# Plasma Induced Variation of Electron Capture and Bound-State $\beta$ Decays

B. Mishra1,2,∗, A. Pidatella1, S. Taioli3,4, S. Simonucci5,6 and D. Mascali1

1Istituto Nazionale di Fisica Nucleare - Laboratori Nazionali del Sud,
Catania, Italy

2Dipartimento di Fisica e Astronomia - Università degli studi di Catania,
Catania, Italy

3European Centre for Theoretical Studies in Nuclear Physics and Related Areas
(ECT*), Bruno Kessler Foundation, Trento, Italy

4Trento Institute for Fundamental Physics and Applications (INFN-TIFPA),
Trento, Italy

5School of Science and Technology, University of Camerino, Camerino, Italy

6Istituto Nazionale di Fisica Nucleare - Sezione di Perugia, Perugia, Italy

## Abstract

The slow neutron capture ($s$-process) synthesises $\sim 50\%$ of all elements
in the universe heavier than iron, whose abundances are determined by the
competition between neutron capture and nuclear $\beta$-decay rates. The
latter are expected to vary inside hot and dense plasmas such as those found
in $s$-process nucleosynthesis sites. Here, we present a new and general
theoretical study of the effect of local and non-local thermodynamic
equilibrium ((N)LTE) plasmas on $\beta$-decays, using orbital electron capture
(EC) decays in 7Be as a model case. We begin from the model of Takahashi and
Yokoi to calculate the lepton phase volume of 7Be as a function of its
ionisation state and excitation level, and consequently, the configuration-
dependent EC decay rate. We then calculate the in-plasma ion charge state
distribution (CSD) and level population distribution (LPD) for a grid of
plasma density and temperatures, using the population kinetics code FLYCHK. By
combining the configuration-dependent EC rate with the CSD and LPD, we
calculate the in-plasma orbital EC rate in 7Be. The results show a strong
correlation between the half-life and thermodynamic conditions of the plasma,
underlining the importance of measuring decay rates in laboratory plasmas and
the relevance of high precision atomic configuration models. The model
discussed in this work is capable of calculating EC and bound state $\beta$
decay (BSBD) rates in both low-density NLTE and high-density LTE plasmas. We
conclude by validating our model with state-of-the-art data in literature on
isotopes of Pr and Dy, and by proposing future extension of the model to
laboratory magnetoplasmas and stellar interiors aimed at improving
nucleosynthesis models.

## Introduction

Understanding the cosmic origin of elements has been a fascinating topic of
study for decades. Since the seminal publications by Burbidge, Burbidge,
Fowler and Hoyle [1] and Cameron [2], the field of nuclear astrophysics has
made great strides in identifying the various processes constituting
nucleosynthesis, and in quantifying the abundance of the elements. Among these
reactions, the $s$-process occurs in asymptotic giant branch (AGB) and massive
stars, synthesising $\sim 50\%$ of all elements between the iron peak and
208Pb. The neutron density in these astrophysical sites $N_{n}\sim
10^{6}\,\mathrm{cm^{-3}}$, leading to neutron captures on the same time scale
as $\beta$-decays (days to few years). Consequently, the nucleosynthesis
pathway proceeds along the valley of $\beta$-stability on the nuclear chart
[3]. Stellar nucleosynthesis models developed for calculating $s$-process
branching of heavy nuclei are thus sensitive to both neutron capture cross
sections $\sigma(n,\gamma)$ and nuclear $\beta$-decay rates $\lambda$ [4].
Uncertainties in either quantity can strongly affect the balance between the
competing processes, leading to mismatch in observed and predicted elemental
abundances [5, 6]. While there are experimental facilities already operational
for measuring and/or updating relevant $\sigma(n,\gamma)$ to high precision
[7, 8, 9], nucleosynthesis models still rely on decay half-lives $t_{1/2}$
measured in neutral atoms which may be significantly different from their
stellar plasma counterparts.

Modification of $\lambda$ due to changes in the atomic environment of
radioactive nuclei has been investigated for decades [10]. The first studies
on the subject involved measuring the rates in extreme physical environments,
such as inside steel-encased cordite bombs [11] or at temperatures of liquid
hydrogen [12]. However, only negligible changes ($<0.05\%$) were observed.
Further research found relatively larger changes on modifying the chemistry of
the material containing the decaying nuclei, such as using a fluoride or oxide
of the isotope instead of the metal [13, 14], or implanting the nucleus into
different compounds [15, 16, 17]. This effect was attributed to perturbation
of the electron density on the nuclear surface which affected electron capture
rates, in line with the the hypotheses of Daudel and Segre [18, 19]. The same
phenomenon was also observed in internal conversion rates of isomers such as
99mTc and 90mNb [20, 21].

A much stronger effect of the atomic environment on $\lambda$ was observed in
highly charged ions (HCIs) circulating in storage rings. The first such
experiment was performed by Jung, Bosch and coworkers [22] who investigated
the decay of fully-stripped 163Dy66+ ions. They discovered that the isotope
became suddenly unstable (neutral 163Dy is stable) and rapidly transmuted into
163Ho through bound state $\beta$ decay (BSBD), making it the first
observation of this phenomenon. Soon after, Bosch, Faestermann and coworkers
discovered BSBD in fully-stripped 187Re [23] and measured a dramatic reduction
in $t_{1/2}$ from $42\,\mathrm{Gyr}$ in neutral 187Re to $32.9\pm
2\,\mathrm{yr}$ in 187Re75+. Since the 187Re/187Os pair is used as a
cosmochronometer due to its long $t_{1/2}$, a nine order increase in $\lambda$
meant re-evaluation of the galactic age based on its actual $t_{1/2}$ in the
stellar interior. A recent review by Litvinov and Chen [24] lists all the HCIs
whose $t_{1/2}$ has been measured in storage rings, and offers valuable
insights into BSBD and electron captures (EC) in single-charge species.
Various aspects of the phenomenon have been studied theoretically over the
years, and consequently, several formulations have been developed to calculate
changes in $t_{1/2}$ in different materials [25, 26, 27] and in fully-stripped
ions [28, 29, 30].

The interior of stars where nucleosynthesis occurs is a naturally occurring
dense plasma which contains isotopes in various ionisation stages surrounded
by an energetic electron cloud. It can thus be expected that decay rates will
vary strongly, depending on the properties of the plasma. Palmerini [5] and
Busso [6] speculated on possible plasma-induced variation of decay rates in
isotopes of Cs, Lu, Nb, Zr and Kr - among others - to quantify their impact on
modelling the abundance of $s$-process elements. Their work was complemented
by Taioli and Simonucci who developed ab-initio Dirac-Hartree-Fock (DHF)
models to predict $t_{1/2}$ of 134Cs and 135Cs as a function of plasma
temperature and density [31]. In addition to the aforementioned radio-
isotopes, in-plasma decay rate of 7Be has attracted a great deal of attention
from researchers. Bahcall, Iben and Gruzinov worked extensively on EC decays
in 7Be [32, 33, 34, 35, 36, 37], analysing in detail their relative
contribution compared to continuum capture rates in the interior of the sun.
The topic witnessed a renewed interest in recent years, with new predictions
about the $t_{1/2}$ of 7Be calculated using updated ab-initio models and its
impact on the solar neutrino production discussed [38, 39, 40, 41, 42].
Coincidentally, many of the experiments investigating chemically-induced
changes in $\lambda$ mentioned earlier also involved 7Be, underlining the
historical role played by this isotope in this field. Studying 7Be is also
important for the cosmological lithium problem, where the abundance of 7Li is
calculated from a series of interconnected reactions, one of which is
7Be($\mathrm{e}^{-},\nu_{e}$)7Li [37, 42, 43].

