\mathfrak{e}^{\prime}$ and then apply the adjuster. Despite the asymmetry of
their forms, unless there are reasons _a priori_ to prefer one e-process over
the other, we recommend using $\gamma=1/2$ for reasons described earlier in
Section 3.1.

Figure 2 gives a visual illustration of the $\mathfrak{e}$-lifting procedures
given by Theorem 2 and Corollary 2. It also highlights that
$(\mathsf{A}(\mathfrak{e}_{t}^{\prime*}))_{t\geq 0}$ is an e-process w.r.t.
$\mathbb{F}$, while $(\mathfrak{e}_{t}^{\prime})_{t\geq 0}$ is not.

##  4 Three illustrative use cases

Given the $\mathfrak{e}$-lifting result (Theorem 2), we now discuss three
illustrative problems in which the technique can be used to achieve validity
in the finer filtration.

###  4.1 Lifting and combining e-processes for testing exchangeability
(revisited)

(a) E-processes and their stopping times

(b) Histogram of stopped e-values (vertical line: mean)

Figure 3: Unlike the conformal test martingale (solid orange), which is not
$\mathbb{F}$-anytime-valid, its $\mathfrak{e}$-lifted version (dashed green)
using Theorem 2 is $\mathbb{F}$-anytime-valid. Similarly to Figure 1, we
compare these two processes at the $\mathbb{F}$-stopping time
$\tau^{\mathbb{F}}$ in (2) (first time seeing five consecutive zeros in the
data). The histogram shows that the lifted e-process has an estimated mean of
$0.47\pm 0.003$ (vertical dashed green) at the stopping time, contrasting with
the conformal test martingale which has an estimated mean of $1.33\pm 0.016$
(vertical solid orange). The lifted e-process can then be combined seamlessly
with any other $\mathbb{F}$-anytime-valid e-process, such as the UI e-process.

We start by revisiting the problem of combining e-processes for testing
exchangeability (Example 1). As we showed earlier, the conformal test
martingale is not valid w.r.t. the data filtration $\mathbb{F}$, unlike the UI
e-process. Using Theorem 2, we can now lift the conformal variant into
$\mathbb{F}$, such that it is valid at $\mathbb{F}$-stopping times and thus
can be combined with the UI e-process via averaging (Corollary 2).

Figure 3 shows plots of the conformal test martingale (the same one used in
Figure 1) alongside its $\mathfrak{e}$-lifted version that uses the adjuster
$\mathsf{A}$ from (11). The plots are drawn analogously to Figure 1, showing
the e-processes in Figure 3(a) as well as the histograms and estimated means
of the stopped e-values in Figure 3(b), over $10,000$ repeated runs. We use
the same $\mathbb{F}$-stopping time $\tau^{\mathbb{F}}$ as before (the first
time seeing five consecutive zeros in the data). The data is once again
sampled from an i.i.d. $\mathsf{Ber}(0.3)$.

The plots show that, unlike the conformal test martingale, its
$\mathfrak{e}$-lifted version is mostly no greater than 1, and the histogram
further shows that the estimated mean of the stopped e-values for the lifted
e-process (vertical dashed green) is $0.47\pm 0.003$, suggesting that the
lifted e-process meets the validity definition (3) at this
$\mathbb{F}$-stopping time. This gives evidence that the lifted e-process can
be combined with the UI e-process via averaging, such that the combined
version is an e-process w.r.t. $\mathbb{F}$.

(a) First-order Markov: $\mathsf{Markov}(0.5,0.4)$

(b) Changepoints: $\mathsf{Ber}(0.5)\Leftrightarrow\mathsf{Ber}(0.25)$

Figure 4: UI (dashed blue), conformal (dotted orange), and the combined (solid
green; Corollary 2) e-processes for testing exchangeability in two scenarios
where the binary data is sampled from a non-exchangeable distribution (left:
first-order Markov; right: changepoints). The combined e-process grows large
under both alternatives, while the UI and conformal e-processes only grow
large under the Markov and changepoint alternatives, respectively. Each line
is averaged over 100 repeated runs.

As a follow-up experiment, we demonstrate how the combined e-process via
$\mathfrak{e}$-lifting and averaging can be useful for leveraging the
statistical power of two e-processes that are powerful against different types
of alternative hypotheses. In Figure 4, we compare the UI, conformal, and
their combined e-processes on a simulated, _non-exchangeable_ binary data
sequence. We combine the UI e-process $(\mathfrak{e}_{t})_{t\geq 0}$ and the
conformal e-process $(\mathfrak{e}_{t}^{\prime})_{t\geq 0}$ via
$\mathfrak{e}$-lifting and simple averaging (Corollary 2), i.e.,
$\bar{\mathfrak{e}}_{t}^{\mathsf{adj}}=\frac{1}{2}(\mathfrak{e}_{t}+\mathsf{A}(\mathfrak{e}_{t}^{\prime*}))$,
$\forall t$, where $\mathsf{A}$ is the adjuster in (11). We use two
alternative distributions for which the UI and conformal procedures are each
known to be powerful: a first-order Markov chain for the UI e-process and a
Bernoulli sequence with multiple changepoints in the mean parameter for the
conformal test martingale.

In Figure 4(a), we generate data from a first-order Markov chain with
$p_{1|0}=0.5$ and $p_{1|1}=0.4$, where $p_{k|j}$ denotes the transition
probability from state $j$ to state $k$, for $j,k\in\\{0,1\\}$, and we plot
the three e-processes, each averaged over $100$ repeatedly sampled Markovian
data sequence of length $T=2,000$. The plot confirms that the UI e-process
(dashed blue) grows quickly while the conformal test martingale (dotted
orange) does not; more importantly, it shows that the combined e-process
(“eLift+Avg”; solid green) closely tracks the fast-growing UI e-process. This
is unsurprising because the combined e-process
$\bar{\mathfrak{e}}_{t}^{\mathsf{adj}}$ is effectively one-half of the
exponentially growing UI e-process $\mathfrak{e}_{t}$.

In Figure 4(b), we generate data from a drifting distribution that is
typically an i.i.d. $\mathsf{Ber}(0.5)$ (“usual state”), but after every $800$
steps it switches to an i.i.d. $\mathsf{Ber}(0.25)$ (“unusual state”) for
$200$ steps, before returning to the usual state. The resulting data contains
multiple changepoints across the sequence length of $T=2,000$. As mentioned
earlier, conformal test martingales are known to be powerful against
changepoint alternatives, unlike the UI e-process. The plot, showing the three
e-processes averaged over $100$ repeated runs, confirms this: the conformal
test martingale grows large when encountering the first unusual state, then
decreases slowly as the data reverts to its usual state, and then grows large
again when encountering the second unusual state; the UI e-process does not
grow large throughout the sequence. Analogous to the Markovian case, the
combined e-process also grows large whenever the conformal test martingale
grows large.

Unlike in Figure 4(a), however, in Figure 4(b) we additionally observe two
peculiarities stemming from the $\mathfrak{e}$-lifting procedure. The first is
that the combined e-process utilizes the _adjusted_ conformal test martingale,
meaning that it will always be at a smaller scale (e.g., $\approx 10^{4}$ at
time $1,000$) than the original e-process ($\approx 10^{6}$). In our test
case, the adjusted evidence is still much larger than a typical threshold for
a “strong”444Game-theoretically, an e-value of $20$ would mean that the
skeptic was able to multiply their wealth by $20$ times by betting against the
null. For a sequential test, a significance level of $\alpha=0.05$ would
correspond to a threshold of $1/\alpha=20$. evidence (10, 20, or 100). The
second is that, as a byproduct of using an adjuster, the combined e-process
utilizes the _running maximum_ of the conformal test martingale, explaining
why the combined e-process remains constant (and not decrease) during times
$1,001$ to $1,800$ when the data reverts to the usual state. Thus, this useful
byproduct allows the combined e-process to insure against losing evidence, the
one from the e-process in the sub-filtration, over time.

###  4.2 Combining e-processes across different sub-filtrations: the case of
comparing forecasts with lags

The next class of examples concerns a scenario where different e-processes are
constructed w.r.t. different sub-filtrations of the data filtration. Formally,
we consider cases where, for each $k=1,2,\dotsc$, an e-process
$\mathfrak{e}^{[k]}$ is constructed w.r.t. a sub-filtration
$\mathbb{F}^{[k]}\subseteq\mathbb{F}$.

Here, we specifically highlight the case of sequentially comparing lag-$h$
forecasters that make probability forecasts for an outcome $h$ time steps
ahead of its occurrence (e.g., 7-day weather forecasters). Recent work on
anytime-valid forecast comparison develop e-processes and sequential tests for
testing forecast dominance (Henzi and Ziegel,, 2022), i.e., whether one
forecaster uniformly outperforms the other over time; testing whether a
forecaster is calibrated (Arnold et al.,, 2023); and estimating the mean
forecast score difference between two forecasters (Choe and Ramdas,, 2023).
Interestingly, these results implicitly utilize the $\mathfrak{p}$-lifting
result, and the third result essentially utilizes the same strategy as
$\mathfrak{e}$-lifting, except for its use of a p-merging function instead of
an e-merging function.

