# Knowledge from Probability

Jeremy Goodman School of Philosophy
University of Southern California, USA<EMAIL_ADDRESS>Faculty of
Philosophy
University of Oxford, UK Bernhard Salow Faculty of Philosophy
University of Oxford, UK<EMAIL_ADDRESS>
###### Abstract

We give a probabilistic analysis of inductive knowledge and belief and explore
its predictions concerning knowledge about the future, about laws of nature,
and about the values of inexactly measured quantities. The analysis combines a
theory of knowledge and belief formulated in terms of relations of comparative
normality with a probabilistic reduction of those relations. It predicts that
only highly probable propositions are believed, and that many widely held
principles of belief-revision fail.

How can we have knowledge that goes beyond what we have observed – knowledge
about the future, or about lawful regularities, or about the distal causes of
the readings of our scientific instruments? Many philosophers think we can’t.
Nelson Goodman, for example, disparagingly writes that “obviously the genuine
problem [of induction] cannot be one of attaining unattainable knowledge or of
accounting for knowledge that we do not in fact have” [21, p. 62]. Such
philosophers typically hold that the best we can do when it comes to inductive
hypotheses is to assign them high probabilities. Here we argue that such
pessimism is misplaced. We give a purely probabilistic analysis of inductive
knowledge and (rational) belief, and motivate it by drawing out its attractive
predictions about a range of cases.

Our analysis builds on two recent strands of research. The first strand is the
idea that knowledge and (rational) belief can be analyzed in terms of a notion
of _normality_ : among the possibilities that are compatible with an agent’s
evidence, their knowledge rules out those that are sufficiently less normal
than their actual circumstances, and their beliefs rule out those that are
sufficiently less normal than some other evidential possibilities.111 See [45,
46, 47, 18, 22, 12, 20, 19, 7, 34, 5, 8, 35, 17] for related ideas about
knowledge, [40, 41, 42, 43, 44] for related ideas about justified belief, and
[28, 36] and references therein for related ideas about non-monotonic
reasoning. The second strand is that what a person knows or believes is always
relative to a contextually supplied question.222See [39, 56, 26] for
precedents in the case of knowledge, and [30, 57, 25, 6] for precedents in the
case of belief. Our guiding observation is this: _there is a natural way to
define the comparative normality of evidential possibilities in terms of the
probabilities of the answers to a question_. This fact allows us to give an
analysis of knowledge and belief in terms of probability and evidence, two
notions that even skeptics about inductive knowledge typically accept.

Here is our plan. We begin by presenting a version of the theory of knowledge
and belief in terms of comparative normality that we have defended elsewhere
[20, 19]. We next explain how comparative normality can be reduced to
evidential probability in a question-relative way. We then use this framework
to model knowledge and belief about ongoing chancy processes (section 3),
lawful regularities (section 4), multiple independent subject matters (section
5), and the values of quantities measured using instruments that are subject
to random noise (section 6).333See [11, 3, 4, 42, 43, 20, 19] for recent
discussion of the kind of cases in $\S\S$3-5, and [51, 52, 54, 55, 9, 18, 48,
46, 7, 13, 8, 19] for recent discussion of the kind of cases in $\S 6$. As
these case studies will illustrate, the framework is conservative in its
synchronic predictions but revisionist in its diachonronic ones: what an agent
knows and believes is closed under entailment and always has a high
probability of being true, but getting new evidence can lead to changes in
what one believes that violate widely endorsed principles about belief-
revision. Three appendices explain ways in which the framework can be extended
in order to model a wider range of cases of inductive knowledge.

##  1 The Normality Framework

Our models of knowledge and belief will be a version of Hintikka semantics for
a single agent.444We can model the knowledge/beliefs of $n$ agents by
generalizing clause 2 of the definition of a normality structure so
$\mathcal{E}\subseteq\mathcal{P}(S)^{n}$ (giving the possible patterns of
bodies of evidence among the agents) and modify the remaining definitions
accordingly. Both knowledge and belief are given by accessibility relations,
in the sense that an agent knows/believes $p$ in a world $w$ iff $p$ is true
in all worlds epistemically/doxastically accessible from $w$. What is
distinctive of the framework is how these accessibility relations are defined
in terms of other relations between worlds, encoding the agent’s evidence and
worlds’ comparative normality.

Let a _normality structure_ be a tuple $\langle
S,\mathcal{E},W,\succcurlyeq,\ggcurly\rangle$ such that:

  1. 1.

$S$ is a non-empty set (of _states_),

  2. 2.

$\mathcal{E}\subseteq\mathcal{P}(S)\backslash\\{\emptyset\\}$ (the _possible
bodies of evidence_)

  3. 3.

$W=\\{\langle s,E\rangle:s\in E\in\mathcal{E}\\}$ (the set of _(centered)
worlds_),

  4. 4.

$\succcurlyeq$ is a preorder on $W$ (read ‘$w\succcurlyeq v$’ as ‘$w$ is _at
least as normal as_ $v$’),

  5. 5.

is a well-founded relation on $W$ (read ‘$w\ggcurly v$’ as ‘$w$ is
_sufficiently more normal than_ $v$’), such that, for any worlds
$w_{1},w_{2},w_{3},w_{4}$:

    1. (a)

If $w_{1}\ggcurly w_{2}$, then $w_{1}\succcurlyeq w_{2}$;

    2. (b)

If $w_{1}\succcurlyeq w_{2}\ggcurly w_{3}\succcurlyeq w_{4}$, then
$w_{1}\ggcurly w_{4}$.

The intuitive idea behind modeling worlds as state/set-of-state pairs is that
we are only modelling the agent’s knowledge and beliefs about a certain
subject matter – the state of the world – and for this purpose we may idealize
and treat worlds as individuated by the state of the world together with the
agent’s evidence about the state of the world, modelled as the set of states
compatible with their evidence. As we will understand it, a person’s evidence
is a subset of their knowledge, and hence is true; this is why in any world
the actual state is a member of the set of states compatible with the agent’s
evidence. We use $R_{e}$ to denote the function mapping each world to the set
of worlds that are _evidentially accessible_ from it, in the sense of being
compatible with the agent’s evidence:

  * $R_{e}(\langle s,E\rangle):=\\{\langle s^{\prime},E\rangle:s^{\prime}\in E\\}$

Next, we define a function $R_{b}$ for _doxastic accessibility_ ,
characterizing the set of worlds compatible with what the agent believes in
any given world. The idea is that what the agent believes goes beyond what is
entailed by their evidence: the doxastic possibilities are those evidential
possibilities that are not sufficiently less normal than any other. Formally,

  * $R_{b}(w):=\\{v\in R_{e}(w):\neg(\exists u\in R_{e}(w):u\ggcurly v)\\}$

Finally, we define a function $R_{k}$ for _epistemic accessibility_ ,
characterizing the set of worlds compatible with what the agent knows in any
given world. There are two natural definitions here: one for those who follow
Stalnaker [45, 46, 47] in thinking that epistemic accessibility is a
transitive relation (in which case knowing $p$ entails knowing that you know
$p$), and another for those who follow Williamson [49, 51, 53, 54] in thinking
that epistemic accessibility cannot be transitive because knowledge requires a
margin for error. The difference concerns whether worlds that are doxastically
inaccessible should be epistemically accessible when they are less normal but
not sufficiently less normal than actuality: the Stalnakerian answers “no”
(the agent knows those worlds don’t obtain), while the Williamsonian answers
“yes” (for all the agent knows, those worlds obtain). Formally, these answers
correspond to the following respective definitions:

  * $R_{k_{\text{}}}(w):=R_{b}(w)\cup\\{v\in R_{e}(w):v\succcurlyeq w\\}$ (Stalnakerian)

  * $R_{k}(w):=R_{b}(w)\cup\\{v\in R_{e}(w):v\succcurlyeq w\\}$ $\cup$ $\overbrace{\\{v\in R_{e}(w):w\succcurlyeq v\wedge\neg(w\ggcurly v)\\}}^{\text{the margin for error}}$ (Williamsonian)

The results we will be exploring in this paper don’t depend on which
definition of knowledge we adopt.555Note that the Williamsonian definition is
equivalent to the much simpler definition $R_{k}(w):=\\{v\in
R_{e}(w):\neg(w\ggcurly v)\\}$ in normality structures where $\succcurlyeq$ is
a _total_ preorder on $E$ for all $E\in\mathcal{E}$, which includes all
normality structures generated from probability structures in the way
described below; see [19] (which considers a slightly more complicated
Williamsonian definition to ensure that the set of epistemically accessible
evidential possibilities is closed under $\succcurlyeq$). Although we will
remain neutral on which definition is preferable by focusing mainly on belief,
we believe that the normality framework is recommended in large part by its
ability to integrate an anti-skeptical theory of knowledge with a non-trivial
theory of inductive belief; we make this case at greater length in [19].