The first comprehensive theories on $\beta$-decay in stellar plasma were
proposed by Bahcall [44, 45, 46], and covered continuum decays and captures in
great detail. A full theoretical analysis of all in-plasma $\beta$-decay
processes was made by Takahashi and Yokoi [47] (hereafter TY83) who put
forward elegant expressions for calculating the lepton phase volume associated
with each decay channel and showed how the stellar plasma may cause
enhancement/suppression of some transitions. Their formulations were also used
to predict BSBD rates in fully-ionised nuclei by Gupta [28] and Chang [29] and
their coworkers, reproducing the observations from storage rings [22] in the
process.

In order to benchmark TY83 and study plasma-induced changes in $\beta$-decay
rates experimentally, a new facility named PANDORA (Plasmas for Astrophysics,
Nuclear Decay Observations and Radiation for Archaeometry) is under
realisation at INFN-LNS in Catania, Italy [48]. The facility aims to use an
electron cyclotron resonance ion source to trap a compact magnetoplasma that
can emulate certain aspects of the stellar interior like the presence of a
charge state distribution (CSD) in ions and a cloud of energetic electrons
forming a continuum. By injecting radioisotopes into the plasma and measuring
decay rates as a function of density and temperature, it would be possible to
estimate stellar $t_{1/2}$ with greater accuracy than through experiments in
storage rings or electron beam ion traps [49, 50] that cannot reproduce plasma
effects [51].

In its original formalism, the TY83 model predicts $\beta$-decay rates in
spatially homogeneous plasmas under local thermodynamic equilibrium (LTE)
wherein ion CSD, level population distribution (LPD), and electron properties
are respectively defined by the Saha, Boltzmann and Maxwell distributions
through a single unified temperature. In this work, to generalise TY83 for any
kind of plasma - including low density non-LTE (NLTE) laboratory
magnetoplasmas - we have split the model into two components:

  * •

An atomic component, which calculates the decay rate of the radio-isotope as a
function of its charge state i and excitation level j (hereafter referred to
as the configuration-dependent decay rate $\lambda^{*}(ij)$)

  * •

A plasma component, which calculates the CSD and LPD associated with the
radio-isotope as a function of plasma density and temperature (hereafter
denoted by the probability factor $p_{ij}$)

The in-plasma decay rate can then be obtained by combining $p_{ij}$ with
$\lambda^{*}(ij)$. By disentangling the two components, it is possible to
analyse the fundamental coupling of atomic and nuclear properties independent
of the plasma, and then incorporate the effects of the latter through the
$p_{ij}$ factors.

Differently from LTE plasmas, $p_{ij}$ in NLTE plasmas are obtained using
population kinetics codes which generate a full reaction rate matrix coupling
all charge states and excitation levels through a collision-radiative (CR)
model, and iteratively solve it till plasma steady state is reached. In this
work, we investigate the variation of $\beta$-decay rates in a generic plasma
using the coupled atomic and plasma components. We discuss our approach
through the example of orbital EC decay in 7Be, but the formalism is identical
for BSBD as well. The choice of this isotope is motivated by its simple atomic
nature which facilitates analysis, but it also has fundamental importance in
several astrophysical models, such as in calculating the solar neutrino flux
mentioned earlier.

## Methods and Results

The methodology used to investigate the plasma effects on the $\beta$-decay
rate consist of multiple steps: first, we calculate $\lambda^{*}(ij)$
following the technique outlined in TY83 and then use the population kinetics
code FLYCHK [52] to calculate $p_{ij}$ under different plasma conditions. We
then calculate in-plasma orbital EC decay rates as a function of plasma
electron density ($n_{e}$) and temperature ($kT_{e}$), in both low-density
NLTE (corresponding to laboratory conditions) and high-density LTE systems
(corresponding to stellar interiors). We benchmark our results with
independent, ab-initio calculations, as well as with experimental data.
Finally, we conclude by highlighting the versatility of the model, and with
perspectives on how to adapt it for the PANDORA facility.

### Configuration-Dependent EC Decay Rate of 7Be

7Be decays into 7Li through the electron capture process

$^{7}\mathrm{Be}+e^{-}\rightarrow^{7}\mathrm{Li}+\nu_{e}$ (1)

where the captured $e^{-}$ may belong to either the continuum or an atomic
orbital. In case of the former, the decay is classified as continuum capture,
while the latter is termed as orbital capture. Under neutral conditions, the
decay is spontaneous with $Q$-value $Q_{0}=861.815\,\mathrm{keV}$ for the
ground state $3/2^{-}\rightarrow 3/2^{-}$ and
$Q_{1}=Q_{0}-477.6=384.215\,\mathrm{keV}$ for the excited $3/2^{-}\rightarrow
1/2^{-}$ transition. These values represent the difference between the atomic
mass energies of the parent and daughter systems, and therefore already
include a small contribution from the binding energy of the electrons. Both
are allowed transitions, with the former being mixed Fermi and Gamow-Teller
and the latter being pure Gamow-Teller. They are characterised by comparative
half-lives $\log{f_{0}t}=3.324$ and $\log{f_{1}t}=3.556$, respectively [53].
The comparative half-lives are simply a product of the lepton phase volume
$f$, and the half-life $t_{1/2}$, and depend only on the nuclear matrix
element fixed by the levels of parent and daughter nuclei. The transitions are
henceforth indexed by $m$ for brevity, where $m=0$ and $m=1$ represent the
decay to the ground and excited states of 7Li, respectively.

The lepton phase volume $f$ describes the phase space accessible to the
electron and neutrino during the decay. It varies with the atomic
configuration $(ij)$ of the radio-isotope ion and the type of transition $m$.
According to the formulation of TY83, the configuration-dependent phase
volumes $f_{m}^{*}(ij)$ of orbital EC and BSBD can both be calculated as

$f_{m}^{*}(ij)=\sum_{x}\sigma_{x}\frac{\pi}{2}[g_{x}\,\mathrm{or}\,f_{x}]^{2}(Q_{m}(ij)/m_{e}c^{2})^{2}S_{(m)x}$
(2)

where $x$ represents the atomic orbital from where the electron is captured
(in case of EC) or into which it is emitted (in case of BSBD), $Q_{m}(ij)$ is
the decay $Q$-value which varies with the ion configuration, $m_{e}c^{2}$ is
the electron rest mass energy, $[g_{x}\,\mathrm{or}\,f_{x}]^{2}$ is the larger
of electron radial wavefunction evaluated at the nuclear radius, and
$S_{(m)x}$ is the shape factor. The quantity $\sigma_{x}$ represents the
occupancy of the orbital $x$.

The configuration-dependent EC decay rate in 7Be can be calculated as

$\lambda^{*}(ij)=\ln{2}\bigg{(}\frac{f^{*}_{0}(ij)}{f_{0}t}+\frac{f^{*}_{1}(ij)}{f_{1}t}\bigg{)}$
(3)

where $f_{0}t$, $f_{1}t$ are the aforementioned $ft$ values.