To further elucidate the connections between $\mathfrak{e}$-lifting and
anytime-valid forecast comparison, we proceed with a concrete example based on
the formulations of Choe and Ramdas, (2023, appendix E) as well as Arnold et
al., (2023, section 4). In the following, let $h>1$ be an integer lag, and
suppose that at each time step $t=1,2,\dotsc$, two forecasters $\mathsf{p}$
and $\mathsf{q}$ make respective probability forecasts $p_{t},q_{t}\in[0,1]$
for an eventual binary outcome $y_{t+h-1}\in\\{0,1\\}$, say, whether it will
rain on the $(t+h-1)$th day. The underlying distribution for the outcome
sequence is not known. Let $\mathbb{F}=(\mathcal{F}_{t})_{t\geq 0}$ denote the
filtration such that both $p_{t}$ and $q_{t}$ are
$\mathcal{F}_{t-1}$-measurable—that is, $\mathcal{F}_{t-1}$ represents the
information set available to each forecaster before making their forecasts on
$y_{t+h-1}$.

######  Example 2 (Combining e-processes for comparing lag-$h$ forecasters).

Suppose that we evaluate the two lag-$h$ forecasters according to the Brier
score $\mathsf{s}(p,y)=(p-y)^{2}$ (Brier,, 1950). For each time step $t\geq
h$, define

$\Delta_{t}^{[k]}=\frac{1}{\left\lvert I_{t}^{[k]}\right\rvert}\sum_{i\in
I^{[k]}}\mathbb{E}\left[{\mathsf{s}(p_{i},y_{i+h-1})-\mathsf{s}(q_{i},y_{i+h-1})\left.\middle|\right.\mathcal{F}_{t-1}}\right],\quad\forall
k\in\\{1,\dotsc,h\\},$ (17)

where $I_{t}^{[k]}$ is an index set that includes every $h$th time step
starting at $k+1$ up to $t-h+1$. The parameter $\Delta_{t}^{[k]}$ thus denotes
the average conditional expectation of score differences between the two
lag-$h$ forecasters over time. Importantly, we condition the scores of $p_{i}$
and $q_{i}$ on $\mathcal{F}_{i-1}$, in order to adequately capture the
information set available to the forecasters _at the time of their
forecasting_.

Now, suppose we seek to test the null hypothesis that asserts, for each time
offset $k=1,\dotsc,h$ (e.g., day of the week), forecaster $\mathsf{p}$
outperforms forecaster $\mathsf{q}$ on average according to $\mathsf{s}$:

$\mathcal{H}_{0}^{\mathsf{pw}}:\Delta_{t}^{[k]}\leq 0,\quad\forall t\geq
1,\;\forall k\in\\{1,\dotsc,h\\}.$ (18)

For each offset $k\in\\{1,\dotsc,h\\}$, Choe and Ramdas, (2023) construct an
e-process $\mathfrak{e}^{[k]}=(\mathfrak{e}_{t}^{[k]})_{t\geq 0}$ for
$\mathcal{H}_{0}^{\mathsf{pw}}$ w.r.t. some sub-filtration
$\mathbb{F}^{[k]}\subseteq\mathbb{F}$, which updates its information every $h$
steps starting at time $k$ (for example, if $h=3$, then
$\mathbb{F}^{[1]}=(\mathcal{F}_{0},\mathcal{F}_{1},\mathcal{F}_{1},\mathcal{F}_{1},\mathcal{F}_{4},\mathcal{F}_{4},\mathcal{F}_{4},\mathcal{F}_{7},\mathcal{F}_{7},\mathcal{F}_{7},\dotsc)$;
this sub-filtration design is due to Arnold et al., (2023)). Even though it is
an e-process w.r.t. $\mathbb{F}^{[k]}$, $\mathfrak{e}^{[k]}$ is not generally
valid at $\mathbb{F}$-stopping times. Because the e-process
$\mathfrak{e}^{[k]}$ only contains evidence from the forecasts made for the
offset $k$, we would want to eventually combine the evidence for all values of
$k\in\\{1,\dotsc,h\\}$, even though each e-process $\mathfrak{e}^{[k]}$ is
only valid w.r.t. its own sub-filtration $\mathbb{F}^{[k]}$.

For each $k\in\\{1,\dotsc,h\\}$, we can leverage the $\mathfrak{e}$-lifting
strategy of Theorem 2 to lift $\mathfrak{e}^{[k]}$ to $\mathbb{F}$, and then
we can average all $h$ of them to obtain one combined e-process for
$\mathcal{H}_{0}^{\mathsf{pw}}$ w.r.t. $\mathbb{F}$:

$\mathfrak{e}_{t}=\frac{1}{h}\sum_{k=1}^{h}\mathsf{A}\left(\sup_{i\leq
t}\mathfrak{e}_{i}^{[k]}\right),\quad\forall t.$ (19)

This e-process is expected to be bounded by 1 at any $\mathbb{F}$-stopping
time, as long as forecaster $\mathsf{p}$ outperforms $\mathsf{q}$ on average
across all offsets (that is, under $\mathcal{H}_{0}^{\mathsf{pw}}$), but it
may grow large if $\mathsf{q}$ outperforms $\mathsf{p}$ on at least one offset
(e.g., on Sundays). Our general strategy mirrors Choe and Ramdas, (2023)’s
strategy, although one difference is that here we combine evidence in the
e-domain via averaging, whereas Choe and Ramdas, (2023) follows Henzi and
Ziegel, (2022)’s strategy to first calibrate the e-processes to p-process,
(implicitly) apply $\mathfrak{p}$-lifting, and then use the harmonic average
p-merging function (Vovk and Wang, 2020a, ) before finally applying a
$\mathfrak{p}$-to-$\mathfrak{e}$ calibration.

###  4.3 Combining e-processes constructed using $k$-wise examples: the case
of sequential independence testing

The third class of examples concerns e-processes constructed w.r.t. a
coarsening of the data filtration that observes $k$ data points each time.
Given a stream of observations $(Z_{t})_{t\geq 1}$ and its natural filtration
$\mathbb{F}=(\mathcal{F}_{t})_{t\geq 0}$,
$\mathcal{F}_{t}=\sigma(Z_{1},\dotsc,Z_{t})$, the e-process is constructed
w.r.t. the coarse filtration $\mathbb{F}^{[k]}=(\mathcal{F}_{t}^{[k]})_{t\geq
0}\subseteq\mathbb{F}$ that is updated only every $k$ steps, that is,
$\mathcal{F}_{t}^{[k]}=\mathcal{F}_{\left\lfloor t/k\right\rfloor k}$ for each
$t$.

A topical example is the case of sequential (nonparametric) independence
testing (Balsubramani and Ramdas,, 2016; Shekhar and Ramdas,, 2023; Podkopaev
et al.,, 2023; Podkopaev and Ramdas,, 2023; Henzi and Law,, 2023), which we
describe below.

######  Example 3 (Combining e-processes for testing independence).

Suppose we have an i.i.d. stream of paired observations $(X_{t},Y_{t})\sim
P_{XY}$, for $t\geq 1$, and we would like to sequentially test whether the
joint distribution factorizes, that is,

$\mathcal{H}_{0}^{\mathsf{indep}}:P_{XY}=P_{X}\times
P_{Y}\quad\text{vs.}\quad\mathcal{H}_{1}^{\mathsf{indep}}:P_{XY}\neq
P_{X}\times P_{Y}.$ (20)

Following, e.g., Podkopaev et al., (2023), we can construct an e-process
$\mathfrak{e}^{\mathsf{pair}}=(\mathfrak{e}_{t}^{\mathsf{pair}})_{t\geq 0}$
for $\mathcal{H}_{0}^{\mathsf{indep}}$ w.r.t. $\mathbb{F}^{[2]}$ that bets on
_pairs_ of incoming data points (“pairwise betting”):
$\mathfrak{e}_{0}^{\mathsf{pair}}=1$ and

$\mathfrak{e}_{t}^{\mathsf{pair}}=\prod_{i=1}^{\left\lfloor
t/2\right\rfloor}\left\\{1+\lambda_{i}f_{i}\left((X_{2i-1},Y_{2i-1}),(X_{2i},Y_{2i})\right)\right\\},\quad\forall
t\geq 1,$ (21)

where $\lambda_{i}\in[-1,1]$ is ($\mathcal{F}_{2i-2}$)-measurable and
$f_{i}\in[-1,1]$ is a payoff function that takes a pair of data points as
inputs. The payoff is expected to be zero whenever $X_{2i}$ and $Y_{2i}$ (as
well as $X_{2i-1}$ and $Y_{2i-1}$) are independent. This e-process is only
adapted w.r.t. the coarse sub-filtration $\mathbb{F}^{[2]}$ but not the data
filtration $\mathbb{F}$. We want to be able to combine this e-process with
another e-process for independence testing, say, Henzi and Law, (2023)’s test
martingale
$\mathfrak{e}^{\mathsf{rank}}=(\mathfrak{e}_{t}^{\mathsf{rank}})_{t\geq 0}$
for $\mathcal{H}_{0}^{\mathsf{indep}}$, which itself is constructed w.r.t. a
different sub-filtration $\mathbb{G}\subseteq\mathbb{F}$ based on the
sequential ranks of the data.