The framework also allows us to model the dynamics of knowledge and belief
_about the state of the world_ in response to new evidence about the state of
the world. To make this idea precise, we first introduce the projection
functions $\pi_{i}$ such that $\pi_{i}(\langle
x_{1},\dots,x_{n}\rangle)=x_{i}$. We then define which states are accessible
from a world $w$ as $\mathcal{R}_{*}(w)=\\{\pi_{1}(v):v\in R_{*}(w)\\}$ where
$*\in\\{e,b,k\\}$. (So, e.g., $\mathcal{R}_{e}(w)=\pi_{2}(w)$.) For any set of
states $p$ and pair of worlds $w$ and $v$, we say that $v$ _is the result of
discovering $p$ in_ $w$ iff $\pi_{1}(v)=\pi_{1}(w)$ and
$\pi_{2}(v)=p\cap\pi_{2}(w)$. Although $R_{b}(w)\cap R_{b}(v)=\emptyset$
whenever $v$ is the result of (non-trivially) discovering $p$ in $w$ (non-
trivially in the sense that $w\neq v$), the dynamics relating
$\mathcal{R}_{b}(w)$ and $\mathcal{R}_{b}(v)$ (and $\mathcal{R}_{k}(w)$ and
$\mathcal{R}_{k}(v)$) are more interesting, as we will explore below. Note
that, unlike standard models of belief-revision, there may be no $v$ that is
the result of discovering $p$ in $w$ – for example, if $p$ is incompatible
with the state of the world in $w$. And since only truths can be discovered,
normality structures allow us to easily model iterated discoveries (in
contrast to formally similar models of theories of belief-revision like AGM
[2], first developed in [23], which do not handle iterated belief-
revision).666Normality structures are also related to models in dynamic
epistemic logic, since instead of modeling accessibility as a relation between
worlds, we could equivalently treat it as family of relations between states
indexed by bodies of evidence, much like how accessibility is relativized to
propositions in dynamic epistemic logic. Another formal precedent is [10], in
which formulas are evaluated relative to a pair of a world and a set of worlds
containing it. (Thanks to Aybüke Özgün for drawing our attention to this
work.) Note that what “world” the agent is in changes as they get new
evidence; those who prefer to reserve the word “world” for something
unchanging can substitute “situation”, “case”, or “centered world”.,777The
framework presented here sides with [19] over [20] by treating normality
relations as holding between worlds rather than states. But it sides with [20]
by presupposing that worlds can be factored into state/evidence pairs, which
in turn implies that evidential accessibility is an equivalence relation.
Appendix A explains how to modify the definitions below in order to model
scenarios in which evidential accessibility is not an equivalence relation.

##  2 Reducing normality to probability

Appealing to notions of comparative normality (or comparative plausibility)
is, by now, a familiar idea in theorizing about knowledge and belief. The main
advance of this paper is to explore the consequences of an analysis of these
notions in terms of the result of conditioning a prior probability
distribution on the agent’s evidence.

Let us begin with the at-least-as-normal relation $\succcurlyeq$. An initially
attractive idea is that $w\succcurlyeq v$ iff $w$ is at least as probable as
$v$. Unfortunately, this simple proposal faces a number of problems. For
example, the probability of a world depends on how finely we individuate
worlds in our model in ways that intuitively shouldn’t make a difference to
what an agent knows or believes; also, natural ways of individuating worlds
often make them all have the same probability, thereby trivializing inductive
knowledge and belief. For these reasons, as well as others explained in
section 5, we will model knowledge and belief as relative to a contextually
supplied question about the state of the world. Doing so allows for a more
robust characterization of normality in terms of probability, as we will now
explain.

Let a _probability structure_ be a tuple $\langle
S,\mathcal{E},W,Q,P,t\rangle$ such that:

  1. 1.

$S,\mathcal{E},W$ satisfy clauses 1-3 of the definition of normality
structures,

  2. 2.

$Q$ (the _question_) is a partition of $S$,

  3. 3.

$P$ (the _prior_) is a probability distribution over $S$ such that $P(q|E)$ is
defined for all $q\in Q$ and $E\in\mathcal{E}$,

  4. 4.

$t\in[0,1]$ (the _threshold_).

We will now explain how to generate a normality structure from a probability
structure. We identify the normality of a world with the evidential
probability at that world of the true answer to $Q$ at that world. Formally,
letting $[s]_{Q}$ (the _answer to_ $Q$ in $s$) be the cell of $Q$ containing
$s$, $P_{w}$ (the _evidential probability_ at $w$) be $P(\cdot|\pi_{2}(w))$,
and $\lambda(w)$ (the _likeliness_ of $w$) be $P_{w}([\pi_{1}(w)]_{Q})$, we
adopt the following definition of one world being at least as normal as
another:

  * normality as likeliness: $w\succcurlyeq v:=\lambda(w)\geq\lambda(v)$ and $v\in R_{e}(w)$

Next, let the _typicality_ of a world be the evidential probability, at that
world, that things are no more normal than they are at that world: formally,
$\tau(w)=P_{w}(\\{\pi_{1}(v):w\succcurlyeq v$ and $v\in R_{e}(w)\\})$. We will
adopt the following definition of one world being sufficiently more normal
than another:

  * sufficiency: $w\ggcurly v:=1-\frac{\tau(v)}{\tau(w)}\geq t$ and $v\in R_{e}(w)$

It is easy to verify that, so defined, $\langle
S,\mathcal{E},W,\succcurlyeq,\ggcurly\rangle$ is a normality structure.888 The
requirement that $\succcurlyeq$ and only relate evidentially accessible worlds
is needed to validate conditions 5a and 5b of the definition of a normality
structure, since worlds with the same likeliness but different evidence can
have different typicality. For example, let
$S=\\{1,\dots,7\\},\mathcal{E}=\\{\\{1,2,3\\},\\{4,5,6,7\\}\\},Q=\\{\\{s\\}:s\in
S\\},P(1)=P(2)=P(4)=.2;P(3)=P(5)=P(6)=P(7)=.1;t=.5$: Let $w=\langle
3,\\{1,2,3\\}\rangle$ and $v=\langle 5,\\{4,5,6,7\\}\rangle$.
$\lambda(w)=\lambda(v)=.2$, but $\tau(w)=.2\neq\tau(v)=.6$. In this normality
structure, what the agent believes about the state of the world is the
strongest disjunction of answers to $Q$ that (i) includes the most probable
answers, (ii) includes all answers at least as probable as any it includes,
and (iii) has total probability at least $t$.999Compare the theory of belief
in [25], which implies (i) and a slight weakening of (ii) (with “more
probable” in place of “at least as probable”), but not (iii). Rather than
seeing this view as a competitor to ours, we prefer to see it as concerned
with a weaker notion of belief that is less closely tied to knowledge. [27]
has independently developed a theory of knowledge and belief very close to the
one presented here, which he applies to the preface and St. Petersburg
paradoxes. (This will also be what the agent knows about the state of the
world if one of the most probable answers to $Q$ is true.) normality as
likeliness ensures (i) and (ii), while sufficiency ensures (iii), which may be
more precisely stated as follows:101010sufficiency does make some somewhat
surprising predictions. Suppose Alice holds 999 tickets in a fair lottery and
each of the million other entrants holds 1000 tickets. Assume $P$ gives each
ticket an equal probability of being chosen given the setup, that the question
$Q$ is _who will win_ , and the threshold $t=.99$, and consider worlds in
which the agent’s evidence is that this is the setup. If Alice will lose, can
the agent know this? sufficiency predicts they can, which seems odd. (This
might not seem so odd at first, since we don’t deny the agent can know this
relative to a different question $Q^{\prime}$ = _how many tickets does the
winner have_. To bring out the oddity, we can modify the example by adding
another loser Bob who holds 1001 tickets; it seems odd that the agent would
know that Alice will lose but not that Bob will lose.) To avoid this
prediction, we might add a requirement that a world is sufficiently more
normal than another only if it is also sufficiently more _likely_. (Compare
the discussion of having no “ _appreciably_ stronger reason” (emphasis ours)
to believe one entrant to a lottery will win compared to any other in [24,
p.16].) We might implement this idea by adopting the stronger principle:
sufficiency+: $w\ggcurly v:=1-\frac{\tau(v)}{\tau(w)}\geq t$ and $v\in
R_{e}(w)$ and $1-\frac{\lambda(v)}{\lambda(w)}\geq t$ This definition also
determines a normality structure; while we actually find it more attractive
than sufficiency, we will ignore it in the main text to simplify our
presentation.

  * threshold: $P_{w}(\mathcal{R}_{b}(w))\geq t$ for all $w\in W$.

##  3 Inductive knowledge about the future

In this section and the next we’ll review two cases of inductive knowledge
involving coin flips from Dorr, Goodman and Hawthorne [11] to help illustrate
the framework. The first case illustrates the possibility of inductive
knowledge about the future:

> Flipping for Heads: A coin flipper will flip a fair coin until it lands
> heads. Then he will flip no more.

Assume that the agent’s evidence entails that this is the setup, and that they
are watching the experiment unfold. We can model the case using the
probability structure in which $S=\\{1,2,\dots\\}$,
$\mathcal{E}=\\{\\{n,n+1,n+2,\dots\\}:n\in S\\}$, $P$ is the probability
function such that $P(\\{n\\})=2^{-n}$ for all $n$, $Q=\\{\\{s\\}:s\in S\\}$,
and $t=.99$. Intuitively, $n$ is the state in which the coin lands heads on
the $n$th flip, the possible bodies of evidence are those compatible with
having watched some initial sequence of zero or more flips that all landed
tails, and $Q$ is the question _on what flip does the coin land heads_. In the
normality structure generated from this probability structure, $\langle
n,E\rangle\succcurlyeq\langle m,E\rangle$ iff $n\geq m$, and $\langle
n,E\rangle\ggcurly\langle m,E\rangle$ iff $n\geq m+7$.111111$\tau(\langle
n,E\rangle)=2^{1-n}$, and $1-2^{-6}<.99<1-2^{-7}$, so $1-\frac{\tau(\langle
m,E\rangle)}{\tau(\langle n,E\rangle)}\geq.99$ iff $n\geq m+7$. So if the
agent has seen the coin land tails $x$ times, what they believe is that it
will land heads within the next 7 trials – i.e., on trials $x+1$ to $x+7$. The
predictions about knowledge match those of [11], on the Williamsonian
definition of epistemic accessibility, and those of [20], on the Stalnakerian
definition.