The configuration-dependent $Q_{m}(ij)$ can be evaluated as

$Q_{m}(ij)=Q_{m}+[B^{*}_{{}^{7}Be^{i}}-B^{*}_{{}^{7}Li^{i^{\prime}}}]+[e^{*}_{{}^{7}Be^{i,j}}-e^{*}_{{}^{7}Be^{i^{\prime},j^{\prime}}}]$
(4)

where $Q_{m}=Q_{0}$ or $Q_{1}$,
$B^{*}_{{}^{7}Be^{i}}-B^{*}_{{}^{7}Li^{i^{\prime}}}$ is difference in
electronic binding energies when the atoms are in the charge states $i$ and
$i^{\prime}$ respectively, and
$e^{*}_{{}^{7}Be^{i,j}}-e^{*}_{{}^{7}Li^{i^{\prime},j^{\prime}}}$ is the
difference in energy of excitation levels $j$ and $j^{\prime}$. The bracketed
terms denote the contribution of atomic ionisation and excitation to the decay
$Q$-value, and when $(ii^{\prime},jj^{\prime})\rightarrow 0$,
$Q_{m}(ij)\rightarrow Q_{m}$. This formalism is an extended version of the one
used by Gupta and Chang [28, 29, 30], applicable to any $ij$-configuration,
and not just fully-stripped atoms.

In its present form, Eq. 4 allows calculating the modification of $Q_{m}$ due
to a single coupling between parent configuration $ij$ and daughter
configuration $i^{\prime}j^{\prime}$. However, given a configuration $ij$ of
7Be, any $i^{\prime}j^{\prime}$-configuration of 7Li may be synthesised, as
long as the effective $Q_{m}(ij\rightarrow i^{\prime}j^{\prime})>0$.
Naturally, the $ij$-configuration should also have electrons in the relevant
shell to allow capture. Consequently, $Q_{m}(ij)$ should be calculated by
fixing the configuration $ij$ of 7Be consistent with the type of capture, and
coupling it with all configurations of 7Li and then averaging the results. In
this way, Eq. 4 can be re-written as

$Q_{m}(ij)=Q_{m}+\bar{\Delta}\epsilon^{ij}$ (5)

where $\bar{\Delta}\epsilon^{ij}$ now represents the total modification in
$Q_{m}$ on account of the $ij$-configuration coupling. The term is calculated
as

$\bar{\Delta}\epsilon^{ij}=\epsilon^{*}_{{}^{7}Be^{i,j}}-\frac{1}{N}\sum_{i^{\prime}j^{\prime}}\epsilon^{*}_{{}^{7}Li^{i^{\prime},j^{\prime}}}$
(6)

where the summation $\sum_{i^{\prime}j^{\prime}}$ is over all daughter
configurations resulting in positive energy difference and $N$ represents the
total number of those levels. Here $\epsilon^{*}_{{}^{7}Be^{i,j}}$ and
$\epsilon^{*}_{{}^{7}Li^{i^{\prime},j^{\prime}}}$ represent the total energy
of their respective configurations, obtained by summing the ionisation
potentials of all preceding charge states $i/i^{\prime}=0,1,...,i-1$ and the
energy of level $j/j^{\prime}$ relative to the ground state. This formalism is
equivalent to that of Eq. 4, with the exception that the binding energies are
replaced by the ionisation potentials. For sake of simplicity, it is assumed
that the decay does not result in a change in the charge state of 7Li such
that $i=i^{\prime}$ and only bound levels within the same ionisation state are
coupled (no internal conversion).

Figure 1: Atomic level coupling scheme to calculate
$\bar{\Delta}\epsilon^{ij}$ for (a) K-shell and (b) L-shell capture. The
continuum band represents all levels whose energy exceeds the ionisation
potential, and are excluded from the coupling. The arrows indicate which
levels of 7Bei+ can contribute to the capture, and which levels of 7Lii+ can
be produced. The level energies are placed on the left of each coloured box,
while the level index is placed on the right. Detailed information on the
level notations can be found in the Appendix.

We use the atomic level descriptions in the population kinetics code FLYCHK
[52] to generate a configuration-coupling scheme capable of calculating
$\bar{\Delta}\epsilon^{ij}$ as a function of $(ij)$. A detailed description of
the nomenclature, spectroscopic notation and electronic configuration of
FLYCHK levels is provided in the Appendix. An example of the level coupling
scheme and aforementioned averaging procedure is shown in Fig. 1 using K-shell
capture in 7Be0+ and L-shell capture in 7Be1+. It can be seen that in some
cases, only certain configurations of the parent nucleus can contribute to the
capture - e.g., L-capture requires the presence of at least one electron in
the L-shell and thus only the first two levels in 7Be1+ can contribute to the
decay.

(a)

(b)

Figure 2: $Q_{0}(ij)-Q_{0}$ for different $j$ in (a)
${}^{7}\mathrm{Be}^{0,1+}$ and (b) ${}^{7}\mathrm{Be}^{2+,3+}$. No plot is
shown for fully-ionised ${}^{7}\mathrm{Be}$ because it is void of electrons
for capture. For sake of brevity, only a few levels in each ionisation stage
are shown, but important levels are marked in the plots. $Q_{m}(ij)$ is
basically the energy of the emitted neutrino.

Fig. 2 shows the variation in $Q_{0}(ij)$ for various configurations of 7Be,
which is basically the energy of the neutrino $\nu_{e}$. The X-axes in the
plots list a few level names, while the Y-axes show the difference between
$Q_{0}(ij)$ and $Q_{0}$. Since $Q_{0}=861.885\,\mathrm{keV}$, none of the
configurations prohibit the decay because their Q-variations are on the
$\mathrm{eV}$ scale, but there are fluctuations nonetheless. Deviations on the
$\mathrm{eV}$ scale are anyway acceptable because the neutrino created in the
decay process carries away the energy and balances the energetics of the
reaction. The overlap between the K- and L-capture plots shows that in neutral
7Be, all levels participate in both K- and L-capture but only certain levels
can do the same from $i=1^{+}$ onward. These points are also detailed in the
plots.

The shape factor $S_{(m)x}$ is defined by the type of transition and orbital
from which the electron is captured [47]. For allowed transitions such as in
7Be, $S_{(m)x}=1$ when capturing electrons from
$x={1s_{1/2},2s_{1/2},2p_{1/2}}$ and $S_{m(x)}=0$ for $x=2p_{3/2}$. Electrons
beyond the L-shell have low capture probability and are consequently
neglected. The shape factors are born from the selection rules and
conservation of spin-parity between nucleon and lepton wavefunctions and are
consistent with the explanation provided by Folan and Tsifrinovich [56]. Here
we use the spectroscopic notation $x=nl_{j=l+s}$ to denote the atomic
orbitals.