Similar to tests of exchangeability, it is impossible to construct nontrivial
test martingales adapted to the data filtration $\mathbb{F}$ for independence
testing (Henzi and Law,, 2023, theorem 1); furthermore, to the best of our
knowledge, all known nontrivial e-processes are constructed w.r.t. some
coarsening of $\mathbb{F}$. In particular, e-processes constructed using the
pairwise betting strategy are only valid at even, $\mathbb{F}^{[2]}$-stopping
times, while Henzi and Law, (2023)’s test martingale is only valid when
peeking at the rank statistics of the data (but not the data itself). This
implies that $\mathfrak{e}$-lifting is necessary for combining such different
processes. Since the two e-processes reside in different sub-filtrations of
$\mathbb{F}$, the combined e-process w.r.t. $\mathbb{F}$ would involve
adjustments on both processes:

$\bar{\mathfrak{e}}_{t}=\frac{1}{2}\left(\mathsf{A}(\mathfrak{e}_{t}^{\mathsf{pair}})+\mathsf{A}^{\prime}(\mathfrak{e}_{t}^{\mathsf{rank}})\right),\quad\forall
t,$ (22)

for two adjusters $\mathsf{A}$ and $\mathsf{A}^{\prime}$.

We close with the note that recent work by Saha and Ramdas, (2024) also
leverages the pairwise betting strategy but for testing exchangeability; the
analogous procedures would give ways to combine its e-process with, e.g., the
UI and conformal e-processes in Example 1.

##  5 Randomized adjustments of e-processes across filtrations

Statistical methods often leverage external randomization to improve their
power, and generally speaking, these methods can be readily incorporated into
our $\mathfrak{e}$-lifting procedures. The conformal test martingale for
testing exchangeability in Example 1, for example, utilizes randomization to
break ties (see Section A for details); Ramdas and Manole, (2023, section
10.2) discuss many other known examples in statistics. In this section, we
focus on one class of external randomization that can be incorporated into the
$\mathfrak{e}$-lifting procedure itself. This is based on recent work that
demonstrated a simple use of an independent random variable
$U\sim\mathsf{Unif}[0,1]$, which we refer to here as “$U$-randomization.”

There are two relevant recent results for $U$-randomization. First, in the
non-sequential setup, Ignatiadis et al., (2023) remark that, if $e$ is an
e-value and $U$ is independent of $e$, then $\tilde{p}=(U/e)\wedge 1$ is a
valid, $U$-randomized p-value. In fact, the _$U$ -randomized e-to-p
calibrator_ $e\mapsto(U/e)\wedge 1$ dominates the e-to-p calibrator
$e\mapsto(1/e)\wedge 1$ of (7), which is the only admissible _deterministic_
e-to-p calibrator (Vovk and Wang,, 2021). Second, in the sequential setup,
Ramdas and Manole, (2023) show that we can analogously incorporate
$U$-randomization by applying it at a stopping time, increasing the
statistical power of the stopped process.

In this section, we show that Ramdas and Manole, (2023)’s $U$-randomization
strategy is naturally applicable to our sequential setup by first applying the
$\mathfrak{e}$-lifting procedure (Theorem 2) and then applying
$U$-randomization at a $\mathbb{F}$-stopping time. As for Ignatiadis et al.,
(2023)’s $U$-randomized e-to-p calibration approach, we provide a preliminary
analysis that gives a loose upper bound on the type I error of the resulting
sequential test in our lifting setup.

###  5.1 The lift-then-randomize procedure

Given an e-process $\mathfrak{e}=(\mathfrak{e}_{t})_{t\geq 0}$ for
$\mathcal{P}$ w.r.t. $\mathbb{G}\subseteq\mathbb{F}$, we showed in Theorem 2
that we can apply $\mathfrak{e}$-lifting to obtain a lifted e-process
$(\mathsf{A}(\mathfrak{e}_{t}^{*}))_{t\geq 0}$ w.r.t. $\mathbb{F}$, for any
choice of an adjuster $\mathsf{A}$. This implies that, for any
$\mathbb{F}$-stopping time $\tau$, we have

$\mathbb{E}_{P}[\mathsf{A}(\mathfrak{e}_{\tau}^{*})]\leq 1,\quad\forall
P\in\mathcal{P}.$ (23)

Next, by the uniformly-randomized Markov’s inequality (Ramdas and Manole,,
2023, theorem 1.2),

$P\left(\mathsf{A}(\mathfrak{e}_{\tau}^{*})\geq\frac{U}{\alpha}\right)\leq\alpha,\quad\forall
P\in\mathcal{P},\;\forall\alpha\in(0,1).$ (24)

One technical remark is that each distribution $P\in\mathcal{P}$ now
additionally describes the external random variable $U$, which we (the
statistician) know to be independent of $\mathbb{F}$ (and of any
$\mathbb{G}$-stopping time). In practice, this means that when to stop the
experiment cannot depend on the external variable $U$.

This two-step _lift-then-randomize (ltr)_ procedure can be summarized by the
following diagram:

$\mathfrak{e}_{\tau}\quad\overset{\text{$\mathfrak{e}$-lifting}}{\underset{\text{(e-to-e)}}{\longrightarrow}}\quad\mathsf{A}(\mathfrak{e}_{\tau}^{*})\quad\overset{\text{$U$-rand.}}{\underset{\text{(e-to-p)}}{\longrightarrow}}\quad\frac{U}{\mathsf{A}(\mathfrak{e}_{\tau}^{*})}\wedge
1$ (25)

for any adjuster $\mathsf{A}$. Here, while $\mathfrak{e}_{\tau}$ is only an
e-value when $\tau$ is a $\mathbb{G}$-stopping time,
$\mathsf{A}(\mathfrak{e}_{\tau}^{*})$ is an e-value for any
$\mathbb{F}$-stopping time $\tau$, and the last random variable,
$(U/\mathsf{A}(\mathfrak{e}_{\tau}^{*}))\wedge 1$, is a p-value for any
$\mathbb{F}$-stopping time $\tau$. Since $U<1$ almost surely, the
$U$-randomization step strictly increases (a.s.) the power of the resulting
sequential test. We summarize this result as a proposition.

######  Proposition 1 (Uniformly-randomized p-value from a lifted e-process).

If $\mathfrak{e}=(\mathfrak{e}_{t})_{t\geq 0}$ is an e-process for
$\mathcal{P}$ w.r.t $\mathbb{G}\subseteq\mathbb{F}$ and $\mathsf{A}$ is an
adjuster, then

$\text{for any $\mathbb{F}$-stopping time
$\tau$,}\quad\tilde{p}_{\tau}^{\mathsf{ltr}}:=\frac{U}{\mathsf{A}(\mathfrak{e}_{\tau}^{*})}\wedge
1\quad\text{is a p-value for $\mathcal{P}$}.$ (26)

In other words, for any $\mathbb{F}$-stopping time $\tau$,
$P(\tilde{p}_{\tau}^{\mathsf{ltr}}\leq\alpha)\leq\alpha$ for any
$P\in\mathcal{P}$ and $\alpha\in(0,1)$.

Proposition 1 yields an $\mathbb{F}$-anytime-valid sequential test for
$\mathcal{P}$ at level $\alpha\in(0,1)$: stop at a $\mathbb{F}$-stopping time
$\tau$, sample an independent random variable $U\sim\mathsf{Unif}[0,1]$,
compute $\tilde{p}_{\tau}^{\mathsf{ltr}}$, and reject $\mathcal{P}$ if
$\tilde{p}_{\tau}^{\mathsf{ltr}}\leq\alpha$. However, notice that
$\tilde{p}_{\tau}^{\mathsf{ltr}}$ is _not_ a p-process w.r.t. $\mathbb{F}$
because it is not $\mathbb{F}$-adapted—it depends on the external random
variable $U$. More generally, when using $U$-randomization, we do not expect
to obtain processes w.r.t. $\mathbb{F}$ because of $U$.

The caveat here is that we only obtain a p-value, even after an adjustment,
and not a p-process. If we were to combine $\tilde{p}_{\tau}^{\mathsf{ltr}}$
with a stopped e-process w.r.t. $\mathbb{F}$, then we would need to apply an
extra p-to-e calibration (likely sacrificing more evidence) before being able
to average the two. Alternatively, if we were to combine the p-value with a
stopped p-process w.r.t. $\mathbb{F}$, then we can, e.g., use an admissible
p-merging function (Vovk et al.,, 2022; Vovk and Wang, 2020b, ) to obtain a
combined p-value.

###  5.2 Lifting the randomized Ville’s inequality

Typically, a p-process with stopping time validity can further yield a time-
uniform inference procedure that can be monitored continuously due to the
equivalence lemma (recall Section 3.1). However, because the lift-then-
randomize procedure of Proposition 1 only yields a “stopped p-value” and not a
p-process, we cannot directly obtain a sequential inference procedure that can
be monitored continuously over time. Fortunately, it follows straightforwardly
from Ramdas and Manole, (2023)’s _randomized Ville’s inequality_ that, when it
comes to deriving a level-$\alpha$ sequential test, we can still monitor the
original e-process $\mathfrak{e}$ at the threshold $1/\alpha$ before reaching
a stopping time, at which we can use the $U$-randomized p-value.