Notice that this model involves a kind of non-monotonic belief revision (which
is prohibited by AGM). Let $w=\langle 2,\\{1,2,\dots\\}\rangle$, $v=\langle
2,\\{2,3,\dots\\}\rangle$, and $p=\\{2,3,\dots\\}$. So $v$ is the result of
discovering $p$ in $w$. At the start of the experiment, the agent is in $w$;
after the first trial, the coin lands tails and the agent is in $v$ (and,
unbeknownst to them, the coin is about to land heads).
$\mathcal{R}_{b}(w)=\\{1,\dots,7\\}$, which is compatible with $p$, yet
$\mathcal{R}_{b}(w)\cap p\neq\mathcal{R}_{b}(v)=\\{2,\dots,8\\}$. Discovering
something compatible with their prior beliefs leads the agent to give up some
those beliefs. We think this is exactly the right prediction.121212Even more
surprising behavior is possible if we modify the case by allowing the agent to
get partial information about the result of the experiment after it is over,
so that $\\{1,\dots,7\\}\in\mathcal{E}$. Consider $w=\langle
1,\\{1,2,\dots\\}\rangle$, $v=\langle 1,\\{1,\dots,7\\}\rangle$, and
$p=\\{1,\dots,7\\}$. $\mathcal{R}_{k}(w)=\mathcal{R}_{b}(w)=p$ and $v$ is the
result of discovering $p$ in $w$, yet
$\mathcal{R}_{k}(v)=\mathcal{R}_{b}(v)=\\{1,\dots,6\\}$: you can gain new
beliefs and knowledge by discovering (i.e., gaining _evidential_ knowledge of)
something you already (inductively) knew.

##  4 Inductive knowledge of laws

The second case from [11] provides a simple model of inductive knowledge of
lawful regularities:

> Heading for Heads: You know a bag contains two coins: one fair, one double-
> headed. Without looking, you reach in and select a coin. You decide to flip
> it 100 times and observe how it lands.

If inductive knowledge of lawful regularities is ever possible, it should be
possible here: that is, you can learn that the coin is double headed by seeing
it land heads 100 times in a row. Our framework predicts this. Consider the
probability structure with $2^{100}+1$ states, encoding the pattern of heads
and tails and whether the coin is fair or double-headed. Let $c$ be the state
where the coin is fair but lands heads every time by coincidence, and $d$ be
the state where it is double-headed. For every state there are two worlds,
corresponding to your evidence before and after flipping, so
$\mathcal{E}=\\{S\\}\cup\\{c,d\\}\cup\\{\\{s\\}:s\in S\backslash\\{c,d\\}\\}$.
$P(\\{d\\})=.5$ and $P(\\{s\\})=.5^{101}$ for $s\in S\backslash\\{d\\}$.
$Q=\\{\\{s\\}:s\in S\\}$ and $t=.9999999$. As desired, this structure predicts
that, after seeing the coin land heads 100 times, you can know it is double
headed.131313The theory similarly predicts that one can come to know how many
sides of a die are painted red by observing the outcomes of a sufficient
number of rolls of the die. It thus explains not only knowledge of law-like
generalizations, but also knowledge of the objective chances associated with
different physical processes.

Notice that, although $\langle c,\\{c,d\\}\rangle\not\in R_{k}(\langle
d,\\{c,d\\}\rangle)$, nevertheless $\langle c,S\rangle\in R_{k}(\langle
d,S\rangle)$.141414$1-\frac{\tau(\langle c,\\{c,d\\}\rangle)}{\tau(\langle
d,\\{c,d\\}\rangle)}=\frac{2^{100}}{2^{100}+1}\geq.9999999$; but
$1-\frac{\tau(\langle c,S\rangle)}{\tau(\langle d,S\rangle)}=.5<.9999999$. In
other words: although you know the coin is double-headed after seeing it land
heads every time, nevertheless, before flipping it, for all you knew the coin
was fair and about to land heads every time by coincidence. This fact
highlights a notable feature of the present framework that departs from our
models in [20]: the comparative normality of two worlds is not a function only
of those worlds’ underlying states, but also depends on the agent’s evidence.
(This is perhaps more intuitive if we gloss $\succcurlyeq$ and as relations of
comparative _plausibility_ : what hypotheses about the state of the world are
more or less plausible depends on what your evidence about the state of the
world is.) The case also illustrates some more extreme departures from AGM-
style dynamics of knowledge and belief, since discovering $p$ (that the coin
landed heads every time) can allow one to know $q$ (that it isn’t fair) even
if, prior to the discovery, $p\wedge\neg q$ was an epistemic possibility.
While such behavior is unfamiliar (and claimed in [11] to be impossible), we
again submit that in this case it is a plausible prediction.151515Note that
the fact that, for all you know at the outset, the coin is fair and will land
heads every time by coincidence, depends on the choice of question. Consider
instead $Q^{\prime}=$ is the coin fair and how many times will it land heads.
Relative to this question, you do know at the outset that, if the coin is
fair, it won’t land heads every time. This is because the change in question
from $Q$ to $Q^{\prime}$ changes the typicality of $\langle c,S\rangle$ from
$.5$ to $.5^{-100}$ – this is because fair-and-all-heads is less normal than
all other states relative to $Q^{\prime}$ except for fair-and-all-tails (which
is equally normal).

##  5 Inductive knowledge about multiple subject matters

Theories of inductive knowledge that accept threshold face a well-known
challenge. Assuming inductive knowledge is possible at all, we should be able
to know propositions whose evidential probability is less than 1. Now consider
many independent subject matters about which we have such inductive knowledge,
and the conjunction of everything we know concerning any one of these subject
matters. If knowledge is closed under conjunction, as Hintikka semantics
predicts, it follows that we know this conjunction. But if we are pooling
knowledge across enough independent subject matters, then this conjunction is
liable to have low evidential probability, even if the evidential probaility
of each of its conjuncts is high, thereby violating threshold.

Our proposal responds to this challenge by maintaining that “know” (and
“believe”) are context-sensitive, with the question $Q$ being the relevant
parameter of context-sensitivity. For any given question, knowledge relative
to that question is closed under conjunction. But there need be no single
question relative to which an agent knows every proposition that they know
relative to some question or other, and hence no question relative to which
they know the conjunction of all such propositions. We will illustrate this
aspect of the proposal using the following case we discuss in [20]:

> Racing for Heads: Each of $n$ coin flippers has a fair coin. Each will flip
> their coin until it lands heads.

If what the agent knows/believe about each coin flipper is like what they
know/believe about the single coin flipper in Flipping for Heads, then as $n$
grows the totality of what they know/believe can have arbitrarily low
probability, violating threshold.161616[50] defends the existence of such
threshold violations; [20] shows how they can be modeled using normality
structures. We will now explain our preferred treatment of the case, in terms
of probability structures. We will show how its predictions about the agent’s
knowledge and beliefs depend on the contextually supplied question.

For concreteness, we will investigate the version of the case with 10 coin
flippers. The probability structure is the one defined in the obvious way like
in Flipping for Heads, with states modeled as sequences of 10 positive
integers, indicating how many trials it will take for each of the 10 coins to
land heads, and $P$ corresponding to the chances of various outcomes prior to
the experiment.

What about $Q$? In reflecting on Racing for Heads a number of natural
questions suggest themselves. Some of these are the $10$ different questions
of the form _how many times will this particular coin land heads_. Relative to
any such question, your knowledge is exactly like that in Flipping for Heads
with respect to this particular coin, and trivial concerning every other coin.
But other natural questions include (i) what will the exact outcome of the
whole experiment be; (ii) what will the shape of the outcome be – that is, how
many coins will take how long to land (the exact outcome up to isomorphism);
(iii) how many total tails will there be in the experiment as a whole; (iv)
how long will it be before all the coins have landed heads; and (v) how many
of the coins will ever land heads at the same time. For each of these
question, we can ask what you know and believe about a number of issues, such
as how many total tails there will be, how long the experiment as a whole will
last, and whether all the experiments will end on the same flip (a claim
labelled ‘same end’ below). The table below records what the agent believes at
the start of the experiment for different choices of $Q$, for thresholds
$t=.75$ and $t=.95$. This will also be what the agent knows in the most normal
worlds.

$Q$ |  which worlds are most normal |  $t$ |  min tails |  max tails |  min trials |  max trials |  same end?
---|---|---|---|---|---|---|---
(i) exact outcome |  all coins land heads first time |  .75 |  0 |  13 |  1 |  14 |  maybe
.95 |  0 |  18 |  1 |  19 |  maybe
(ii) outcome shape |  $6\times 1$ flip, $3\times 2$ flips, $1\times 3$ flips |  .75 |  1 |  15 |  2 |  8 |  no
.95 |  0 |  22 |  1 |  12 |  maybe
(iii) how many total tails |  8 or 9 total tails [tied] |  .75 |  5 |  14 |  1 |  15 |  maybe
.95 |  2 |  18 |  1 |  19 |  maybe
(iv) how long until over |  ends on 4${}^{\text{th}}$ trial |  .75 |  2 |  50 |  3 |  6 |  maybe
.95 |  1 |  70 |  2 |  8 |  maybe
(v) how many end together |  5 flippers get heads at once |  .75 |  3 |  $\infty$ |  2 |  $\infty$ |  no
.95 |  2 |  $\infty$ |  2 |  $\infty$ |  no

The table illustrates a general difference between relatively fine-grained
questions (such as exact outcome or outcome shape) and more coarse-grained
ones (such as how many total tails, how long until over, and how many end
together). When the topic of our knowledge aligns with a coarse-grained
question, we will generally know more relative to that question than we know
relative to a more fine-grained question: for example, we know more about how
many tails there will be relative to how many total tails than we do relative
to exact outcome or outcome shape. But this increase in knowledge comes at a
cost, since relative to a coarse-grained question we will know very little
about the many topics that are orthogonal to that question: for example, we
know little about how many tails there will be relative to how long until over
or how many end together. These two facts share a common explanation. By
treating all worlds that agree on the answer to a coarse-grained question as
equally normal, we make it easier to exceed $t$ as we add probabilities along
the normality order while staying within a relatively restricted class of
answers to that question, thus generating a lot of knowledge about that
question. But in doing this, we will be including some worlds amongst the
relatively normal ones in which things unfold in the least probable way they
might with regard to some orthogonal subject matter.

##  6 Inductive knowledge from instrument readings

We often learn about the values of continuous quantities like weight and
temperature by measuring them using less than perfectly reliable scales,
thermometers, and so on. Modeling such knowledge requires two generalizations
of the present framework. One concerns cases where there are a continuum of
possible bodies of evidence, so $P(E)=0$ for some $E\in\mathcal{E}$. To handle
such cases, we relax the requirement that evidential probabilities are always
the result of conditioning a prior probability distribution on your evidence.
Instead, we directly associate every $E\in\mathcal{E}$ with a probability
distribution $P_{E}$ over $E$. A second problem concerns cases where $Q$ has a
continuum of answers all of which have evidential probability 0, yet we want
to allow for non-trivial inductive knowledge. Here the natural solution is to
generalize the operative notion of probability to _probability density_ ; see
appendix B for the technical details. Our example in this section will
illustrate both of these issues. Again, we will show that the framework allows
us to derive (from purely probabilistic considerations) models of agents’
knowledge and beliefs that have been defended in the literature on independent
grounds.