The quantities $f_{x}$ and $g_{x}$ describe the overlap between the electronic
orbital and the nuclear surface, assuming that the capture takes place within
a thin, shell-like region at the nuclear boundary. In principle, the capture
may take place inside the entire nucleus, but a rigorous analysis of the
overlap between nucleons and electrons in the nuclear interior by Morresi and
coworkers [57] produced no significant difference with respect to pure surface
capture alone, and thus the assumption still holds. $f_{x}$ and $g_{x}$ are
calculated through the expressions

$\displaystyle f_{x}=f_{\kappa}(r)|_{r=R}=f_{\kappa}(R)$ (7) $\displaystyle
g_{x}=g_{\kappa}(r)|_{r=R}=g_{\kappa}(R)$ (8)

where $f_{\kappa}(r)$, $g_{\kappa}(r)$ are radial wavefunctions associated
with the orbital $x$, the parameter $\kappa$ is related to the spin and
angular momentum of the same orbital [58], and $R$ is the phenomenological
nuclear radius $R=R_{0}A^{1/3}$ expressed in atomic units, with
$R_{0}=1.2\,\mathrm{fm}$. The radial wavefunctions can be calculated as

$\displaystyle f_{\kappa}(r)$
$\displaystyle=\bigg{(}\frac{\hbar}{m_{e}ca_{0}}\bigg{)}^{3/2}\frac{P_{\kappa}(r)}{r}$
(9) $\displaystyle g_{\kappa}(r)$
$\displaystyle=\bigg{(}\frac{\hbar}{m_{e}ca_{0}}\bigg{)}^{3/2}\frac{Q_{\kappa}(r)}{r}$
(10)

where $a_{0}$ is the Bohr radius and $P_{\kappa}(r)$, $Q_{\kappa}(r)$ are the
eigenfunctions of the radial component of the Dirac equations in a central
Coulomb field of potential $V$

$\displaystyle\frac{\mathrm{d}P_{\kappa}(r)}{\mathrm{d}r}$
$\displaystyle=-\frac{\kappa}{r}P_{\kappa}(r)-\bigg{(}2c^{\prime}+\frac{V-\epsilon}{c^{\prime}}\bigg{)}Q_{\kappa}(r)$
(11) $\displaystyle\frac{\mathrm{d}Q_{\kappa}(r)}{\mathrm{d}r}$
$\displaystyle=\frac{\kappa}{r}Q_{\kappa}(r)+\bigg{(}\frac{V-\epsilon}{c^{\prime}}\bigg{)}P_{\kappa}(r)$
(12)

where $c^{\prime}$ is the speed of light in atomic units and $\epsilon$ is the
energy of the orbital. The coefficient $\hbar/m_{e}ca_{0}$ in Eqs. 9 and 10
converts $P_{\kappa}(r),Q_{\kappa}(r)$ from atomic to natural units.

Being eigenfunctions of the radial component alone, $P_{\kappa}(r)$,
$Q_{\kappa}(r)$ obey the normalisation condition

$\int_{0}^{\infty}[P_{\kappa}(r)^{2}+Q_{\kappa}(r)^{2}]\mathrm{d}r=1$ (13)

where $[P(r)^{2}+Q(r)^{2}]\mathrm{d}r$ represents the radial charge
probability, i.e. the probability of finding an electron at a certain distance
$r$ from the nucleus. On the other hand, $f_{\kappa}(r)$ and $g_{\kappa}(r)$
obey the normalisation condition

$\int_{0}^{\infty}[f_{\kappa}(r)^{2}+g_{\kappa}(r)^{2}]r^{2}\mathrm{d}r=1$
(14)

because $f_{\kappa}(r)^{2}+g_{\kappa}(r)^{2}$ represents the charge
probability density. This means that while certain orbitals may have non-zero
probability density at the centre of the nucleus
($f_{\kappa}(r)^{2}+g_{\kappa}(r)^{2}\neq 0$ for $r\rightarrow 0$), the
probability of electrons being there is zero because
$[f_{\kappa}(r)^{2}+g_{\kappa}(r)^{2}]r^{2}\rightarrow 0$.

We use the analytical solutions to Eqs. 11 and 12 as derived in Ref [58] to
calculate $f_{x}$ and $g_{x}$. Fig. 3(a) shows the radial charge probability
as a function of $r$ for different orbitals in 7Be. The plots corresponding to
$x=2p_{1/2}$ and $2p_{3/2}$ overlap with each other on account of their
similarity. It can be seen that all orbitals have probability maxima, but in
addition to this, the $x=2s_{1/2}$ also has a probability minimum, which
denotes the node of the wavefunction. The larger of $f_{\kappa}^{2}$ or
$g_{\kappa}^{2}$ as a function of $r$ is shown in Fig. 3(b). The dotted
vertical line in the same plot denotes the nuclear surface, and the
intersection of the two represents $f_{x}^{2}$ or $g_{x}^{2}$. It can be
easily seen that compared to $x=1s_{1/2}$ and $x=2s_{1/2}$, outer orbitals
penetrate extremely weakly into the nucleus. K-captures are the strongest
processes followed by L-captures which are almost entirely carried out by
electrons in the $x=2s_{1/2}$ orbitals.

(a)

(b)

Figure 3: (a) Radial charge probability as a function of distance from the
centre of the atom in 7Be and (b) the larger of $f(x)^{2}$ or $g(x)^{2}$ at
the surface of 7Be nucleus for various orbitals. The brown, vertical, dashed
line indicates the phenomenological radius of the nucleus. The $2p_{3/2}$
orbital curve falls too rapidly to be shown here.

The occupancy $\sigma_{x}$ is a number between $0$ and $1$ which describes the
completeness of an orbital, with the former meaning completely empty and the
latter meaning completely full. It is calculated according to the usual Pauli
exclusion principles for atomic shell filling. To use Eq. 2 identically for
both EC and BSBD, $\sigma_{x}$ can be generalised as

$(\sigma_{x})_{EC}=1-(\sigma_{x})_{BSBD}$ (15)

We calculate the occupancy of the K- and L-shell orbitals approximately using
the level descriptions of FLYCHK whose spectroscopic notations are often a
direct representation of electron distribution in various orbitals.

By combining all the above quantities with Eqs. 2 and 3, we obtain the
configuration-dependent EC rate $\lambda^{*}(ij)$ in 7Be. To demonstrate the
effect of the ionic configuration better, we calculate the percentage change
in EC decay rate $\delta\lambda^{*}(ij)$ as

$\delta\lambda^{*}(ij)=\frac{\lambda^{*}(ij)-\lambda(0)}{\lambda(0)}\times
100\%$ (16)

where $\lambda(0)$ is the decay rate of neutral, ground state 7Be. The results
are shown in Fig. 4.

(a)

(b)

Figure 4: Percentage change in decay rate with respect to neutral, ground
state 7Be as a function of configuration $(ij)$ for (a) $i=0^{+},1^{+}$ and
(b) $i=2^{+},3^{+}$. As in Fig. 2, only a few levels per ionisation stage are
shown for brevity, but important levels are marked.

It is evident that the 7Be EC decay rate varies for all excitation levels, but
the variation is strongest in doubly and triply-ionised states. For the
neutral and $i=1^{+}$ ionisation states, $\delta\lambda^{*}(ij)$ is driven by
$Q(ij)$, as can be corroborated by Fig. 2. The small spike in capture rate at
$\mathrm{li}2s$ is a model-artifact, arising from the fact that
$\sigma_{2s}=0.5$ for this level, whereas the same for levels in neutral 7Be
is smaller and subject to greater uncertainty. This is due to the lack of
precision in the electronic configuration of neutral atoms in FLYCHK data, as
is described in more detail in the Appendix. The deviations are quite small,
however, and may be neglected. For $i=2^{+}$ and $i=3^{+}$, the major factor
is the occupancy of the K and L-shells. Since K-shell captures carry more
weight compared to their L-shell counterparts, a rapid drop from
$\sigma_{1s}=100\%$ in $\mathrm{he}1s$ to $\sigma_{1s}=50\%$ in
$\mathrm{he}2st$ (Fig. 4(b)) results in an almost equal drop in decay rates.
The rate decreases again going from $\mathrm{he}2ss$ to $\mathrm{he}2pt$ when
L-captures switch from $2s$ to $2p_{1/2}$ orbitals. The final sharp drop by
$50\%$ in 7Be$3^{+}$ when going from $\mathrm{hy}1$ to $\mathrm{hy}2$ is
explained by the excitation of the one attached electron from the K- to
L-shell.