First, to recap, Ramdas and Manole, (2023) develop randomized versions of
Markov’s inequality, as well as other concentration bounds, by utilizing the
independent uniform random variable. Given an e-process
$\mathfrak{e}=(\mathfrak{e}_{t})_{t\geq 0}$ w.r.t. $\mathbb{G}$, the
_randomized Ville’s inequality_ (corollary 4.1.1 in their work) states that,
for any $P\in\mathcal{P}$ and for any $\mathbb{G}$-stopping time $\tau$,

$P\left(\exists
t\leq\tau:\mathfrak{e}_{t}\geq\frac{1}{\alpha}\quad\text{OR}\quad\mathfrak{e}_{\tau}\geq\frac{U}{\alpha}\right)\leq\alpha,$
(27)

where $U\sim\mathsf{Unif}[0,1]$ is independent of $\mathbb{G}$ (and thus of
$\mathfrak{e}$ and $\tau$). The inequality yields the following level-$\alpha$
sequential inference procedure for $\mathcal{P}$ w.r.t. $\mathbb{G}$:

  * •

Collect data and continuously monitor the e-process w.r.t. the filtration
$\mathbb{G}$.

  * •

If $\mathfrak{e}_{t}$ ever crosses $1/\alpha$, stop and reject the null.
Otherwise, stop at any $\mathbb{G}$-stopping time $\tau$.
(This is the first part of the “OR” clause in (27).)

  * •

Draw an independent random variable $U\sim\mathsf{Unif}[0,1]$. Reject the null
if $\mathfrak{e}_{\tau}\geq U/\alpha$.
(This is the second part of the “OR” clause in (27).)

The second threshold of $U/\alpha$ is almost surely smaller than the level
$1/\alpha$, meaning that it gives an additional chance of rejecting the null
under the alternative at a more lenient threshold than the usual $1/\alpha$.
Note that, unlike the original Ville’s inequality, the $U$-randomized version
depends on the filtration $\mathbb{G}$ through the $\mathbb{G}$-stopping time
$\tau$.

Extending this procedure to $\mathbb{F}$-stopping times, we show that we can
continuously monitor the e-process, and then apply $U$-randomization to the
_adjusted running maximum_ at an $\mathbb{F}$-stopping time. As before, $U$
must be independent of $\mathbb{F}$, so the stopping time cannot depend on
$U$.

######  Corollary 3 ($\mathfrak{e}$-lifted randomized Ville’s inequality).

Let $\mathbb{G}\subseteq\mathbb{F}$, and let
$\mathfrak{e}=(\mathfrak{e}_{t})_{t\geq 0}$ be an e-process for $\mathcal{P}$
w.r.t. $\mathbb{G}$. Suppose that $U\sim\mathsf{Unif}[0,1]$ is a uniform
random variable independent of $\mathbb{F}$.

Then, given an adjuster $\mathsf{A}$ and any $\mathbb{F}$-stopping time
$\tau$, the e-process $\mathfrak{e}$ satisfies the randomized Ville’s
inequality in the following way: for any $P\in\mathcal{P}$ and
$\alpha\in(0,1)$,

$P\left(\exists
t\leq\tau:\mathfrak{e}_{t}\geq\frac{1}{\alpha}\quad\text{OR}\quad\mathsf{A}(\mathfrak{e}_{\tau}^{*})\geq\frac{U}{\alpha}\right)\leq\alpha.$
(28)

The proof follows directly from Proposition 1 by choosing the
$\mathbb{F}$-stopping time

$\rho=\tau\wedge\eta,\quad\text{where}\quad\eta=\inf\\{t\geq
1:\mathfrak{e}_{t}\geq 1/\alpha\\},$ (29)

for any $\mathbb{F}$-stopping time $\tau$. Note that we can equivalently say
$\eta=\inf\\{t\geq 1:\mathfrak{e}_{t}^{*}\geq 1/\alpha\\}$ since the time that
a process first exceeds $1/\alpha$ is also when its running maximum first
exceeds $1/\alpha$.

Corollary 3 suggests the following level-$\alpha$ sequential inference
procedure w.r.t. $\mathbb{F}$:

  * •

Collect data and continuously monitor the e-process $\mathfrak{e}$ w.r.t.
$\mathbb{F}$.

  * •

If $\mathfrak{e}$ ever crosses $1/\alpha$, stop and reject the null.
Otherwise, stop at any $\mathbb{F}$-stopping time $\tau$.

  * •

Draw an independent random variable $U\sim\mathsf{Unif}[0,1]$. Reject the null
if $\mathsf{A}(\mathfrak{e}_{\tau}^{*})\geq U/\alpha$, or equivalently, if
$\tilde{p}_{\tau}^{\mathsf{ltr}}\leq\alpha$.

The key difference from (27) is that the stopping time $\tau$ can be chosen
w.r.t. the _lifted_ filtration $\mathbb{F}$, even though $\mathfrak{e}$ is an
e-process only w.r.t. $\mathbb{G}$.

The main takeaway here is that the randomized Ville’s inequality, and its
corresponding sequential testing procedure involving the running maximum and
the adjustment, can be applied to e-processes w.r.t. $\mathbb{G}$ at any
$\mathbb{F}$-stopping times in a specific way: we can track the e-process up
until $\tau$ but then use its adjusted version at the stopping time before
applying $U$-randomization.

###  5.3 Can a randomize-then-lift procedure also be $\mathbb{F}$-anytime-
valid?

An alternative proposal to the lift-then-randomize approach is to first apply
$U$-randomization and then apply a lifting procedure. The potentially
interesting connection here is that, if we can first obtain a $U$-randomized
p-value at a stopping time, then perhaps similarly to Ignatiadis et al.,
(2023), we can utilize a p-to-e calibrator $\mathsf{C}$ for lifting. As shown
in Section 2.4, we know that each $\mathsf{C}$ has a one-to-one correspondence
with an adjuster $\mathsf{A}$. While we leave the anytime-validity of this
proposed procedure as an open problem, we provide a formal description and
give a preliminary analysis that establishes a loose upper bound on the type I
error at an $\mathbb{F}$-stopping time.

First, we formally describe the _randomize-then-lift (rtl)_ procedure. Given
an e-process $\mathfrak{e}$ w.r.t. a sub-filtration
$\mathbb{G}\subseteq\mathbb{F}$ and any $\mathbb{G}$-stopping time $\tau$, we
apply the $U$-randomized e-to-p calibrator, which maps $\mathfrak{e}_{\tau}$
to a $\mathbb{G}$-stopped p-value of
$\tilde{\mathfrak{p}}_{\tau}=(U/\mathfrak{e}_{\tau})\wedge 1$. Note that, for
an arbitrary $\mathbb{F}$-stopping time $\tau^{\mathbb{F}}$
$\tilde{\mathfrak{p}}_{\tau^{\mathbb{F}}}$ need not be a p-value. We also do
not take the running maximum here. Then, we apply any p-to-e calibrator
$\mathsf{C}$ to obtain an $\mathbb{G}$-stopped e-value. The following diagram
summarizes this process:

$\mathfrak{e}_{\tau}\quad\overset{\text{randomized
e-to-p}}{\underset{U\sim\mathsf{Unif}[0,1]}{\longrightarrow}}\quad\tilde{\mathfrak{p}}_{\tau}^{\mathsf{rtl}}:=\frac{U}{\mathfrak{e}_{\tau}}\wedge
1\quad\overset{\text{p-to-e}}{\longrightarrow}\quad\tilde{\mathfrak{e}}_{\tau}^{\mathsf{rtl}}:=\mathsf{C}\left(\frac{U}{\mathfrak{e}_{\tau}}\wedge
1\right).$ (30)

While $\tilde{\mathfrak{e}}_{\tau}^{\mathsf{rtl}}$ is a valid e-value for
$\mathcal{P}$ at any $\mathbb{G}$-stopping time $\tau$, it remains unclear
whether this validity extends to $\mathbb{F}$-stopping times. The technical
challenge here is that, once $U$ is included in the process, any event
involving the resulting sequence is no longer adapted to the original
filtration ($\mathbb{G}$). This precludes the use of the lifting lemma (Lemma
1), which requires an adapted sequence of events. As before, when to stop the
experiment cannot depend on the external random variable $U$.

Now, we describe our preliminary result. Ideally, what we want to show is that
$\tilde{\mathfrak{p}}_{\tau^{\mathbb{F}}}^{\mathsf{rtl}}$ is a p-value for
$\mathcal{P}$ for any $\mathbb{F}$-stopping time $\tau^{\mathbb{F}}$, that is,
$P(\tilde{\mathfrak{p}}_{\tau^{\mathbb{F}}}^{\mathsf{rtl}}\leq\alpha)\leq\alpha$
for any $P\in\mathcal{P}$ and $\alpha\in(0,1)$. The level $\alpha$ would then
be an upper bound on
$\sup_{P\in\mathcal{P}}P(\tilde{\mathfrak{p}}_{\tau^{\mathbb{F}}}^{\mathsf{rtl}}\leq\alpha)$,
which is the type I error. Here, the following lemma instead gives a loose
upper bound of $\tilde{\alpha}=\alpha+\alpha\log(1/\alpha)$ on the type I
error. We refer to this result as the “$U$-lifting lemma” as it plays the
analogous role to the lifting lemma (Lemma 1).