Consider the kind of case made famous by Williamson [54]. You are going to
glance at an unmarked modernist clock, with only an hour hand.
$S=[0,2\pi)\times[0,2\pi)$, where $\langle x,y\rangle$ is a state in which the
hand’s orientation is $x$ (so that, e.g., $\frac{\pi}{2}$ represents
3-o-clock) and its apparent orientation when you look at it is $y$. We assume
that, before looking at the clock, you have no idea how it will look or what
time it is; after looking at the clock, your evidence is exhausted by how it
appeared. That is, $\mathcal{E}=\\{S\\}\cup\\{\\{s\in
S:\pi_{2}(s)=y\\}:y\in[0,2\pi)\\}$, and $P_{S}$ is uniform concerning both the
hand’s real and apparent orientations, in the sense that, for any interval
$I=[a,b]\subseteq[0,2\pi)$, $P_{S}(\\{s:\pi_{1}(s)\in
I\\})=P_{S}(\\{s:\pi_{2}(s)\in I\\})=\frac{b-a}{2\pi}$. Since your evidence
after seeing the clock (i.e., that the apparent orientation was $y$) had prior
probability 0, your new evidential probabilities cannot be given by
conditioning your prior evidential probabilities on your new discovery.
Moreover, if the question is how the hand is oriented – i.e., if $Q=\\{\\{s\in
S:\pi_{1}(s)=x\\}:x\in[0,2\pi)\\}$ – then, since your eyesight is imperfect,
each of its answers will still have probability 0 after looking at the clock.
The case thereby illustrates both issues described in the last paragraph.

To allow for non-trivial inductive knowledge concerning the position of the
hand, we must modify normality as likeliness so that evidentially accessible
worlds can differ in normality after looking at the clock. Fortunately, there
is a natural way to do this. The key observation is that, after looking, not
all _intervals_ of orientations are on a par – their probabilities are no
longer given merely by their length. Their probabilities are given instead by
a non-uniform _probability density_ function, a “bell curve” centered on the
apparent orientation $y$. The area under this curve between two points gives
the probability that the hand’s true orientation is in that interval. This
fact suggests modifying the definition of $\succcurlyeq$ as follows: rather
than ordering worlds according to the respective _probabilities_ of their
answers to $Q$, we can instead order them by the respective _probability
densities_ (i.e., heights of the “bell curve”) of their answers to $Q$. The
formal details are given in appendix B.

The normality structures generated in this way determine epistemic and
doxastic accessibility relations of the same kind that have been defended in
the literature. The agent’s beliefs about the hand’s orientation will be
characterized by a non-trivial interval centered on its apparent orientation.
Their knowledge will also be characterized by such an interval, in a way that
may or may not always leave a “margin for error”, depending on whether we
adopt a Williamsonian or a Stalnakerian definition of epistemic
accessibility.171717Williamson [54] and Stalnaker [46] respectively defend
such models of our knowledge about the unmarked clock. In [19], we discuss how
other models in the literature arise from various combinations of definitions
of epistemic accessibility and claims about the comparative normality of the
relevant possibilities. We believe that the present framework is strongly
recommended by its ability to vindicate natural and anti-skeptical models of
our knowledge in cases of this kind.

##  7 Conclusion

In this paper we have offered a new framework for modeling inductive knowledge
and (rational) belief using resources congenial to philosophers in the
Bayesian tradition. We did so by showing how the relations of comparative
normality that have recently been used to model knowledge and belief can
themselves be analyzed in probabilistic terms. The framework offers a unified
account of our knowledge about chancy processes (section 3), lawful
regularities (section 4), and imprecisely measured quantities (section 6). By
positing a certain kind of context-sensitivity in “know” and “believe”, it
offers a way of avoiding inductive skepticism while maintaining that only
highly probable propositions are ever known or rationally believed (section
5). An urgent question for further research is how the contextually-supplied
question that features in the probabilistic analysis of normality is
determined.181818Note that relative to the question _is it true that $p$_,
knowing that $p$ requires only that $p$ is true and has a high enough
evidential probability. This might be considered objectionable for familiar
reasons to do with Gettier cases [14]. If so, that would be one reason to deny
that such questions are supplied by any context – although see [26] for
arguments that there are contexts in which “knowledge” is this easy to come
by.

## Acknowledgements

Thanks to Cian Dorr and three anonymous referees for TARK for comments on a
draft of this material, and to Kevin Dorst, John Hawthorne, Ben Holguín, and
Harvey Lederman for very helpful discussion. Special thanks to Dorst for help
with the coding required to calculate the values in the table in section 5.

##  Appendix A Primitive evidential accessibility

Rather than modeling worlds as state/set-of-state pairs, we could treat them
as unstructured points, and explicitly specify an evidential accessibility
relation on them. This no longer allows us to model the notion of discovery,
but it avoids the presupposition that there is a principled way of factoring
worlds into state/set-of-state pairs. Moreover, it allows us to model
evidential accessibility as an arbitrary reflexive relation. To handle cases
where evidential accessibility is not an equivalence relation, we will need to
relativize relations of comparative normality to a reference world – the world
whose evidential probabilities are being used to assess the comparative
normality of two other worlds.191919[49] argues that evidential accessibility
is not an equivalence relation; [33] and [47] maintain that it is. [7] and
[35] argue for a kind of world-relativity of (non-comparative) normality; see
[19] for discussion in the context of comparative normality; ordering
semantics for counterfactuals is a formal precedent, _cf._ [31].

Let a _relativized normality structure_ be a tuple $\langle
W,R_{e},\succcurlyeq,\ggcurly\rangle$ such that:

  1. 1.

$W$ is a non-empty set,

  2. 2.

$R_{e}:W\to\mathcal{P}(W)$ such that $w\in R_{e}(w)$ for all $w\in W$,

  3. 3.

For each $w\in W$, $\succcurlyeq_{w}$ is a preorder on $W$,

  4. 4.

For each $w\in W$, w is a well-founded relation on $W$ such that, for any
worlds $w_{1},w_{2},w_{3},w_{4}$:

    1. (a)

If $w_{1}{}_{w}w_{2}$, then $w_{1}\succcurlyeq_{w}w_{2}$;

    2. (b)

If $w_{1}\succcurlyeq_{w}w_{2}{}_{w}w_{3}\succcurlyeq_{w}w_{4}$, then
$w_{1}{}_{w}w_{4}$.

We define $R_{b}(w)$ and $R_{k}(w)$ as in section 1, replacing $\succcurlyeq$/
with $\succcurlyeq_{w}$/w.

Now let a _worldly probability structure_ be a tuple $\langle
W,R_{e},Q,P,t\rangle$ such that:

  1. 1.

$W,R_{e}$ satisfy 1 and 2 in the definition of a relativized normality
structure,

  2. 2.

$Q$ is a partition of $W$,

  3. 3.

$P$ is a probability distribution over $W$ such that $P(q|R_{e}(w))$ is
defined for all $q\in Q$ and $w\in W$,

  4. 4.

$t\in[0,1]$.

Let $\lambda_{w}(v):=P([v]_{Q}|R_{e}(w))$ and
$\tau_{w}(v):=P(\\{u:v\succcurlyeq_{w}u$ and $u\in R_{e}(w)\\}|R_{e}(w))$.
With these world-relative notions of likeliness and typicality in hand, we can
now define $v\succcurlyeq_{w}u:=\lambda_{w}(v)\geq\lambda_{w}(u)$ and
$v{}_{w}u:=1-\frac{\tau_{w}(u)}{\tau_{w}(v)}\geq t$. It is easy to verify that
these definitions yield a relativized normality structure that also obeys
threshold (reformulated with $R$ in place of the now ill-defined
$\mathcal{R})$.

##  Appendix B Probability densities

Let a _density structure_ be a tuple $\langle
S,\mathcal{E},W,P,Q,m,f,t\rangle$ such that:202020It would also be natural to
require that, where possible, $P$ behaves as if it were the result of
conditioning a prior on the agent’s evidence – i.e. that, for all
$E,E^{\prime}\in\mathcal{E}$, if $P_{E}$ is defined on $E\cap E^{\prime}$,
then $P_{E}(\cdot|E\cap E^{\prime})=P_{E^{\prime}}(\cdot|E\cap E^{\prime})$.

  1. 1.

$S$, $\mathcal{E}$, $W$, $Q$, and $t$ are as in a probability structure and
$t>0$,

  2. 2.

For all $E\in\mathcal{E}$:

    1. (a)

$P_{E}$ is a probability distribution over $E$,

    2. (b)

$P_{E}(q)=0$ for all $q\in Q$,

  3. 3.

$m:Q\to\mathbb{R}$ (the _measuring function_),

  4. 4.

$f:\mathcal{E}\to\mathbb{R}^{\mathbb{R}}$ such that $f_{E}$ is the _density of
$P_{E}$ relative to $m$_.

The intuitive idea behind a probability density function is that of a curve
the area under which gives the associated probabilities. So, in particular,
$\int_{a}^{b}f_{E}(x)dx=P_{E}(\bigcup\\{q\in Q:m(q)\in[a,b]\\})$. A formal
characterization of $f$ is given in a footnote; the role of $m$ will be
illustrated in appendix C.212121Formally, $f_{E}$ is _the density of $P_{E}$
with respect to the reference measure $\mu^{\prime}$_ (defined as usual in
terms of the Radon-Nikodym derivative), where
$\mu^{\prime}(p)=\mu\\{m(q):q\subseteq p\\}$ for all $p$ on which $P_{E}$ is
defined, and $\mu$ is the Lebesgue measure on $\mathbb{R}$.