Figure 5: 7Be configuration-dependent $\lambda^{*}(ij)$ calculated using our
model and DHF [31, 41, 57].

The $\lambda^{*}(ij)$ calculated here are validated against independent
results from the relativistic Dirac-Hartree-Fock model [31, 41, 57] and the
match between the two rates can also be appreciated in Fig. 5. The small
mismatch at $\mathrm{li}2s$ is due to the same overestimated capture rate
predicted by our model as explained above, but the difference is quite small.
When applied to neutral 7Be in ground state (denoted by level index $04g02$),
the model predicts $t_{1/2}=53.44\,\mathrm{days}$ and branching ratio
$10.45\%$ which are which is well within the error limits reported in the
ENSDF database [53] ($t_{1/2}=53.22\pm 6\,\mathrm{d}$ and BR=$10.44\%$). It
should be mentioned that the $\log{ft}$ values had to be slightly altered to
get this degree of match - $\log{f_{0}t}=3.371$ and $\log{f_{1}t}=3.650$ were
used - but the differences are only $1.4\%$ and $2.6\%$ respectively with
respect to ENSDF data, and are well within the accepted uncertainty on these
parameters.

### In-Plasma EC Decay Rate of 7Be

The configuration-dependent decay rates $\lambda^{*}(ij)$ can be converted
into the in-plasma decay rates through the equation

$\lambda^{*}=\sum_{ij}p_{ij}\lambda^{*}(ij)$ (17)

where $p_{ij}$ indicates the probability of finding the ion in the
$(ij)$-configuration. The purpose of the plasma component of our model is to
customise the formalism to evaluate $p_{ij}$ according to the geometry and
nature of the plasma.

For uniform plasmas (homogeneous and isotropic), we use FLYCHK to obtain
$p_{ij}$ in a large range of $n_{e}$ and $kT_{e}$ values. The calculations can
be run in both LTE and NLTE modes, which allows us to explore plasma ion
dynamics under a variety of situations and study their effect on
$\lambda^{*}$.

Figure 6: $\langle Z\rangle$ of 7Be as a function of $kT_{e}$ for
$n_{e}=10^{12}\,\mathrm{cm^{-3}}$ (blue curves) and
$n_{e}=10^{25}\,\mathrm{cm^{-3}}$ (red curves), in LTE (squares) and NLTE
(diamonds) plasmas.

Fig. 6 shows the mean charge $\langle Z\rangle$ of 7Be as a function of
$kT_{e}$ for two different densities - $n_{e}=10^{12}\,\mathrm{cm^{-3}}$ and
$n_{e}=10^{25}\,\mathrm{cm^{-3}}$ \- under both LTE and NLTE conditions. The
lower $n_{e}$ is on the same order of magnitude as laboratory magnetoplasmas
such as PANDORA, whereas the higher $n_{e}$ describes stellar interiors. It
can be noticed that at low densities, LTE and NLTE plasmas show strong
differences in behaviour which vary according to the plasma temperature. At
low $kT_{e}$, the predicted $\langle Z\rangle$ are close, and they converge
again at high $kT_{e}$. This convergence can be attributed to increased inter-
level reaction rates, signalling the onset of equilibrium. At low densities,
NLTE predictions can be believed to be more reliable because the plasma is not
collisional enough to establish an equilibrium. High collisionality and
reaction rates are present in high $n_{e}$ plasmas at all temperatures, which
explains the perfectly superposed LTE and NLTE $\langle Z\rangle$ curves at
$n_{e}=10^{25}\,\mathrm{cm^{-3}}$. These trends highlight the fact LTE-based
models like TY83 cannot be directly validated in low density magnetoplasmas
and therefore justifies the need to generalise the same to NLTE systems.

(a)

(b)

Figure 7: (a) $\lambda^{*}$ vs $kT_{e}$ for LTE and NLTE systems, compared
with DHF results (for LTE alone) and (b) variation in EC $t_{1/2}$ vs $kT_{e}$
in NLTE plasma. The plasma density is fixed to
$n_{e}=10^{12}\,\mathrm{cm^{-3}}$.

We use the $p_{ij}$ predicted by FLYCHK for $n_{e}=10^{12}\,\mathrm{cm^{-3}}$
and a set of $kT_{e}\in[2,12,22,32,42]\,\mathrm{eV}$ to calculate
$\lambda^{*}$ according to Eq. 17. Fig. 7(a) shows the in-plasma orbital EC
decay rate of 7Be as a function of $kT_{e}$, in LTE and NLTE conditions. The
effect of $\langle Z\rangle$ on $\lambda^{*}$ is clear - at
$kT_{e}=2\,\mathrm{eV}$, our model predicts the same $\lambda^{*}$ for LTE and
NLTE which have similar $\langle Z\rangle$ (Fig. 6). The deviation between LTE
and NLTE $\langle Z\rangle$ on increasing $kT_{e}\rightarrow 42\,\mathrm{eV}$
spills over into $\lambda^{*}$ which decreases by orders of magnitude under
LTE conditions, but shows a much lower suppression in NLTE plasmas. The decay
rates can be expected to converge again at $kT_{e}>60\,\mathrm{eV}$, according
to the trends in Fig. 6. Our results are also benchmarked by comparing our LTE
results against independent DHF calculations [31, 41, 57], which are in solid
agreement. The first principle simulations of the LTE plasma using the DHF
approach were performed with the in-house relativistic code DIRECT [59].

Fig. 7(b) shows the variation of $t_{1/2}$ of 7Be in the NLTE plasma for low
density $n_{e}=10^{12}\,\mathrm{cm^{-3}}$. It can be observed that $t_{1/2}$
increases with $kT_{e}$ because hotter plasmas generate more $2^{+}$ and
$3^{+}$ ions which strongly suppress capture rates (as seen in Figs. 4 and 5).
At $kT_{e}=42\,\mathrm{eV}$, there is already a five-fold increase in
$t_{1/2}$, and it may be expected to rise even more till all ions are pushed
into the $i=4^{+}$ state. At this point, orbital captures will stop completely
and only continuum captures will contribute to the decay of 7Be. The precise
balance between orbital and continuum captures depends strongly on the density
and temperature of the plasma, and has been a topic of intense study over the
past few years [33, 35, 38, 41]. Preliminary DHF calculations indicate that
for $n_{e}\sim 10^{12-13}\,\mathrm{cm^{-3}}$, continuum captures in 7Be will
be suppressed on account of low electron density at the nuclear surface, and
thus the radio-isotope may become completely stable. This will have a strong
bearing on the planned experiments with 7Be in PANDORA, and may have some
influence on primordial nucleosynthesis models as well. The configuration-
dependent EC decay rate of 7Be as predicted by our model can also provide
theory support to the experimental results obtained so far at the ERNA
facility [60], and upcoming measurements.