######  Lemma 2 (The $U$-lifting lemma for e-processes in a sub-filtration).

Let $\mathfrak{e}=(\mathfrak{e}_{t})_{t\geq 0}$ be an e-process for
$\mathcal{P}$ w.r.t. $\mathbb{G}\subseteq\mathbb{F}$, and let
$U\sim\mathsf{Unif}[0,1]$ be independent of $\mathbb{F}$. Fix
$\alpha\in(0,1)$. Then,

$\text{for any $\mathbb{F}$-stopping time $\tau^{\mathbb{F}}$,}\quad
P\left(\mathfrak{e}_{\tau^{\mathbb{F}}}\geq\frac{U}{\alpha}\right)\leq\alpha+\alpha\log(1/\alpha),\quad\forall
P\in\mathcal{P}.$ (31)

Equivalently,

$\text{for any $\mathbb{F}$-stopping time $\tau^{\mathbb{F}}$,}\quad
P\left(\tilde{\mathfrak{p}}_{\tau^{\mathbb{F}}}^{\mathsf{rtl}}\leq\alpha\right)\leq\alpha+\alpha\log(1/\alpha),\quad\forall
P\in\mathcal{P}.$ (32)

The lemma says that the “pseudo-p-value” obtained via $U$-randomized lifting,
$\tilde{\mathfrak{p}}_{\tau^{\mathbb{F}}}^{\mathsf{rtl}}$, has its type I
error controlled at an inflated level of
$\tilde{\alpha}=\alpha+\alpha\log(1/\alpha)>\alpha$. For example, if
$\alpha=0.05$, then $\tilde{\alpha}$ is approximately $0.2$. To control the
type I error at the level $\tilde{\alpha}=0.05$, we need to use a more
stringent level of $\alpha\approx 0.0087$ for
$\tilde{\mathfrak{p}}_{\tau^{\mathbb{F}}}^{\mathsf{rtl}}$, but then this may
offset gains in the statistical power from the randomization procedure when
compared with the non-randomized $\mathfrak{p}$-lifting procedure (Theorem 1).

The proof of the $U$-lifting lemma, included in Section C.3, only depends on
the classical Markov’s inequality and the equivalence lemma (Ramdas et al.,,
2020; Howard et al.,, 2021). It remains to be seen whether it is possible to
remove the extra $+\alpha\log(1/\alpha)$ term for any $\mathbb{F}$-stopping
time $\tau^{\mathbb{F}}$. If $\tilde{\mathfrak{p}}_{\tau^{\mathbb{F}}}$ could
be shown to be a p-value for $\mathcal{P}$, then it would follow that
$\tilde{\mathfrak{e}}_{\tau^{\mathbb{F}}}=\mathsf{C}(\tilde{\mathfrak{p}}_{\tau^{\mathbb{F}}})$
is an e-value for $\mathcal{P}$ for any p-to-e calibrator $\mathsf{C}$, and
thus we could further derive a continuous monitoring procedure via randomized
Ville’s inequality involving $\tilde{\mathfrak{e}}_{\tau^{\mathbb{F}}}$,
analogous to the one in Section 5.2.

##  6 Discussion

To conclude, this work (i) studied methods for combining e-processes across
filtrations via $\mathfrak{e}$-lifting, which applies an adjuster to any
e-process that is only valid in a sub-filtration; (ii) illustrated three
topical use cases in sequential inference; and (iii) developed extensions that
leverage external randomization to improve the power of a lifted e-process at
a stopping time. We close with discussions of additional topics, including
implications on when powerful e-processes exist for a testing problem and on
designing e-processes by coarsening the filtration; other variations of the
setup; and a game-theoretic interpretation of our main result.

###  6.1 A theoretical implication on when powerful e-processes exist for a
testing problem

The lifting result of Theorem 2 establishes the answer to a theoretical
question that, to the best of our knowledge, was not previously known in its
generality. To motivate the question, we first recall that, when it comes to
test martingales (_not_ e-processes), there are testing problems for which
there exist no powerful test martingales w.r.t. the data filtration while
there do exist powerful test martingales w.r.t. a coarsening of the
filtration, such as Vovk, 2021b ’s conformal test martingale for
exchangeability and Henzi and Law, (2023)’s rank-based test martingale for
independence.

We may now ask the analogous question for e-processes:

Are there testing problems for which there is no powerful e-process in the
data filtration but there exist ones in a coarser filtration?

Theorem 2 implies that the answer is “no”: if we can find a powerful e-process
in a sub-filtration, then we can always $\mathfrak{e}$-lift it into the data
filtration with an adjuster. Since any adjuster $\mathsf{A}$ is an increasing
function that satisfies
$\lim_{\mathfrak{e}\to\infty}\mathsf{A}(\mathfrak{e})=\infty$ by definition
(10), we can be assured that the lifted e-process is also powerful (against
the same alternative). We also note that adjusting a test martingale w.r.t. a
sub-filtration yields an e-process but _not_ a test martingale w.r.t. the data
filtration (follows from definition (9)), so this result does not contradict
prior results on how there exist no powerful test martingales for composite
nulls like exchangeability and independence.

Thus, when designing a sequential inference procedure for a highly composite
null, for which it is challenging to construct a powerful e-process w.r.t. the
data filtration, coarsening the filtration to find a powerful e-process may be
the only viable strategy.

###  6.2 A practical implication on designing e-processes by coarsening the
filtration

One motivation for our study has been that an e-process (or a test martingale)
designed in a coarse filtration is not _truly_ immune to data peeking, even
though the proposed key benefit of using an anytime-valid method in the first
place is (arguably) the ability to freely peek at the data and deciding when
to stop. Yet, if we were to lift an e-process from a coarse filtration to
achieve anytime-validity at the data level, we need to go through an adjuster
(according to Theorem 2), which may hurt the power when there is only small
data or weak evidence. In other words, depending on the use case, designing
e-processes w.r.t. a sub-filtration does not come without a cost, as it limits
the flexibility of the experimenter’s actions during a sequential experiment.
Of course, some coarsening approaches like pairwise betting strategies
(Podkopaev et al.,, 2023; Saha and Ramdas,, 2024) can avoid this issue in
practice, and for certain composite nulls (e.g., the hypothesis of
independence in Section 4.3), coarsening the filtration remains the only known
way of sequentially testing them.

###  6.3 Other variations of the setup

Although we focused on combining two e-processes for the same (composite) null
$\mathcal{P}$, it is not difficult to imagine more complex variations of this
setup. The extension to $K>2$ e-processes is straightforward: if
$\mathfrak{e}^{(k)}$ is an e-process for $\mathcal{P}$ w.r.t. filtration
$\mathbb{F}^{(k)}$, then we can combine them in the smallest “envelope”
filtration encompassing all filtrations
$\mathbb{F}^{(1)},\dotsc,\mathbb{F}^{(k)}$, and $\mathfrak{e}$-lift each
$\mathfrak{e}^{(k)}$ into the envelope filtration. Additionally, if, say, one
of the filtrations $\mathbb{F}^{(k)}$ is the data filtration, then the
envelope filtration would simply be the data filtration and
$\mathfrak{e}^{(k)}$ would not need to be adjusted.

A slightly more involved setup is when the two e-processes test related but
distinct hypotheses. For example, suppose that $\mathfrak{e}$ is an e-process
for $\mathcal{P}$ w.r.t. $\mathbb{G}$, $\mathfrak{e}^{\prime}$ is an e-process
for $\mathcal{Q}$ w.r.t. $\mathbb{F}$, and $\mathbb{G}\subseteq\mathbb{F}$.
Can we meaningfully combine the two e-processes if $\mathcal{P}$ and
$\mathcal{Q}$ are related in some way? One possibility is when $\mathcal{P}$
and $\mathcal{Q}$ share a common subset of hypotheses, i.e.,
$\mathcal{P}\cap\mathcal{Q}=\mathcal{H}\neq\emptyset$. Then, since both
$\mathfrak{e}$ and $\mathfrak{e}^{\prime}$ are e-processes for $\mathcal{H}$,
we can combine them via Corollary 2 to obtain an e-process for $\mathcal{H}$.
This procedure can be further extended to the case where there is a cascade of
multiple hypotheses,
$\mathcal{P}^{(1)}\subseteq\dotsb\subseteq\mathcal{P}^{(K)}$, for which
e-processes $\mathfrak{e}^{(1)},\dotsc,\mathfrak{e}^{(K)}$ are defined w.r.t.
their own filtrations $\mathbb{F}^{(1)},\dotsc,\mathbb{F}^{(K)}$,
respectively. It remains to be seen whether there are motivating examples for
which these generalizations can be useful.

###  6.4 A game-theoretic protocol for combining evidence: Cooperative
skepticism

While our main results are stated in the language of measure-theoretic
probability, they can also be stated in the language of game-theoretic
probability and statistics (Shafer and Vovk,, 2005, 2019; Ramdas et al.,,
2023). The starting point is the connection that a test martingale represents
the wealth of a gambler who tries to discredit a forecaster by betting against
their proposed forecasts (the null hypothesis) over a sequence of outcomes.
Specifically, in the _testing-by-betting_ framework (Shafer,, 2021), the
wealth of a gambler (“skeptic”) who plays against a point null $P$ is a test
(super)martingale for $P$, such that large wealth for the skeptic indicates
evidence against $P$. For a composite null $\mathcal{P}$, the gambler is
evaluated by the _minimum_ wealth of all skeptics, each of whom plays against
each member of the null $P\in\mathcal{P}$. An e-process for $\mathcal{P}$, a
composite generalization of a test martingale, corresponds to this minimum
wealth against each $P\in\mathcal{P}$ (Ramdas et al.,, 2022). Definition (3)
translates to saying that an e-process can only grow large if the data
discredits every possible null scenario.