To understand this definition, let us return to the unmarked clock. As
described in section 6, $P$ satisfies clause 2 (where $Q$ is _what is the
hand’s orientation_). Let $m([s]_{Q})=\pi_{1}(s)$ – i.e., it maps answers to
$Q$ (understood as sets of states) to corresponding real numbers in
$[0,2\pi)$. $f_{S}(x)=\frac{1}{2\pi}$, the constant function. By contrast,
$f_{\\{s\in S:\pi_{2}(s)=y\\}}$ – the probability density function determining
your evidential probabilities after discovering that the hand’s apparent
orientation is $y$ – will be a ‘bell curve’ centered on and symmetric around
$y$ (e.g. a ‘wrapped normal distribution’). In cases like this, rather than
generating a normality order via normality as likeliness from a probability
function, we instead generate it from this probability _density_ function. Let
$d(w)$ (the _density_ of $w$) be $f_{\pi_{2}(w)}(m([\pi_{1}(w)]_{Q}))$, and
define being at least as normal as follows:

  * normality as density: $w\succcurlyeq v:=d(w)\geq d(v)$ and $v\in R_{e}(w)$

As before, is defined by sufficiency.222222Requiring that $t>0$ ensures that
is well-founded; for an illustration of why this is needed, see $d^{I}$ in
appendix C.

To see this definition in action, we will consider a modification of the clock
case that allows us to work with familiar Gaussian probability density
functions (also known as ‘normal distributions’). We do so by considering a
continuous quantity whose values are not confined to $[0,2\pi)$, such as the
difference in weight of two objects. Suppose $Q$ is the question _how much
heavier is this apple than this orange_ , and our scale reads $\mu$ grams.
Suppose, as a first approximation, that our evidential probabilities are now
characterized by a Gaussian probability distribution over weight-difference in
grams, with mean $\mu$ and standard deviation $\sigma$. $m$ will be the
function from answers to $Q$ to corresponding real numbers of grams and
$f_{E}(x)=\frac{1}{\sigma\sqrt{2\pi}}e^{-\frac{1}{2}(\frac{x-\mu}{\sigma})^{2}}$,
where our evidence $E=\\{\langle x,\mu\rangle:x\in\mathbb{R}\\}$. (We model
states as ordered pairs of actual and measured values of the quantity, as
before.) By setting $t=.9545\dots$, we predict that
$\mathcal{R}_{b}(\langle\langle x,\mu\rangle,E\rangle)=\\{\langle
x^{\prime},\mu\rangle:|x^{\prime}-\mu|\leq 2\sigma\\}$ for all
$x\in\mathbb{R}$: we will believe that the scale reads $\mu$ grams and that
the true weight-difference is within two standard deviations of that.
Predictions concerning knowledge depend on which of the two clauses for
$R_{k}$ are adopted, but in either case our knowledge will be non-trivial (and
will coincide with what we believe when the scale is perfectly accurate –
i.e., when $x=\mu$). This model (simplified from [18]) shows how, at least in
favorable circumstances, we can see familiar uses of confidence intervals in
inferential statistic as corresponding to non-trivial knowledge and (rational)
beliefs about the values of imprecisely measured quantities.232323In the case
of belief, our models correspond to the “minimum likelihood” method for
generating confidence intervals (which yields the shortest possible intervals
that have probability $t$). The idea of ordering possibilities by probability
density for this purpose goes back to [37]. This procedure has been criticized
in the case of asymmetric distributions; in particular, in the case of
$\chi^{2}$ distributions, where it yields different verdicts from ordinary
$\chi^{2}$ tests [16, 38, 29]. We lack the space to address these criticisms
here, except to note that the Decay example in appendix C illustrates why we
find appealing the distinctive predictions that the minimum likelihood
approach makes in the case of certain asymmetric distributions.

Our view is not that normality relations are _always_ determined by density
structures. Sometimes they are determined by probability structures. It
depends on whether the answers to $Q$ have positive probabilities or only
probability densities. When answers differ in this regard, we advocate a
hybrid approach, with $\succcurlyeq$ determined by $\lambda$ among pairs of
answers one of which has positive probability and by $d$ among pairs of
answers with well-defined probability densities. Another generalization of
density structures is also needed to handle multidimensional probability
densities. In the case of $n$ dimensions, we will then have
$f_{E}:\mathbb{R}^{n}\to\mathbb{R}$. This generalization is needed when the
relevant quantity is multidimensional (e.g., where a dart will land on a
dartboard) and/or when $Q$ concerns the outcomes of multiple independent noisy
measurements of a given quantity (which will be modeled as a vector of real
numbers).

##  Appendix C Normality _de dicto_ and _de se_

In the main text, we modelled a question as a partition of $S$. However, some
natural questions cannot be modelled in this way, because they don’t merely
concern the state of the world. In Flipping for Heads, we might, for example,
wonder how many more times a coin will be flipped – two worlds in which it is
flipped a total of 5 times can differ on the answer to this question, because
in one it is part of your evidence that the coin has already been flipped
(only) 2 times while in another it is part of your evidence that it has been
flipped 3 times. Following [32], we will think of these as de se questions,
concerning not only the history of the world but also your place in it.
Formally, we implement this idea by modeling $Q$ as a partition of $W$, as in
appendix A – recall that members of $W$ should be thought of as _centered_
worlds, two of which can agree on the complete history of the world while
disagreeing on your evidence (because they concern your evidential situation
at different times in that history).

Unlike appendix A, for present purposes we may keep $P$ defined on subsets of
$S$ – this is natural in cases where we want $P$ to conform to the prior
objective chances which are (plausibly) only defined over histories of the
world (i.e., states). Since $P$ is now not defined on answers to $Q$ (these
being sets of worlds), we need to modify the definition of $\lambda$. For
$q\in Q$, we let $q_{E}=\\{\pi_{1}(w):w\in q\text{ and }\pi_{2}(w)=E\\}$; then
$Q_{E}:=\\{q_{E}:q\in Q\\}$ is a partition of $E$. So we can redefine
$\lambda(\langle s,E\rangle)$ as $P([s]_{Q_{E}}|E)$.242424This definition is a
proper generalization of the previous one since, if we start with a partition
$Q^{S}$ of $S$ and use it to generate a partition $Q^{W}$ of $W$ in the
obvious way, defining likeliness as above using $Q^{W}$ yields the same
results as the original definition of likeliness using $Q^{S}$.

We will illustrate this generalization by showing how it allows us to model
the following case (also discussed in [19]):

> Decay: A radioactive atom is created; eventually, it will decay. The average
> time for an atom of this isotope to decay is one year.

Assume that the agent’s evidence entails that this is the set-up, that they
are keeping track of time, and that, when the atom decays, they will be
alerted of this fact. So states can be modeled as positive real numbers,
specifying how many years from its creation it will be before the atom decays,
and possible bodies of evidence (in which the atom has yet to decay) are
intervals $(t,\infty)$ for positive real $t$.

What should the agent believe about how many years after its creation the atom
will decay? A natural thought is that, at every time $t$ before the atom
decays, the agent’s doxastic possibilities will be characterized by the
interval $[t+a,t+b]$, for some $0<a<1<b$. (Compare other events that tends to
happen unexpectedly, like when you will next have an urge to sneeze, or when
you will next get an email from an intermittent correspondent: you think they
won’t happen in the next second, you think they will have happened within a
decade, and the doxastically possible times form an interval, at least to a
first approximation.) To ensure that the shape of this interval is the same at
every time, we model the case using the de se question how long after the
current time will the atom decay. To predict that the agent believes that the
atom will take at least $a$ years to decay, we need to choose the measuring
function $m$ of our density structure to reflect the fact that extremely short
times to decay can be as far from average, in the relevant sense, as extremely
long times to decay.

We will now make these ideas precise using the framework of density structures
from appendix B (generalized to allow _de se_ questions). Let
$S=\mathbb{R}^{+}$, $\mathcal{E}=\\{(t,\infty):t\geq 0\\}$, and let
$P_{(t,\infty)}$ be given by the objective chances at $t$ (which are entailed
by the agent’s evidence at $t$, since it entails the setup and the time).
$Q=\\{q_{r}:r\in\mathbb{R}^{+}\\}$, where $q_{r}=\\{\langle
t^{\prime},(t,\infty)\rangle:t^{\prime}-t=r\\}$. But this does not yet specify
a density structure, since it is compatible with different choices of
measuring functions $m$ and corresponding probability density assignments $f$.

How should we associate answers to $Q$ with real numbers? The simplest choice
is the _index_ function $m^{I}$, where $m^{I}(q_{r})=r$. Given this choice,
the corresponding density $f^{I}_{E}(x)=e^{-x}$ (for all $E\in\mathcal{E}$, so
the agent’s beliefs about how much longer it will be before the atom to decays
don’t change if they see the atom hasn’t decayed yet), from which it follows
that $d^{I}(\langle t^{\prime},(t,\infty)\rangle)=e^{t-t^{\prime}}$, since
$[\langle t^{\prime},(t,\infty)\rangle]_{Q}=q_{t^{\prime}-t}$. Since the
shortest lengths of time until decay correspond to the highest densities, it
is always doxastically possible that the atom will decay arbitrarily soon.