## Discussion

To investigate the validity of our model in more detail, we analyse some
aspects of the atomic and plasma components separately. Fig. 8 shows the CSD
$p_{i}$ of 7Be under solar conditions, as predicted by FLYCHK. The plasma
parameters are taken as $n_{e}\sim 3.8\times 10^{25}\,\mathrm{cm^{-3}}$ and
$kT_{e}\sim 1.38\,\mathrm{keV}$, corresponding to mass density
$\rho=150\,\mathrm{g/cm^{-3}}$ and temperature $T=16\times 10^{6}\,\mathrm{K}$
[41]. The temperature is the same for both electrons and ions because $p_{i}$
is calculated under LTE conditions which are identical to NLTE at high $n_{e}$
(Fig. 6). It can be observed that $\sim 20\%$ of 7Be is expected to be in
$i=3^{+}$, which means that in the sun, there is a small but non-zero
contribution of orbital EC to the total decay rate. This is precisely in line
with the established literature on the subject [37] per the predictions by
Iben, Bahcall, Moeller and Grunzinov [33, 34, 35], and reaffirmed by Simonucci
and Taioli [41]. While our model still needs to be updated to calculate the
total capture rate on 7Be in the sun through the addition of continuum capture
and inclusion of screening effects on the electron wavefunctions, the results
of Fig. 8 reinforce the validity of the plasma component of our model because
they reproduce the presence of bound electrons in the plasma.

In addition, the plasma module also reproduces the expected physics of EC
decay in 7Be - as can be seen in Fig. 7(b), $t_{1/2}\rightarrow
53.2\,\mathrm{[d]}$ as $kT_{e}\rightarrow 0$ meaning that the model inevitably
reproduces the terrestrial decay rate of 7Be when the plasma disappears.

Figure 8: $p_{i}$ of 7Be for $n_{e}\sim 3.8\times 10^{25}\,\mathrm{cm^{-3}}$
and $kT_{e}=1.38\,\mathrm{keV}$, corresponding to solar conditions.

The atomic component of the model has been validated against EC and BSBD rates
of certain radio-isotopes in select $(ij)$ configurations. As already noted
before, the $t_{1/2}$ of neutral 7Be in ground state matches the ENSDF data
perfectly well. In addition, we also analyse the situation for 163Dy and 140Pr
whose configuration-dependent decays have been experimentally studied in
storage rings.

163Dy is stable under neutral conditions, but when fully-ionised, it undergoes
BSBD with a measured $t_{1/2}=47^{+5}_{-4}\,\mathrm{d}$ [22]. On applying Eq.
3 to 166Dy66+ and using a single transition marked by $\mathrm{log}ft=4.99$,
we calculate $t_{1/2}=49.77\,\mathrm{d}$, which is not only within the
experimental error limits, but also the same as calculated in Chang and
coworkers [29] ($t_{1/2}=49.52\,\mathrm{d}$).

On the other hand, 140Pr is an unstable element which undergoes decays through
EC and $\beta^{+}$, with an accepted $t_{1/2}=3.39\pm 1\,\mathrm{min}$ under
neutral conditions [61]. Experiments in storage rings have measured
$\lambda^{*}(ij)$ of 140Pr in fully-ionised, one-electron attached (hydrogen-
like) and two-electron attached (helium-like) configurations [62]. We
calculate the EC decay rate for helium-like 140Pr57+ and obtain
$\lambda^{*}=0.00145\,\mathrm{s^{-1}}$, in perfect agreement with measured
rate $0.00147\pm 7\,\mathrm{s^{-1}}$. The authors in Ref. [62] also note that
$\beta^{+}$ decay rates are fairly invariant over different atomic
configurations, and on adding their measured $\lambda^{*}_{\beta^{+}}$ to our
calculated $\lambda^{*}_{EC}$ for neutral 140Pr, we obtain
$t_{1/2}=3.61\,\mathrm{min}$ which also in full agreement with the
aforementioned half-life of 140Pr0+. A list of all isotopes analysed and
results obtained is presented in Table 1. It should be noted that $t_{1/2}$
reported for 140Pr57+ corresponds to EC decays alone, which explains the
factor two difference compared to the neutral case.

Table 1: Comparison between $t_{1/2}$ and branching ratio (BR) of some isotopes as measured in storage rings or known from literature, calculated independently by other groups, and calculated by our model Isotope | Measured | Other works | This work
---|---|---|---
7Be0+ |  $53.22\pm 6\,\mathrm{d}$ [53] |  $54.19\,\mathrm{d}$ (DHF [31, 57]) | $53.44\,\mathrm{d}$
| BR=$10.44\%$ |  | BR=$10.45\%$
163Dy66+ |  $47^{+5}_{-4}\,\mathrm{d}$ [22] |  $49.52\,\mathrm{d}$ (Chang et al [29]) | $49.77\,\mathrm{d}$
140Pr0+ |  $3.39\pm 1\,\mathrm{min}$ [61] |  | $3.61\,\mathrm{min}$
140Pr57+ |  $7.86\pm 0.4\,\mathrm{min}$ [62] |  | $7.97\,\mathrm{min}$
| (Only EC) |  | (Only EC)

In summary, we have developed a general model of in-plasma $\beta$ decay which
can be identically applied to a vast range of plasma parameter space. The
strength of the model lies in its separation of atomic and plasma components,
which allows generating more observables to validate the model against, while
also maintaining versatility when dealing with diverse environments. The model
has tested well with 7Be, 163Dy and 140Pr, and will be applied to many more
radio-isotopes soon.

Currently, we are introducing several upgrades to our model to improve its
applicability. First and foremost is extending the formalism to predict in-
plasma continuum captures and continuum $\beta^{+/-}$ decays, complementary to
the work by Gupta and coworkers [28, 30]. This extension also involves
including excited nuclear states in the parent and daughter atoms and
assessing their impact on the modification of $t_{1/2}$. Since these
additional nuclear couplings lack literature data on their comparative half-
lives, we are looking into shell model and ab-initio calculations to obtain
better precision on $\mathrm{log}ft$ values, in line with the suggestions by
Chang and coworkers [29].

We are also working on incorporating the effect of ionisation potential
depression (IPD) on level energies which will modify $Q_{m}(ij)$ as a function
of $n_{e}$ and $kT_{e}$. With regards to EC decays, Litvinov and coworkers
[62] have already identified the presence of the hyperfine splitting effect in
hydrogen-like 140Pr58+ which potentially enhances the capture rate [56]. We
are working on probing the effect of the same on EC capture decay rates in
140Pr58+ and 7Be3+. And finally, to fulfill the objective of benchmarking TY83
in laboratory magnetoplasmas, we intend to couple our model with a Particle-
in-Cell Monte Carlo (PIC-MC) code to simulate $p_{ij}$ in an electron
cyclotron resonance (ECR) plasma trap so as to predict 3D, space-resolved in-
plasma decay rates for experimental verification by PANDORA.

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## Acknowledgements

The authors gratefully acknowledge the support of $3^{rd}$ National Committee
of INFN and funding from project PANDORA_Gr3. A.P. would like to acknowledge
the financial support from the MUR-PNRR Project PE0000023-NQSTI, financed by
the European Union (NextGeneration EU). S.T. would like to acknowledge the
European Union under grant agreement no. 101046651 for this action. The
authors are grateful to Chang Xu and Shuo Liu at the School of Physics,
Nanjing University, China, for clarifying the meaning of the equations used to
calculate the radial wavefunctions on the nuclear surface.