Following Shafer and Vovk, (2019), we can formalize the game each skeptic
plays against a null forecaster $P=(P_{t})_{t\geq 0}$ as follows. At each
$t=1,2,\dotsc$, the forecaster proposes a probability distribution $P_{t}$ on
the eventual outcome $Y_{t}$, and then a skeptic chooses a bet
$S_{t}:\mathcal{Y}\to\mathbb{R}_{\geq 0}$ that determines the multiplicative
increment on their wealth depending on the outcome. The game then requires
that this bet is expected to be small if the forecaster were correct:

$\mathbb{E}_{{P_{t}}}\left[{S_{t}(Y_{t})}\right]\leq 1.$ (33)

Assuming the skeptic starts with an initial wealth of $\mathcal{K}_{0}=1$, the
skeptic’s wealth at time $t$ would be $\mathcal{K}_{t}=K_{t-1}\cdot
S_{t}(Y_{t})$, forming a test martingale for $P$. By the constraint (33), the
wealth process $(\mathcal{K}_{t})_{t\geq 0}$ can only grow large if the
forecast $P$ does not accurately describe the actual outcomes $(Y_{t})_{t\geq
0}$.

Based on this insight, we can introduce a game-theoretic protocol that
corresponds to the setup for Corollary 2. The setup corresponds to having two
skeptics that are playing against a common forecaster (the shared null), and
their wealth processes correspond to their respective e-processes. In the
general setup for this paper, each skeptic operates with their own information
set, which we will make explicit. For the sake of exposition, we assume that
the two e-processes are simply test supermartingales for a point null
$P=(P_{t})_{t\geq 0}$ over a data sequence $(Y_{t})_{t\geq 0}$, but the
framework can be generalized to e-processes. Inspired by Dawid et al., 2011b
(, protocol 1), we refer to the following game as _cooperative skepticism_ to
highlight the shared effort among the two Skeptics.

######  Protocol 1 (Cooperative skepticism).

Let $\mathcal{K}_{0}^{(1)}=\mathcal{K}_{0}^{(2)}=1$. For rounds
$t=1,2,\dotsc$:

  1. 1.

Forecaster announces a distribution $P_{t}$ on $\mathcal{Y}$.

  2. 2.

Skeptics 1 and 2 _individually_ place their bets:

     * •

Skeptic 1 makes a $\mathcal{F}_{t-1}$-measurable bet
$S_{t}^{(1)}:\mathcal{Y}\to\mathbb{R}_{\geq 0}$ s.t.
$\mathbb{E}_{{P_{t}}}\left[{S_{t}^{(1)}(Y_{t})\left.\middle|\right.\mathcal{F}_{t-1}}\right]\leq
1$.

     * •

Skeptic 2 makes a $\mathcal{G}_{t-1}$-measurable bet
$S_{t}^{(2)}:\mathcal{Y}\to\mathbb{R}_{\geq 0}$ s.t.
$\mathbb{E}_{{P_{t}}}\left[{S_{t}^{(2)}(Y_{t})\left.\middle|\right.\mathcal{G}_{t-1}}\right]\leq
1$.

  3. 3.

Reality announces $y_{t}\in\mathcal{Y}$.

  4. 4.

Each skeptic’s wealth is updated as follows:

$\displaystyle\mathcal{K}_{t}^{(1)}=\mathcal{K}_{t-1}^{(1)}\cdot
S_{t}^{(1)}(y_{t})\quad\text{and}\quad\mathcal{K}_{t}^{(1)}=\mathcal{K}_{t-1}^{(2)}\cdot
S_{t}^{(2)}(y_{t}).$ (34)

_Note:_ $\mathcal{F}_{t-1}$ and $\mathcal{G}_{t-1}$ represent information sets
($\sigma$-fields) with which skeptics 1 and 2 respectively choose their bets
for round $t$. In general, they are distinct (but not necessarily disjoint).

In Protocol 1, we (as the statisticians who stand outside of the game) may
choose to bet against the forecaster by splitting our initial wealth and (say)
use half of our money following skeptic 1’s strategy and the other half
following skeptic 2’s strategy. The protocol suggests that the skeptics’
wealth processes, $(\mathcal{K}_{t}^{(1)})_{t\geq 0}$ and
$(\mathcal{K}_{t}^{(2)})_{t\geq 0}$, are test supermartingales w.r.t. the
respective filtrations, $\mathbb{F}=(\mathcal{F}_{t})_{t\geq 0}$ and
$\mathbb{G}=(\mathcal{G}_{t})_{t\geq 0}$. Then, assuming that we operate with
our own information set, $\mathbb{H}$, that includes both skeptic’s
information sets in each round, i.e.,
$\mathbb{F},\mathbb{G}\subseteq\mathbb{H}$, Theorem 2 translates to saying
that we can construct our own test martingale (wealth process) by
$\mathfrak{e}$-lifting each skeptic’s wealth process and averaging them. Put
differently, in each round $t$, we agree to play against the forecaster and
receive the wealth of
$\frac{1}{2}[\mathsf{A}((\mathcal{K}_{t}^{(1)})^{*})+\mathsf{A}^{\prime}((\mathcal{K}_{t}^{(2)})^{*})]$
for some adjusters $\mathsf{A},\mathsf{A}^{\prime}$. The interpretation
generalizes to e-processes, where each skeptic is now evaluated by their
minimum wealth over multiple games, one against each member (forecaster) of
the null hypothesis. Denoting the minimum wealth after round $t$ as
$\kappa_{t}^{(1)}$ and $\kappa_{t}^{(2)}$ for each skeptic, we would receive
the analogous wealth of
$\frac{1}{2}[\mathsf{A}((\kappa_{t}^{(1)})^{*})+\mathsf{A}^{\prime}((\kappa_{t}^{(2)})^{*})]$.

We close with the remark that, while Protocol 1 describes a game-theoretic
interpretation of Theorem 2 and Corollary 2, it is not itself a result in
game-theoretic probability like Dawid et al., 2011b ’s. Dawid et al., 2011b
defines a perfect-information protocol, with which they formally prove the
equivalence between (9) and (10) to characterize adjusters; Protocol 1 is not
a perfect-information game, and it remains unclear whether Theorem 2 can be
proven solely in the game-theoretic framework (without measure-theoretic
language).

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##  Appendix A Details on e-processes for testing exchangeability

#### Setup

Given a stream of binary random variables $X_{1},X_{2},\dotsc$, we are
interested in sequentially testing whether the data forms an exchangeable
sequence:

$\mathcal{P}^{\mathsf{exch}}:X_{1},X_{2},\dotsc\text{ are exchangeable.}$ (35)

As mentioned in Section 1, any test for exchangeability is also a test for
whether the data is i.i.d. (also referred to as _randomness_ in, e.g., Vovk,
2021b ), and further any nontrivial e-process for testing whether the data is
i.i.d. must be an e-process for testing exchangeability (Ramdas et al.,,
2022). In the below, let $\mathbb{F}=(\mathcal{F}_{t})_{t\geq 0}$ denote the
data filtration such that $\mathcal{F}_{t}=\sigma(X_{1},\dotsc,X_{t})$.

#### Universal inference e-process

Ramdas et al., (2022) derive an e-process for $\mathcal{P}^{\mathsf{exch}}$
w.r.t. the data filtration $\mathbb{F}$ that is powerful against Markovian
alternatives by leveraging the universal inference (UI) approach (Wasserman et
al.,, 2020). The high-level idea of UI is to compute a generalized likelihood
ratio, whose denominator is the maximum likelihood under the null and the
numerator is a mixture (Robbins,, 1970) over a class of alternatives. For the
denominator, it suffices to use the maximum likelihood under the i.i.d. family
$\mathcal{P}^{\mathsf{iid}}=\\{\mathsf{Ber}(p)^{\infty}:p\in[0,1]\\}$, whose
convex hull is precisely $\mathcal{P}^{\mathsf{exch}}$ due to de Finetti,
(1931)’s theorem, and then invoke Ramdas et al., (2020, proposition 15)’s
result that says any e-process for a family of distributions (e.g.,
$\mathcal{P}^{\mathsf{iid}}$) is also an e-process for its convex hull
($\mathcal{P}^{\mathsf{exch}}$). For the numerator, Ramdas et al., (2022)
choose a tractable mixture distribution over all first-order Markovian
alternatives, which are parametrized using two parameters $p_{1|0}$ and
$p_{1|1}$, where $p_{k|j}$ denotes the transition probability from state $j$
to state $k$ for $j,k\in\\{0,1\\}$.