$f^{I}_{(t,\infty)}(x)$

(a) $d^{I}(q_{r})$

$x$

$r$

To avoid this prediction, we must choose a different measuring function. And
there is a natural alternative: the _logarithmic_ measuring function
$m^{\ln}$, where $m^{\ln}(q_{r})=\ln(r)$. This choice corresponds to a
different density function $f^{\ln}_{E}=e^{x}e^{-e^{x}}$ (for all
$E\in\mathcal{E}$), from which it follows that $d^{\ln}(\langle
t^{\prime},(t,\infty)\rangle)$ = $(t^{\prime}-t)e^{t-t^{\prime}}$.252525By
definition, $f_{E}^{\ln}(x)$ is such that
$\displaystyle\int_{-\infty}^{y}f^{\ln}_{E}(x)$
$\displaystyle=P_{E}\Big{(}\bigcup\big{\\{}q_{r}:m^{\ln}(q_{r})\in(-\infty,y)\big{\\}}\Big{)}$
$\displaystyle=\int_{0}^{e^{y}}e^{-x}dx\text{\hskip 14.22636pt (for all
}E\in\mathcal{E})$ $\displaystyle=-e^{-e^{y}}+1$ Differentiating both sides
with respect to $y$ then yields $f_{E}^{\ln}(x)=e^{x}e^{-e^{x}}$. So
$d^{\ln}(\langle
t^{\prime},(t,\infty)\rangle)=f_{(t,\infty)}^{\ln}(m^{\ln}([\langle
t^{\prime},(t,\infty)\rangle]_{Q}))=f_{(t,\infty)}^{\ln}(\ln(t^{\prime}-t))=(t^{\prime}-t)e^{t-t^{\prime}}$.
Intuitively, this choice of measuring function amounts to thinking of
intervals between the present time and the time of decay in terms of the order
of magnitude of their duration. In addition to being mathematically natural,
this choice is psychologically well-motivated, as there is a large body of
psychometric research showing that we perceive temporal duration in this way;
see [15].

$f^{\ln}_{(t,\infty)}(x)$

(b) $d^{\ln}(q_{r})$

$x$

$r$

Unlike $d^{I}(q_{r})$ (which has its highest values as $r$ approaches $0$),
$d^{\ln}(q_{r})$ approaches $0$ as $r$ approaches $0$. This means that, for
$r<1$, smaller values of $r$ make for increasing abnormality, and (centered)
worlds in which the atom is going to decay extremely soon are hence
doxastically inaccessible. More generally, the doxastically possible answers
to $Q$ are those whose value of $d$ falls above a certain horizontal line
(which depends on the probability threshold $t$); inspection of the graph of
$d^{\ln}$ above shows that, as desired, this will always be an interval that
excludes both possibilities in which the atom decays immediately and ones in
which it won’t decay for a very long time.

This model combines two ideas: the appeal to a de se question, and the appeal
to a logarithmic measuring function. One might wonder whether the first of
these is really necessary. Couldn’t we have achieved the same effect by using
a logarithmic measuring function on the de dicto question $Q^{*}$: how long
after its creation does the atom decay? The answer is “no” – doing so achieves
the same effect only at the moment the atom is created. This is because,
unlike the probability densities of answers to $Q$, the probability densities
of answers to $Q^{*}$ change as new evidence becomes available: when the atom
hasn’t decayed 9 months after its creation, the probability that it will decay
within the next year remains unchanged, but the probability that it will decay
within its first year clearly decreases. (In particular, after a year, the
answer to $Q^{*}$ that implies that the atom will decay immediately will have
the highest probability density even with a logarithmic measuring function; so
possibilities in which the atom decays immediately will then be doxastically
accessible.) Intuitively, it is unsurprising that a de se question is needed
for the desired dynamical behavior in Decay. The idea that one is always
entitled to believe that the atom isn’t about to decay relies on there being
something particularly odd about it decaying right now or very soon; but that
oddity is clearly tied to de se notions, and cannot be articulated without
them.

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#  HYDRA-HGR: A Hybrid Transformer-based Architecture for Fusion of
Macroscopic and Microscopic Neural Drive Information

###### Abstract

Development of advance surface Electromyogram (sEMG)-based Human-Machine
Interface (HMI) systems is of paramount importance to pave the way towards
emergence of futuristic Cyber-Physical-Human (CPH) worlds. In this context,
the main focus of recent literature was on development of different Deep
Neural Network (DNN)-based architectures that perform Hand Gesture Recognition
(HGR) at a macroscopic level (i.e., directly from sEMG signals). At the same
time, advancements in acquisition of High-Density sEMG signals (HD-sEMG) have
resulted in a surge of significant interest on sEMG decomposition techniques
to extract microscopic neural drive information. However, due to complexities
of sEMG decomposition and added computational overhead, HGR at microscopic
level is less explored than its aforementioned DNN-based counterparts. In this
regard, we propose the HYDRA-HGR framework, which is a hybrid model that
simultaneously extracts a set of temporal and spatial features through its two
independent Vision Transformer (ViT)-based parallel architectures (the so
called Macro and Micro paths). The Macro Path is trained directly on the pre-
processed HD-sEMG signals, while the Micro path is fed with the p-to-p values
of the extracted Motor Unit Action Potentials (MUAPs) of each source.
Extracted features at macroscopic and microscopic levels are then coupled via
a Fully Connected (FC) fusion layer. We evaluate the proposed hybrid HYDRA-HGR
framework through a recently released HD-sEMG dataset, and show that it
significantly outperforms its stand-alone counterparts. The proposed HYDRA-HGR
framework achieves average accuracy of $94.86$% for the $250$ ms window size,
which is $5.52$ % and $8.22$ % higher than that of the Macro and Micro paths,
respectively.

Index Terms—  Biological Signal Processing, Deep Neural Networks, Human-
Machine Interface, sEMG Decomposition.

##  1 Introduction

In recent years, development of prosthetic Human-Machine Interface (HMI)
systems [1] has greatly improved lives of those suffering from amputated
limb(s) or neuromuscular disorders. Generally speaking, recent HMI systems [2]
are mainly controlled with a learning model developed based on Biological
Signal Processing (BSP) algorithms. Learning-based algorithms developed for
potential use in prosthetic HMI devices are primarily comprised of advanced
Machine Learning (ML) models [3] or Deep Neural Networks (DNNs) [4] that
perform Hand Gesture Recognition (HGR) via surface Electromyogram (sEMG)
signals. To secure high performance and proper control in HGR tasks, attention
is directed to DNNs and/or hybrid architectures because of their proven
capabilities in tackling challenging real-life problems [5, 6, 7]. Such
models, however, are still prone to several major challenges such as high
latency, large training times, undue complexity, and difficulty in finding the
discriminative patterns in sEMG signals of different hand gestures.
Accordingly, designing an accurate HGR method that can be effectively adopted
in an HMI system addressing the above-mentioned issues is the centerpiece of
today’s gesture recognition-based research works.

Literature Review: sEMG signals measure the electrical activities of the
underlying motor units in limb muscles and are collected non-invasively from
the electrodes placed on skin surface [8]. In particular, High Density sEMG
(HD-sEMG) signals are acquired through a two-dimensional (2D) grid with a
large number of closely-located electrodes [9], capturing both temporal and
spatial information of muscle activities. HD-sEMG acquisition, therefore,
provides superior spatial resolution of the neuromuscular system in comparison
to its sparse acquisition counterpart. This has inspired targeted focus on
development of DNN-based HGR methods based on HD-sEMG signals [10, 11, 12,
13]. Broadly speaking, HD-sEMG-based BSP approaches can be classified into the
following two main categories:

  * (i)

Raw HD-sEMG Processing for HGR: Algorithms belonging to this category directly
use raw HD-sEMG signals for the task of HGR. In this context, e.g., Reference
[11] performed instantaneous training of a Convolutional Neural Network (CNN)
using a 2D image of a single time measurement. In [12], Recurrent Neural
Networks (RNNs) are combined with CNNs to create a hybrid attention-based [14]
CNN-RNN architecture, which has improved HGR performance due to joint
incorporation of spatial and temporal features of HD-sEMG signals. Sun et al.
[13] introduced a network of dilated Long Short-Term Memories (LSTMs) to
classify hand gestures from the transient phase of HD-sEMG signals.

  * (ii)

HD-sEMG Decomposition: The focus here is on HD-sEMG decomposition to extract
microscopic neural drive information. HD-sEMG signals have encouraged
emergence of sEMG decomposition algorithms in the last decade [15] as they
provide a significantly high-resolution 2D image of Motor Unit (MU) activities
in each time point. sEMG decomposition refers to a set of Blind Source
Separation (BSS) [16] methods that extract discharge timings of motor neuron
action potentials from raw HD-sEMG data. Single motor neuron action potentials
are summed to form Motor Unit Action Potentials (MUAPs) that convert neural
drive information to hand movements [17]. Motor unit discharge timings, also
known as Motor Unit Spike Trains (MUSTs), represent sparse estimations of the
MU activation times with the same sampling frequency and time interval as the
raw HD-sEMG signals [18]. Extracted MUSTs are used in several domains such as
identification of motor neuron diseases [19], analysis of neuromuscular
conditions [20], and myoelectric pattern recognition [21].

A third category can be identified when the extracted MUSTs in Category (ii)
are used for HGR at microscopic level. HD-sEMG signals are modelled as a
spatio-temporal convolution of MUSTs, which provide an exact physiological
description of how each hand movement is encoded at neurospinal level [22].
Thus, MUSTs are of trustworthy and discernible information on the generation
details of different hand gestures, which leads to adoption of another group
of HGR algorithms that accept MUSTs [23] as their input. Nevertheless, due to
complexities of the decomposition stage and added computational overhead,
microscopic level HGR using MUST is less explored than models of Category (i),
which use HD-sEMG signals at a macroscopic level. There are, however, some
promising works [21, 1, 24] in which MUSTs carrying microscopic neural drive
information are exploited for HGR instead of directly using raw sEMG signals.
To discover a direct connection between different hand gestures and extracted
MUSTs, these methods have suggested estimating MUAPs of the identified sources
and extracting a set of useful features from MUAPs that are unique for each
hand gesture. For instance, in [21], the peak-to-peak (p-to-p) values of MUAPs
are calculated for each MU and each electrode channel separately and a 2D
image of MUAP p-to-p are constructed for all the channels of a single MU.
Afterwards, this 2D image is fed to a CNN architecture and its performance is
compared to that of traditional ML methods. In short, using HD-sEMG
decomposition results for HGR is still in its infancy, and in this paper, we
aim to further advance this domain.