## Appendix - FLYCHK Data

Data on the excitation levels in ions, their energy and electronic
configurations used in this model is taken from the atomic physics database
employed by FLYCHK. FLYCHK is a population kinetics code which can calculate
CSD and LPD of ions for different kinds of plasma in various configurations
[52]. The code is a successor to the FLY suite of codes [54] and is capable of
generating detailed spectra of plasma in a broad wavelength range. In order to
do so, FLYCHK uses an extensive repository of atomic data for elements from
$Z=1-71$ which includes valuable info like level electronic configuration,
energy, oscillation strength and statistical weight.

The nomenclature of atomic levels in FLYCHK is a combination of existing names
from the FLY module [54], the HULLAC database [55] and superconfiguration
states. A detailed explanation of each is provided in Ref. [52] but a brief
summary is provided here. When performing computations for H-, He- and Li-like
ions (meaning one, two and three electrons attached, respectively), FLYCHK
uses the FLY database for $Z\leq 26$, and the HULLAC database for others.
Consequently, levels in 7Be1/2/3+ and 7Li0/1/2+ follow the FLY nomenclature
while those in neutral 7Be0+ follow FLYCHK’s own system of nomenclature
instead. The atomic levels are broadly categorised as bound and autoionising,
depending on their energy - the former are levels with energy less than the
ionisation energy of the atom/ion whereas the latter are free states that
automatically collapse into the next ionisation state. The plasma decay model
described in this work only considers bound levels.

As may be seen from Figs.1 and 2(a), levels in 7Be0+ are named as $04g[n]$
where $[n]$ ranges from $02-10$. The first index $04$ represents $4$ electrons
present in the system, while $g$ represents bound states. $[n]$ denotes the
principal quantum number of the valence electron but no detailed information
on the subshell configuration is available. This is the reason behind the
imprecise estimation of $\sigma_{2s}$ in EC rates in neutral 7Be, which leads
to a relative overestimation in the same at $\mathrm{li}2s$.

Levels in 7Be1+ and 7Li0+ are generally designated as $\mathrm{li}[nl]$ where
$[nl]$ represents the principal and azimuthal quantum numbers respectively.
This detailed notation lasts till $n=5$, after which only $n$ is shown. This
implies that subshell configuration details from $\mathrm{li}6$ onward do not
exist.

In case of 7Be2+ and 7Li1+, the level notations follow the general structure
$\mathrm{he}[nls]$ where $[nls]$ stands for principal and azimuthal quantum
numbers of the valence electron, and total spin state respectively. This level
of detail holds till $n=2$ beyond which the indices collapse to the simpler
form $\mathrm{he}[nps]$.

The levels of 7Be3+ and 7Li2+ are simply represented as the hydrogenic
$\mathrm{hy}[n]$ where $n$ is the principal quantum number of the one attached
electron. A short summary of the level indices and their characteristics is
provided in Table 2.

Table 2: FLYCHK level notations and description Level | Spectroscopic Notation | Shell Description
---|---|---
$04g02$ | $(1s)^{2}(2s2p)^{2}$ | Averaged over $l$ ($K^{2}L^{2}$)
$04g03$ | $(1s)^{2}(2s2p)^{1}(3s3p3d)^{1}$ | Averaged over $l$ ($K^{2}L^{1}M^{1}$)
$04g07$ | $(1s)^{2}(2s2p)^{1}(7s7p7d7f7g7h7i)^{1}$ | Averaged over $l$ ($K^{2}L^{1}Q^{1}$)
$\mathrm{li}2s$ | $(1s)^{2}(2s)^{1}$ | Averaged over $m_{l}$ ($K^{2}L^{1}$)
$\mathrm{li}3p$ | $(1s)^{2}(3p)^{1}$ | Averaged over $m_{l}$ ($K^{2}M^{1}$)
$\mathrm{li}7$ | $(1s)^{2}(7s7p7d7f7g7h7i)^{1}$ | Averaged over $l$ ($K^{2}Q^{1}$)
$\mathrm{he}1s$ | $(1s)^{2}$ | Ground state ($K^{2}$)
$\mathrm{he}2st$ | $(1s)^{1}(2s)^{1}\,\,{}^{3}S$ | Spin triplet ($K^{1}L^{1}$)
$\mathrm{he}2ss$ | $(1s)^{1}(2s)^{1}\,\,{}^{1}S$ | Spin singlet ($K^{1}L^{1}$)
$\mathrm{he}2ps$ | $(1s)^{1}(2p)^{1}\,\,{}^{3}S$ | Spin triplet, averaged over $m_{l}$ ($K^{1}L^{1}$)
$\mathrm{he}7ps$ | $(1s)^{1}(10p)^{1}\,\,{}^{1}S$ | Spin singlet, averaged over $m_{l}$ ($K^{1}Q^{1}$)
$\mathrm{hy}1$ | $(1s)^{1}$ | Ground state ($K^{1}$)
$\mathrm{hy}3$ | $(3s3p3d)^{1}$ | Averaged over $l$ ($M^{1}$)


p({\bm{y}}_{i}|{\bm{x}}_{i},{\bm{\theta}})+const$ (74)

Passing from $p_{t-1}$ to $p_{t}$ amounts to applying an additional discount
factor to the likelihoods, which is equivalent to discounting the entire log
posterior and adding back a fraction of the log prior so that it is not
discounted:

$\displaystyle\log p_{t}({\bm{\theta}}|{\mathcal{D}}_{1:t-1})$
$\displaystyle=\log p({\bm{\theta}})+\sum_{i=1}^{t-1}(1+\alpha)^{-(t-i)}\log
p({\bm{y}}_{i}|{\bm{x}}_{i},{\bm{\theta}})+const$ (75) $\displaystyle=\log
p({\bm{\theta}})+\frac{1}{1+\alpha}\sum_{i=1}^{t-1}(1+\alpha)^{-(t-1-i)}\log
p({\bm{y}}_{i}|{\bm{x}}_{i},{\bm{\theta}})+const$ (76)
$\displaystyle=\frac{1}{1+\alpha}\log
p_{t-1}({\bm{\theta}}|{\mathcal{D}}_{1:t-1})+\frac{\alpha}{1+\alpha}\log
p({\bm{\theta}})$ (77)

The same reasoning applies in the general case with state dynamics. We expand
the log posterior after step $t-1$ as

$\displaystyle\log p_{t-1}({\bm{\theta}}_{t-1}|{\mathcal{D}}_{1:t-1})=\log
p({\bm{\theta}}_{t-1})+\log
p_{t-1}({\mathcal{D}}_{1:t-1}|{\bm{\theta}}_{t-1})+const$ (78)

Passing from $p_{t-1}$ to $p_{t}$ amounts to discounting the data contribution
while preserving the latent predictive prior:

$\displaystyle\log p_{t}({\bm{\theta}}_{t-1}|{\mathcal{D}}_{1:t-1})$
$\displaystyle=\log p({\bm{\theta}}_{t-1})+\frac{1}{1+\alpha}\log
p_{t-1}({\mathcal{D}}_{1:t-1}|{\bm{\theta}}_{t-1})+const$ (79)
$\displaystyle=\frac{1}{1+\alpha}\log
p_{t-1}({\bm{\theta}}_{t-1}|{\mathcal{D}}_{1:t-1})+\frac{\alpha}{1+\alpha}\log
p({\bm{\theta}}_{t-1})+const$ (80)

A similar result was derived in (Kurle et al., 2020).