Put together, the UI e-process $\mathfrak{e}=(\mathfrak{e}_{t})_{t\geq 0}$ for
$\mathcal{P}^{\mathsf{exch}}$ w.r.t. $\mathbb{F}$ is given by the following
closed-form expression: $\mathfrak{e}_{0}=1$, and for each $t\geq 1$,

$\mathfrak{e}_{t}=\frac{\Gamma\left(n_{0|0}+\frac{1}{2}\right)\Gamma\left(n_{0|1}+\frac{1}{2}\right)\Gamma\left(n_{1|0}+\frac{1}{2}\right)\Gamma\left(n_{1|1}+\frac{1}{2}\right)}{2\Gamma\left(\frac{1}{2}\right)^{4}\Gamma\left(n_{0|0}+n_{1|0}+1\right)\Gamma\left(n_{0|1}+n_{1|1}+1\right)}\bigg{/}\left(\left(\frac{n_{1}}{t}\right)^{n_{1}}\left(\frac{n_{0}}{t}\right)^{n_{0}}\right),$
(36)

where given the observed data $X_{1},\dotsc,X_{t}$ up to time $t$, $n_{j}$
denotes the number of observations for the value $j\in\\{0,1\\}$ and $n_{k|j}$
denotes the number of transitions from $j$ to $k$ for $j,k\in\\{0,1\\}$.
$\Gamma$ denotes the usual gamma function.

The UI e-process is the first known nontrivial e-process w.r.t. $\mathbb{F}$
for a composite null for which no nontrivial test supermartingale w.r.t.
$\mathbb{F}$ exists (Ramdas et al.,, 2022).

#### Conformal inference e-process and the “simple jumper” algorithm

Vovk et al., (2003); Vovk, 2021b  introduce a sequential testing procedure for
exchangeability entirely based on conformal p-values. At each time $t$, define
the _nonconformity score_
$\alpha_{t}=\mathcal{A}(X_{t},\\{X_{1},\dotsc,X_{t}\\})\in\mathbb{R}$. For
general data types, $\mathcal{A}$ can be any measure for the “strangeness” of
$X_{t}$ alongside the observed data $\\{X_{1},\dotsc,X_{t}\\}$; for our binary
case, we follow the prior works and consider the identity function, that is,
$\alpha_{t}=X_{t}$ for each $t\geq 1$ (Vovk, 2021a, ). Then, the _conformal
p-value_ $p_{t}$ at time $t$ is defined using the nonconformity scores
$\alpha_{1},\dotsc,\alpha_{t}$ as follows:

$\displaystyle p_{t}$ $\displaystyle=\frac{\\#\left\\{i\leq
t:\alpha_{i}>\alpha_{t}\right\\}+U_{t}\cdot\\#\left\\{i\leq
t:\alpha_{i}=\alpha_{t}\right\\}}{t}$ (37)
$\displaystyle=\begin{cases}U_{t}\cdot f_{1}(t)&\mbox{if }X_{t}=1;\\\
f_{1}(t)+U_{t}\cdot f_{0}(t)&\mbox{if }X_{t}=0,\end{cases}$ (38)

where $f_{k}(t)=t^{-1}\cdot\\#\\{i\leq t:X_{i}=k\\}$ denotes the empirical
frequency of seeing $k\in\\{0,1\\}$ up to time $t$, and
$U_{t}\sim\mathsf{Unif}[0,1]$ is an i.i.d. Uniform random variable that is
independent of $\mathbb{F}$. Equation (37) is a general definition for
conformal p-values, while equation (38) represents a simplification assuming
that the nonconformity score is the identity function and that the data is
binary. Then, by Vovk et al., (2003, 2005), the conformal p-values
$(p_{t})_{t\geq 1}$ are i.i.d. $\mathsf{Unif}[0,1]$ under the null hypothesis.

Next, define the filtration $\mathbb{G}=(\mathcal{G}_{t})_{t\geq 0}$
consisting of the conformal p-values, that is,
$\mathcal{G}_{t}=\sigma(p_{1},\dotsc,p_{t})$. Then, w.r.t. $\mathbb{G}$, we
can construct test martingales (e-processes) for $\mathcal{H}_{0}$ of the
following form: $\mathfrak{e}_{0}(\lambda)=1$ and

$\mathfrak{e}_{t}(\lambda)=\prod_{i=1}^{t}\left[1+\lambda\left(p_{i}-\frac{1}{2}\right)\right],\quad\forall
t\geq 1,$ (39)

for any fixed $\lambda\in\mathbb{R}$. The scalar $\lambda$ determines the sign
and amount of the “bet” on the deviation of each $p_{i}$ from its mean of
$1/2$ under the null hypothesis. In Figure 1, we use the value of $\lambda=1$,
corresponding to betting when $p_{i}$ exceeds its expected mean under the null
hypothesis.

In practice, since a weighted average of e-processes is also an e-process,
w.r.t. the same filtration, we can take a mixture of e-processes defined using
different values of $\lambda$. Vovk et al., (2021) defines the “simple jumper”
e-process as

$\mathfrak{e}_{t}^{\mathsf{sj}}=w_{-1}\mathfrak{e}_{t}(-1)+w_{0}\mathfrak{e}_{t}(0)+w_{1}\mathfrak{e}_{t}(1),\quad\forall
t\geq 0,$ (40)

for some fixed weights $w_{-1},w_{0},w_{1}\geq 0$ that sum to 1, or
alternatively notice that (39) is also an e-process for any
$\mathbb{G}$-predictable sequence $(\lambda_{t})_{t\geq 0}$ and use clever
choices of the hyperparameter (e.g., Fedorova et al.,, 2012; Waudby-Smith and
Ramdas,, 2023). The e-process we use in the main paper corresponds to the
simple jumper with weights $w_{1}=1$ and $w_{0}=w_{-1}=0$; similar results to
Figure 1 can be obtained with a variation where $w_{-1}=w_{1}=1/2$ and the
stopping time is now the first time when $k$ consecutive zeros _or_ ones are
observed. In our testing, Vovk, 2021a ’s original version with
$w_{-1}=w_{0}=w_{1}=1/3$ does not empirically violate the stopping time
validity w.r.t. the data filtration $\mathbb{F}$, despite having no
theoretical guarantee of $\mathbb{F}$-anytime-validity.

##  Appendix B Additional example: Testing a scale-invariant Gaussian mean

Here we give another concrete example of an e-process defined w.r.t. a coarser
filtration $\mathbb{G}\subseteq\mathbb{F}$ that is not valid at a
$\mathbb{F}$-stopping time. This example is due to Pérez-Ortiz et al., (2022).

######  Example 4 (Sequentially testing a scale-invariant Gaussian mean;
Pérez-Ortiz et al., (2022)).

Suppose that the data $X_{1},X_{2},\dotsc$ is sequentially sampled from a
Gaussian $\mathcal{N}(\mu,\sigma^{2})$ with unknown parameters,
$\mu\in\mathbb{R}$ and $\sigma\in\mathbb{R}^{+}$. Consider the following
composite null and composite hypotheses:

$\mathcal{H}_{0}:\frac{\mu}{\sigma}=\delta_{0}\quad\text{vs.}\quad\mathcal{H}_{1}:\frac{\mu}{\sigma}=\delta_{1},$
(41)

for some $\delta_{0}\neq\delta_{1}$. The parameter of interest
$\delta:=\mu/\sigma$ is scale-invariant, in the sense that it is invariant to
the mapping $(\mu,\sigma)\mapsto(c\mu,c\sigma)$ for any $c\in\mathbb{R}^{+}$.
The null and alternative hypotheses corresponding to (41) can be parametrized
as $\Theta_{j}=\\{(\delta_{j}\sigma,\sigma):\sigma\in\mathbb{R}^{+}\\}$ for
$j=0,1$, respectively (making them composite hypotheses).

Let $\mathbb{F}=(\mathcal{F}_{t})_{t=1}^{\infty}$ denote the canonical data
filtration, i.e., $\mathcal{F}_{t}=\sigma(X_{1},\dotsc,X_{t})$, and let
$\mathbb{G}=(\mathcal{G}_{t})_{t=1}^{\infty}$ denote the following scale-
invariant coarsening of $\mathbb{F}$:

$\mathcal{G}_{t}=\sigma\left(\frac{X_{1}}{\left\lvert
X_{1}\right\rvert},\dotsc,\frac{X_{t}}{\left\lvert
X_{1}\right\rvert}\right),\quad\forall t\geq 1.$ (42)

Intuitively, $\mathbb{G}$ removes from $\mathbb{F}$ the scale information
about the initial data point $X_{1}$ and retains only its sign. Then, the test
statistic defined by the ratio of likelihood mixtures (Cox,, 1952; Lai,,
1976),

$\mathfrak{e}_{t}^{\mathsf{MaxInv}}=\frac{\int_{\sigma>0}p_{\delta_{1},\sigma}(m_{t}(X_{1},\dotsc,X_{t}))d\rho(\sigma)}{\int_{\sigma>0}p_{\delta_{0},\sigma}(m_{t}(X_{1},\dotsc,X_{t}))d\rho(\sigma)}=\frac{\int_{\sigma>0}\frac{1}{\sigma}\exp\left[-\frac{n}{2}\left\\{\left(\frac{\hat{\mu}_{t}}{\sigma}-\delta_{1}\right)^{2}+\frac{\hat{\sigma}_{t}^{2}}{\sigma^{2}}\right\\}\right]d\sigma}{\int_{\sigma>0}\frac{1}{\sigma}\exp\left[-\frac{n}{2}\left\\{\left(\frac{\hat{\mu}_{t}}{\sigma}-\delta_{0}\right)^{2}+\frac{\hat{\sigma}_{t}^{2}}{\sigma^{2}}\right\\}\right]d\sigma},$
(43)

is an e-process for $\mathcal{H}_{0}$ w.r.t. the sub-filtration $\mathbb{G}$
(Pérez-Ortiz et al.,, 2022, proposition 1.3), where
$m_{t}(X_{1},\dotsc,X_{t})=(X_{1}/|X_{1}|,\dotsc,X_{t}/|X_{1}|)$ represents a
$\mathcal{G}_{t}$-measurable and “maximally invariant” statistic that only
depends on the scale-invariant parameter $\delta$. $\rho(\sigma)=1/\sigma$ is
a right Haar measure.555Here, we sidestep group-theoretic definitions that are
not central to our discussion. See Pérez-Ortiz et al., (2022) for an in-depth
discussion of deriving GROW e-values and e-processes for composite nulls and
alternatives that are group-invariant. Under the null, the expectation of
$\mathfrak{e}^{\mathsf{MaxInv}}$ at any $\mathbb{G}$-stopping time
$\tau^{\mathbb{G}}$ is upper-bounded by 1.