Contributions: In this paper, for the first time to the best of our knowledge,
we introduce the idea of integrating the macroscopic and microscopic neural
drive information obtained from HD-sEMG data into a hybrid framework for HGR.
As indicated in [5, 10], Vision Transformers (ViTs) [25] are proved to surpass
the most popular DNN architectures like CNNs and hybrid CNN-RNN models in HGR
tasks, using either sparse or HD-sEMG datasets. Being motivated by this fact,
here, we propose a framework that consists of two independent ViT-based
architectures that work on the basis of the attention mechanism [10]. While
one of these ViTs is trained directly on the preprocessed HD-sEMG data, the
other one is fed with the p-to-p values of the extracted MUAPs of each source.
Thereby, a set of temporal and spatial features of various hand movements are
extracted from HD-sEMG signals and MUAPs, which are then fused via a Fully
Connected (FC) fusion layer for final classification. We show that by
extracting a very small number of underlying sources (maximum of $7$ sources
in this case), the proposed method improves the overall classification
accuracy compared to its counterparts. In brief, contributions of the paper
are as follows:

  * •

Introducing, for the first time to the best of our knowledge, the idea of
integrating macroscopic and microscopic neural drive information through a
hybrid DNN framework for HGR.

  * •

Simultaneous utilization of raw HD-sEMG signals and extracted MUSTs from HD-
sEMG decomposition via parallel construction of two ViT-based architectures.

##  2 Materials and Method

Fig. 1: The proposed HYDRA-HGR: (a) The ViT-based models in the Macro and
Micro paths are trained based on 3D, HD-sEMG and 2D, p-to-p MUAP images,
respectively. The final Micro and Macro class tokens are concatenated and
converted to a $1,024$-dimensional feature vector, which is fed to a series of
FC layers. (b) The transformer encoder.

In this section, first, we introduce the dataset used to develop the proposed
hybrid HYDRA-HGR . Afterwards, we briefly describe the transformer
architecture as its building block.

2.1. Dataset

The utilized dataset [26] in this study is a HD-sEMG collection of $20$ non-
disabled participants that performed $65$ isometric hand movements in $5$
repetitions. One of the movements is performed twice, therefore, we have $66$
gestures in total. The dataset is a set of $1$-, $2$\- and multiple-degree of
freedom hand gestures. There are two high-density grids, each with $64$
($8\times 8$) electrodes and an inter-electrode distance of $10$mm, placed on
the flexor and extensor muscles to record the sEMG signals with the sampling
frequency of $2,048$ Hz (Quattrocento, OT Bioelettronica, Torino, Italy).

2.2. Transformer Architecture

The proposed framework is originated from the ViT architecture in which the
attention mechanism is deployed to find the similarities among different parts
of the input data. In a ViT, the input, which is normally a three-dimensional
(3D) image, is first divided into several patches and the sequence of patches
are then fed as the main input. In the next stage, the 3D input patches are
first flattened and then go through a patch embedding block, which converts
each small flattened patch to an embedded vector to make them clearly
understandable by the transformer network. This process is referred to as the
linear projection of patches. Moreover, to encapsulate the learned information
obtained during the training phase in a single vector, a zero vector (called
the class token) is concatenated to the embedded patches similar to the BERT
architecture [27]. In this way, the final classification decision is made only
based on the class token, which is attributed as the network’s placeholder
that prevents making decisions that are biased towards any of the patches as
such reduces the number of trainable parameters in the model. ViTs, in
contrast to RNNs, receive their input data in parallel, which makes it
difficult for the network to identify the correct order in time-series data.
Therefore, a trainable positional embedding vector is added to all the input
patches, helping the network to understand their position more efficiently.
These steps result in matrix $\bm{Z}_{0}$ as the input to the transformer
encoder, which is given by

$\displaystyle\bm{Z}_{0}=[{\bm{x}}^{p}_{0};{\bm{x}}^{p}_{1}\bm{E};{\bm{x}}^{p}_{2}\bm{E};\dots;{\bm{x}}^{p}_{N}\bm{E}]+\bm{E}^{pos},$
(1)

where, $N$ is the number of patches, ${\bm{x}}^{p}_{i}$ represents the
flattened patches that are multiplied by an embedding matrix $\bm{E}$,
prepended with an all-zero class token and eventually added to a positional
embedding matrix $\bm{E}^{pos}$. Note that all the embedded patches before and
after going to the patch embedding block are of size $d$, which is the
embedding dimension of the model that remains constant throughout all the
training steps. Subsequently, $\bm{Z}_{0}$ is fed to the transformer encoder
that contains the Multiheaded Self-Attention (MSA) mechanism in which the
patches are divided into several parallel layers (heads). The scaled dot-
product attention is distinctively found in each head. The corresponding
results for each head are given to a feed forward layer and then, joined
together to form the final decision. The attention metric is calculated based
on different $d$-dimensional Queries ($\bm{Q}$), Keys ($\bm{K}$) and Values
($\bm{V}$) as follows

$\displaystyle\text{$Attention(Q,K,V)$}=\text{$Softmax$}(\frac{\bm{Q}\bm{K}^{T}}{\sqrt{d}})\bm{V}.$
(2)

The output matrix of the transformer encoder is then given by

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\bm{\mathcal{Z}}_{0}=[\bm{z}^{p}_{0};\bm{z}^{p}_{1};\dots;\bm{z}^{p}_{N}]=\textit{Feed-
Forward}(LN(\bm{\mathbb{Z}}_{0}))+\bm{\mathbb{Z}}_{0},$ (3)
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\text{where
}\quad\quad\bm{\mathbb{Z}}_{0}=MSA(LN(\bm{Z}_{0}))+\bm{Z}_{0},$ (4)

and $\bm{z}^{p}_{i}$ is the final output for the $i^{th}$ patch, and $LN$
stands for the layer normalization block. In the last stage, the output class
token ($\bm{z}^{p}_{0}$) is forwarded to a linear layer to form the
classification target. This completes an overview of the preliminary material,
next, we present the proposed HYDRA-HGR framework.

##  3 Proposed HYDRA-HGR Architecture

In this section, we present the proposed hybrid ViT-based architecture that
couples HD-sEMG signals and their extracted MUAPs. The HYDRA-HGR framework
simultaneously extracts a set of temporal and spatial features through its two
independent ViT-based parallel paths. The Macro Path is trained directly on
the pre-processed HD-sEMG signals, while the Micro path is fed with the p-to-p
values of the extracted MUAPs of each source. A fusion path, structured in
series to the parallel ones and consisting of FC layers, then combines
extracted temporal and spatial features for final classification.

3.1. Macro Path: Extraction of Macroscopic Information

In this sub-section, we present the Macro Path of the HYDRA-HGR framework,
which directly learns from raw HD-sEMG signals. As HD-sEMG signals are of $3$
dimensions, i.e., one dimension in time and two dimensions in space, they can
directly be fed to a ViT that is designed for 3D images. The raw HD-sEMG
signals, however, should be first pre-processed to be entirely compatible with
the ViT’s input format. First, the positive envelop of the data is found via a
low-pass butterworth filter (at $1$Hz), which is applied to signals of each
electrode independently [11, 10]. Then, the filtered signals are normalized
through the $\mu$-law algorithm [28], which increases the selectivity power of
the model. Following that, each HD-sEMG repetition is split into windows of
size $512$ ($250$ ms) with a skip step of $256$, which means $50$% of overlap
among consecutive windows. The proposed framework takes each data segment with
shape ($512$,$8$,$16$) as input and outputs the predicted label among $66$
gestures.

3.2. Micro Path: Extraction of Microscopic Neural Drive Info.

Here, first, we present the fundamentals of the developed BSS algorithm for
extracting MUSTs from HD-sEMG signals. Then we elaborate on how the HYDRA-HGR
framework is employed for classification of MUAPs as a by-product of the BSS
method. Multi-channel sEMG signals are generated as a convolutive mixture of a
set of impulse trains representing the discharge timings of multiple MUs,
i.e.,

$\displaystyle{\bm{x}}_{i}(t)=\Sigma_{l=0}^{L-1}\Sigma_{j=1}^{N}h_{ij}(l)\bm{s}_{j}(t-l)+\bm{\nu}_{i}(t),$
(5)

where ${\bm{x}}_{i}(t)$ is the $i^{th}$ channel’s EMG data (from the entire
$M$ channels); $h_{ij}(l)$ is the action potential of the $j^{th}$ MU (from
the entire $N$ extracted MUs) measured at the $i^{th}$ channel;
$\bm{s}_{j}(t)$ is the MUST at the $j^{th}$ MU, and; $\bm{\nu}_{i}$ is the
additive white noise at channel $i$. Additionally, $t$ is the time index; $D$
is the duration of sEMG recordings; and $L$ is the duration of MUAPs. Eq. (5)
is represented in matrix form as

$\displaystyle\bm{X}(t)=\Sigma_{l=0}^{L-1}\bm{H}(l)\bm{S}(t-l)+\underline{\bm{\nu}}(t),$
(6)

where
$\bm{X}(t)\\!=\\![{{\bm{x}}}_{1}(t),{{\bm{x}}}_{2}(t),\dots,{{\bm{x}}}_{M}(t)]^{T}$
and $\bm{S}(t)\\!=\\![{\bm{s}}_{1}(t),{\bm{s}}_{2}(t),$
$\dots,{\bm{s}}_{N}(t)]^{T}$ are the recordings of all the $M$ electrode
channels and the MUSTs of all the $N$ extracted sources at time $t$,
respectively. Term $\bm{H}(l)$ is the ($M\times N$) matrix of action
potentials, which is considered to be constant in duration $D$. The
convolutive equality of Eq. (6) is the basic BSS assumption.

The objective is to find the maximum number of independent matrices
$\bm{S}(t)$ from Eq. (6) if $\bm{X}(t)$ is the only known parameter. Eq. (6)
can be written in an instantaneous form, where the source vectors are extended
with their $L-1$ delayed contributions. Additionally, to adapt the model for
BSS conversions, the observation vectors are extended with their $T$ delayed
versions, resulting in the following final convolutive model

$\displaystyle\widetilde{\bm{X}}(t)=\widetilde{\bm{H}}\hskip
1.42271pt\widetilde{\bm{S}}(t)+\widetilde{\underline{\bm{\nu}}}(t),$ (7)

where each of the $\widetilde{\bm{X}}(t)$, $\widetilde{\bm{H}}$,
$\widetilde{\bm{S}}(t)$, and $\widetilde{\underline{\bm{\nu}}}(t)$ are the
extended versions of the observation, MUAPS, sources, and noise matrices,
respectively. Among the existing BSS approaches [15] suggested for HD-sEMG
decomposition, gradient Convolution Kernel Compensation (gCKC) [29, 30] and
fast Independent Component Analysis (fastICA) [31] are of great prominence and
frequently used in the literature. To achieve better accuracy, the utilized
BSS algorithm [15] is a combination of gCKC [29, 30] and fastICA [31]
algorithms. In the gCKC method, the MUSTs are estimated using a linear Minimum
Mean Square Error (MMSE) estimator as follows

$\displaystyle\hat{\bm{s}}_{j}(t)=\hat{\bm{c}}_{\bm{s}_{j}x}^{T}\bm{C}_{xx}^{-1}{\bm{x}}(t),$
(8)

in which $\hat{\bm{s}}_{j}(t)$ is the estimate of the $j^{th}$ MUST at time
$t$,
$\hat{\bm{c}}_{\bm{s}_{j}x}\approx\bm{E}\\{{\bm{x}}(t){\bm{s}_{j}}^{T}(t)\\}$
is approximation of the unknown cross-correlation vector between the MUSTs and
the observations, and $\bm{C}_{xx}=\bm{E}\\{{\bm{x}}(t){\bm{x}}^{T}(t)\\}$ is
the correlation matrix of observations. Term $\bm{E}\\{\cdot\\}$ indicates the
mathematical expectation. According to Eq. (8), as
$\hat{\bm{c}}_{\bm{s}_{j}x}$ is unknown, a blind estimation of MUSTs is
iteratively found with gradient descent [29]. On the other hand, in the
fastICA, the goal is to estimate separation vectors $w$ such that

$\displaystyle\hat{\bm{s}}_{j}(t)=w_{j}^{T}(k)\bm{Z}(t),$ (9)

where $\hat{\bm{s}}_{j}$ is the $j^{th}$ MUST; $w_{j}$ is the $j^{th}$
separation vector; and $\bm{Z}$ is the whitened matrix of observations. The
separation vectors are identified through the fixed-point optimization
algorithm [31, 15]. Note that term $k$ in Eq. (9) denotes the separation
vector identifying fixed-point iterations. To provide analogous inputs for the
Macro and Micro ViT-based models, we extracted MUSTs for windowed HD-sEMG
signals of shape ($512$,$8$,$16$), separately. The number of iterations for
estimating a new source is set to $7$, therefore, a maximum of $7$ sources
that are either sparse or uncorrelated at time lag zero are found for each
window. The length of MUAPs ($L$) that are computed from extracted MUSTs in
the next step are assumed to be $20$ samples. As stated in [15], extension
factor $T$ in Eq. (7) multiplied by the number of sEMG channels should be
greater than the number of extracted sources multiplied by the length of
MUAPs. Furthermore, it is empirically shown that extension factors greater
than $16$ have almost the same impact on the number and quality of extracted
MUSTs. Therefore, we set extension factor to $20$ to be greater than
$\frac{N\times L}{M}$. The silhouette threshold determining which sources are
of acceptable quality in each iteration is set to $0.92$.

Fig. 2: (a) Diagram of the adopted procedures for obtaining MUAP p-to-p images. (b) MUAPs for a single MU of the first windowed signal corresponding to the first repetition of gesture $1$ (bending the little finger). The MUAPs are estimated/shown for each channel separately. (c) p-to-p values of MUAPs represented as a 2D image. (d) 3D representation of MUAP p-to-p values.  Table 1: Comparison of classification accuracy and STD for each fold and their average. The accuracy and STD for each fold is averaged over $19$ subjects. Model’s Name | Fold1(%) | Fold2 | Fold3 | Fold4 | Fold5 | Average
---|---|---|---|---|---|---
Stand-alone Macro Model |  79.92 ($\pm$3.39) |  91.43 ($\pm$2.48) |  93.84 ($\pm$2.05) |  92.57 ($\pm$2.28) |  88.96 ($\pm$2.83) |  89.34 ($\pm$2.61)
Stand-alone Micro Model |  81.53 ($\pm$3.45) |  88.03 ($\pm$2.66) |  89.63 ($\pm$2.39) |  89.11 ($\pm$4.02) |  84.92 ($\pm$2.97) |  86.64 ($\pm$3.10)
The HYDRA-HGR | 89.38 ($\pm$2.88) | 96.86 ($\pm$1.82) | 96.82 ($\pm$1.75) | 96.65 ($\pm$2.75) | 94.61 ($\pm$1.90) | 94.86 ($\pm$2.22)

After extracting MUSTs for all the windows, the MUAP waveform for each channel
and each extracted MU is calculated with the Spike-Triggered Averaging (STA)
technique [1]. Following the work of [32], MUSTs are divided by a sliding
window of $256$ samples with skip size of $256$, a MUAP of length $20$ is
calculated for each window distinctly and the results are summed to give the
final MUAP waveform. In such a way, $128$ ($8\times 16$) MUAPs are acquired
for each extracted source. In the final stage, the p-to-p values of each MUAP
is calculated and fed to the Micro path’s ViT-based architecture as a 2D
($8\times 16$) image. This 2D image indicates the relative activation level of
the neural drive under the area covered by the electrode grids, which is
unique for each hand gesture. A summary of the adopted procedures from taking
the raw HD-sEMG signals to calculating the MUAP p-to-p images is shown in Fig.
2(a). Fig. 2(b-d) illustrate the extracted MUAPs for a single MU of the first
$512$-sample window of gesture $1$ (bending the little finger), 2D image of
their p-to-p values, and a 3D representation of the p-to-p values,
respectively. As can be seen, the muscles under the electrodes of the extensor
grid were more active in the course of bending the little finger.

3.3. Combining the Two ViT-based Architectures

Before fusing the Macro and Micro neural information (extracted via the two
ViT-based paths), each model is trained separately on its input dataset. Then,
each model’s weights are frozen and the FC layers at the end of each network
that take the class token for classification are removed. Instead, the class
tokens with shape of ($1\times d$) and ($7\times d$) for Macro network and
Micro network, respectively, are concatenated and flattened. Here, $1$
represents the number of $512$-sample HD-sEMG signals, $7$ is the number of
extracted MUs for each window, and $d$ is the model’s embedding dimension. The
final feature vector has a dimension of $1,024$ ($8\times 128$), which is fed
to two consecutive FC trainable layers to form the classification results.

##  4 Experimental Results

Fig. 3: Boxplots and IQR of the $3$ models over all the $19$ subjects.

In this section, we evaluate classification accuracy and standard deviation
(STD) of the Micro and Macro paths and the final fusion accuracy of the HYDRA-
HGR. To facilitate a fair comparison between our study and future research
works, we performed a $5$-fold cross validation, in which one repetition is
opted out from training and used in testing each time. The mean accuracy among
the participants is calculated for all the folds and their average is reported
at the end. As one of the subjects in the dataset had incomplete information,
all the studies are performed based on the recorded signals of $19$ subjects.
In the first experiment, windowed HD-sEMG signals are given to the Macro ViT-
based architecture. In this architecture, the image size and the number of
input channels are set to ($512\times 16$) and $8$, respectively. The patch
size is set to ($8\times 16$) implying that there are $64$ temporal patches
for each window. The model’s embedding dimension ($d$) is $128$ and the
patches are divided into $8$ parallel heads in the transformer encoder. Adam
optimizer is used with learning rate of $0.0001$ and weight decay of $0.001$.
The network is trained with $20$ epochs with each batch containing $128$ data
samples. In the second experiment, 2D MUAP p-to-p images are fed to the Micro
ViT-based architecture. In this architecture, the image size and the number of
input channels are set to ($8\times 16$), $1$ respectively. The patch size is
set to ($8\times 8$) implying that there are $2$ spatial patches for each
p-to-p image, each corresponding to an ($8\times 8$) electrode grid. The
model’s embedding dimension ($d$) and number of heads is the same as its Macro
counterpart. Similarly, Adam optimizer is used with learning rate of $0.0003$
and weight decay of $0.001$. The network is trained with $50$ epochs with each
batch containing $64$ data samples. Each data sample, in this case, has a
maximum of $7$ MUAP p-to-p images. Finally, in the third experiment, both
Macro and Micro ViT-based architectures are frozen, the feature
vectors/matrices from each model are concatenated and fed to two trainable FC
layers that map $1024$ features to $128$ and $128$ to $66$ which is the number
of gestures in our study. This network is trained with $20$ epochs and a batch
size of $128$ samples. Similarly, Adam optimizer is used with learning rate of
$0.0005$ and weight decay of $0.0001$. A comparison of each model’s accuracy
and STD for each fold is represented in Table 1. The box plots showing
accuracy and InterQuartile Range (IQR) measured for $19$ subjects is
represented in Fig. 3 for each model. As can be seen in Table 1, the Macro
Model’s accuracy is, $\approx$ $3-4$% better than that of the Micro Model for
each fold. However, the HYDRA-HGR improves the accuracy of each fold by at
least $3$% and the average accuracy of all the folds by $5.52$%. Additionally,
according to Fig. 3, Micro Model has the least IQR and the HYDRA-HGR stands
significantly higher than the stand-alone models in terms of its accuracy
among $19$ subjects.

##  5 Conclusion

In this paper, a DNN-based model for classification of hand gestures from HD-
sEMG signals is proposed in which two ViT-based architectures are fused to
outperform the case of adopting each architecture singly. The training
procedure in the proposed method comprises two stages; During the former, ViT-
based architectures are trained separately on HD-sEMG signals and p-to-p
values of the extracted MUAPs from raw HD-sEMG signals. In the latter, both
models’ weights are kept fixed and the feature sets obtained in their last
layer are joined and given to two FC layers for final classification. We show
that the classification accuracy of the fused framework is, on average,
$5.52$% and $8.22$% greater than that of Macro and Micro Models, respectively.

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