However, Pérez-Ortiz et al., (2022, appendix B) also notes that
$\mathfrak{e}^{\mathsf{MaxInv}}$ is _not_ upper-bounded by 1 at
$\mathbb{F}$-stopping times in expectation. Specifically, define
$\tau^{\mathbb{F}}=1+\mathds{1}\left(\left\lvert
X_{1}\right\rvert\in[a,b]\right)$ for some constants $0<a<b$, meaning that we
consider stopping immediately if the absolute value of the first data point is
outside a range, and if not we stop after the second data point. Then,
$\tau^{\mathbb{F}}$ is clearly an $\mathbb{F}$-stopping time but is not
adapted w.r.t. $\mathbb{G}$ because it depends on the scale of $X_{1}$. Then,
for the values $\delta_{0}=0$, $a\approx 0.44$, $b\approx 1.70$, and a
suitable choice of mixture over $\mathcal{H}_{1}$, numerical calculations show
that

$\mathbb{E}_{{P}}\left[{\mathfrak{e}_{\tau^{\mathbb{F}}}^{\mathsf{MaxInv}}}\right]\approx
1.19>1,$ (44)

where $P\in\mathcal{H}_{0}$ is a null distribution that posits a standard
normal distribution over the first two data points, i.e.,
$X_{1},X_{2}\overset{iid}{\sim}\mathcal{N}(0,1)$.

##  Appendix C Proofs

###  C.1 Proof of Lemma 1

The direction b $\Rightarrow$ a is immediate. For the lifting direction, a
$\Rightarrow$ b, as alluded to in the main paper, the proof adapts the logic
from Ramdas et al., (2020, lemma 1) and Howard et al., (2021, lemma 3) to
stopping time validity across filtrations $\mathbb{F}$ and $\mathbb{G}$, where
$\mathbb{G}\subseteq\mathbb{F}$.

Consider any $P\in\mathcal{P}$. Given the condition a, choose the
$\mathbb{G}$-stopping time $\tau^{\mathbb{G}}=\inf\\{t\geq 1:\text{$\xi_{t}$
occurs}\\}$ to obtain

$P\left(\bigcup_{t\geq 1}\xi_{t}\right)\leq\alpha.$ (45)

Then, for any random time $T$ taking values in $\mathbb{N}\cup\\{\infty\\}$,
we have that $P(\xi_{T})\leq\alpha$ because

$\xi_{T}=\left(\bigcup_{t\in\mathbb{N}}(\xi_{t}\cap\\{T=t\\}\right)\cup\left(\xi_{\infty}\cap\\{T=\infty\\}\right)\subseteq\bigcup_{t\geq
1}\xi_{t}.$ (46)

Now, given that $\xi_{t}\in\mathcal{G}_{t}\subseteq\mathcal{F}_{t}$, for each
$t\geq 0$, we know that $(\xi_{t})_{t\geq 0}$ is also adapted to $\mathbb{F}$.
Since any $\mathbb{F}$-stopping time $\tau^{\mathbb{F}}$ is a random time, the
result b follows.

###  C.2 Proof of Theorem 2

All of the following statements hold “for $\mathcal{P}$.” Since
$(\mathfrak{e}_{t})_{t\geq 0}$ is an e-process w.r.t. $\mathbb{G}$, we know
from Ville’s inequality for e-processes (Ville,, 1939; Ramdas et al.,, 2020)
that

$\mathfrak{p}_{t}^{*}=\frac{1}{\mathfrak{e}_{t}^{*}}\wedge 1,\quad\forall
t\geq 0,$ (47)

is a p-process w.r.t. $\mathbb{G}$. By $\mathfrak{p}$-lifting (Theorem 1), we
know that $(\mathfrak{p}_{t}^{*})_{t\geq 0}$ is also a p-process w.r.t.
$\mathbb{F}$.

Given an adjuster $\mathsf{A}$, we can find its corresponding p-to-e
calibrator, i.e.,
$\mathsf{C}(\mathfrak{p})=\mathsf{A}(1/\mathfrak{p}),\,\forall\mathfrak{p}\in[0,1]$.
Then, the process $(\mathsf{A}(\mathfrak{e}_{t}^{*}))_{t\geq 0}$ is upper-
bounded by the e-process $(\mathsf{C}(\mathfrak{p}_{t}^{*}))_{t\geq 0}$ w.r.t.
$\mathbb{F}$:

$\mathsf{A}(\mathfrak{e}_{t}^{*})=\mathsf{C}\left(\frac{1}{\mathfrak{e}_{t}^{*}}\right)\leq\mathsf{C}\left(\mathfrak{p}_{t}^{*}\right),\quad\forall
t\geq 0.$ (48)

Note that the inequality holds because $\mathsf{C}$ is nonincreasing, and it
is solely due to the clipping of the p-process in (47) that ensures
$\mathfrak{p}_{t}$ does not exceed 1. Then, for any $P\in\mathcal{P}$ and any
$\mathbb{F}$-stopping time $\tau$, we have that

$\mathbb{E}_{P}\left[\mathsf{A}(\mathfrak{e}_{\tau}^{*})\right]\leq\mathbb{E}_{P}\left[\mathsf{C}\left(\mathfrak{p}_{\tau}^{*}\right)\right]\leq
1,$ (49)

proving that $(\mathsf{A}(\mathfrak{e}_{t}^{*}))_{t\geq 0}$ is an e-process
w.r.t. $\mathbb{F}$.

###  C.3 Proof of Lemma 2

This proof utilizes Markov’s inequality and the equivalence lemma (Ramdas et
al.,, 2020; Howard et al.,, 2021). Compared to the proof of the lifting lemma
(Lemma 1), it involves extra steps to carefully handle the external random
variable $U$ that makes the event $\\{\mathfrak{e}_{\tau}\geq U/\alpha\\}$
non-adapted to $\mathbb{G}$ or $\mathbb{F}$. Because
$\\{\mathfrak{e}_{\tau}\geq U/\alpha\\}$ is not adapted to $\mathbb{G}$ or
$\mathbb{F}$, the usual equivalence between random time validity and stopping
time validity (w.r.t. $\mathbb{G}$ or $\mathbb{F}$) no longer holds.

Given that $\mathfrak{e}$ is an e-process w.r.t. $\mathbb{G}$, we have
$\mathbb{E}_{P}[\mathfrak{e}_{\tau}]\leq 1$. Thus, Markov’s inequality implies
that

$P\left(\mathfrak{e}_{\tau}\geq\frac{\epsilon}{\alpha}\right)\leq\frac{\alpha}{\epsilon},\quad\forall
P\in\mathcal{P},$ (50)

for any fixed $\epsilon\in(\alpha,1)$. (The bound is vacuous when
$\epsilon\leq\alpha$.)

By Ramdas et al., (2020, lemma 2), specifically the (iii) $\Rightarrow$ (ii)
direction, the statement then holds for all random times $T$ (possibly
infinite): for any fixed $\epsilon\in(\alpha,1)$,

$P\left(\mathfrak{e}_{T}\geq\frac{\epsilon}{\alpha}\right)\leq\frac{\alpha}{\epsilon},\quad\forall
P\in\mathcal{P}.$ (51)

(The random time $T$ may be, e.g., a stopping time w.r.t. any filtration.)

Then, for any $\mathbb{F}$-stopping time $\tau$ and any $P\in\mathcal{P}$, we
have

$\displaystyle P\left(\mathfrak{e}_{\tau}\geq\frac{U}{\alpha}\right)$
$\displaystyle=\int_{0}^{1}P\left(\mathfrak{e}_{\tau}\geq\frac{u}{\alpha}\right)du$
(52) $\displaystyle\leq\int_{0}^{1}\left(\frac{\alpha}{u}\wedge 1\right)du$
(53)
$\displaystyle=\int_{0}^{\alpha}du+\int_{\alpha}^{1}\frac{\alpha}{u}du=\alpha+\alpha\log(1/\alpha).$
(54)

Note that we used the independence of $U$ from $\mathbb{F}$ in (52). The
inequality in (53) follows from (51).