We now specialize eq. 80 to LO-FI. Given our initial prior
$p({\bm{\theta}}_{0})=\mathcal{N}({\bm{\theta}}_{0}|\bm{\mu}_{0},\eta_{0}^{-1}\mathbf{I}_{P})$
and dynamics
$p({\bm{\theta}}_{t}|{\bm{\theta}}_{t-1})=\mathcal{N}({\bm{\theta}}_{t}|\gamma_{t}{\bm{\theta}}_{t-1},q_{t}\mathbf{I}_{P})$,
the latent unconditional predictive prior of the dynamical system at time
$t-1$ is

$\displaystyle p({\bm{\theta}}_{t-1})$
$\displaystyle=\mathcal{N}({\bm{\theta}}_{t-1}|\Gamma_{t-1}\bm{\mu}_{0},\eta_{t-1}^{-1}\mathbf{I}_{P})$
(81) $\displaystyle\eta_{t}^{-1}$
$\displaystyle=\gamma_{t}^{2}\eta_{t-1}^{-1}+q_{t}$ (82)
$\displaystyle\Gamma_{t-1}$ $\displaystyle=\prod_{i=1}^{t-1}\gamma_{i}$ (83)

Substituting this and our posterior
$p_{t-1}({\bm{\theta}}_{t-1}|{\mathcal{D}}_{1:t-1})=\mathcal{N}({\bm{\theta}}_{t-1}|\bm{\mu}_{t-1},(\bm{\Upsilon}_{t-1}+\mathbf{W}_{t-1}\mathbf{W}_{t-1}^{\mkern-1.5mu\mathsf{T}})^{-1})$
into eq. 80 yields

$\displaystyle
p_{t}({\bm{\theta}}_{t-1}|{\mathcal{D}}_{1:t-1})=\mathcal{N}\left({\bm{\theta}}_{t-1}\middle|\acute{\bm{\mu}}_{t-1},\left(\acute{\bm{\Upsilon}}_{t-1}+\acute{\mathbf{W}}_{t-1}\acute{\mathbf{W}}_{t-1}^{\mkern-1.5mu\mathsf{T}}\right)^{-1}\right)$
(84)

with

$\displaystyle\acute{\bm{\mu}}_{t-1}$
$\displaystyle=\bm{\mu}_{t-1}+\frac{\alpha\eta_{t-1}}{1+\alpha}\left(\acute{\bm{\Upsilon}}_{t-1}+\acute{\mathbf{W}}_{t-1}\acute{\mathbf{W}}_{t-1}^{\mkern-1.5mu\mathsf{T}}\right)^{-1}(\Gamma_{t-1}\bm{\mu}_{0}-\bm{\mu}_{t-1})$
(85) $\displaystyle\acute{\bm{\Upsilon}}_{t-1}$
$\displaystyle=\frac{1}{1+\alpha}\bm{\Upsilon}_{t-1}+\frac{\alpha\eta_{t-1}}{1+\alpha}\mathbf{I}_{P}$
(86) $\displaystyle\acute{\mathbf{W}}_{t-1}$
$\displaystyle=\frac{1}{\sqrt{1+\alpha}}\mathbf{W}_{t-1}$ (87)

Equation 85 implements a form of regularization toward the prior predictive
mean $\Gamma_{t-1}\bm{\mu}_{0}$, which originates in the log-prior term in eq.
80. Equations 86 and 87 implement inflation of the covariance by a factor of
$1+\alpha$, together with the log-prior correction being added to
$\acute{\bm{\Upsilon}}_{t-1}$. Together these expressions show how the
parameters of the distribution change as we pass from
$p_{t-1}({\bm{\theta}}_{t-1}|{\mathcal{D}}_{1:t-1})$ to
$p_{t}({\bm{\theta}}_{t-1}|{\mathcal{D}}_{1:t-1})$. Notice that we have
incremented the subscript in $p_{t}$ but the random variable is still
${\bm{\theta}}_{t-1}$. Thus
$\acute{\bm{\mu}}_{t-1},\acute{\bm{\Upsilon}}_{t-1},\acute{\mathbf{W}}_{t-1}$
define the “post-inflation” posterior that is passed to the predict step in
section A.1 to obtain the iterative prior, given by
$\bm{\mu}_{t|t-1},\bm{\Upsilon}_{t|t-1},\mathbf{W}_{t|t-1}$.

###  E.2 Simple inflation

A simpler version of inflation can be obtained by discounting the prior as
well as the likelihood. In that case, passing from $p_{t-1}$ to $p_{t}$
amounts to discounting the entire log posterior. Thus instead of eq. 80 we
have

$\displaystyle\log p_{t}({\bm{\theta}}_{t-1}|{\mathcal{D}}_{1:t-1})$
$\displaystyle=\frac{1}{1+\alpha}\log
p_{t-1}({\bm{\theta}}_{t-1}|{\mathcal{D}}_{1:t-1})$ (88)

Substituting
$p_{t-1}({\bm{\theta}}_{t-1}|{\mathcal{D}}_{1:t-1})=\mathcal{N}({\bm{\theta}}_{t-1}|\bm{\mu}_{t-1},(\bm{\Upsilon}_{t-1}+\mathbf{W}_{t-1}\mathbf{W}_{t-1}^{\mkern-1.5mu\mathsf{T}})^{-1})$
yields

$\displaystyle p_{t}({\bm{\theta}}_{t-1}|{\mathcal{D}}_{1:t-1})$
$\displaystyle=\mathcal{N}\left({\bm{\theta}}_{t-1}\middle|\bm{\mu}_{t-1},(1+\alpha)\left(\bm{\Upsilon}_{t-1}+\mathbf{W}_{t-1}\mathbf{W}_{t-1}^{\mkern-1.5mu\mathsf{T}}\right)^{-1}\right)$
(89)

Thus we merely inflate the covariance by $1+\alpha$, as in Alessandri et al.
(2007) and Ollivier (2018). This implies the simple inflation equations

$\displaystyle\acute{\bm{\mu}}_{t-1}$ $\displaystyle=\bm{\mu}_{t-1}$ (90)
$\displaystyle\acute{\bm{\Upsilon}}_{t-1}$
$\displaystyle=\frac{1}{1+\alpha}\bm{\Upsilon}_{t-1}$ (91)
$\displaystyle\acute{\mathbf{W}}_{t-1}$
$\displaystyle=\frac{1}{\sqrt{1+\alpha}}\mathbf{W}_{t-1}$ (92)

###  E.3 Hybrid inflation

Rather than mixing in the latent predictive prior, as in eq. 80, we can mix in
a distribution that uses the prior predictive variance but the posterior mean:

$\displaystyle\log p_{t}({\bm{\theta}}_{t-1}|{\mathcal{D}}_{1:t-1})$
$\displaystyle=\frac{1}{1+\alpha}\log
p_{t-1}({\bm{\theta}}_{t-1}|{\mathcal{D}}_{1:t-1})+\frac{\alpha}{1+\alpha}\log\mathcal{N}({\bm{\theta}}_{t-1}|\bm{\mu}_{t-1},\eta_{t-1}^{-1}\mathbf{I}_{P})+const$
(93)

This approach fits into the more general regularized forgetting framework of
Kulhavý & Zarrop (1993) and can be interpreted heuristically as regularizing
the covariance but not the mean, which may be preferable since $\bm{\mu}_{0}$
is sampled randomly rather than being an informed prior. In this case,
substituting LO-FI’s posterior
$p_{t-1}({\bm{\theta}}_{t-1}|{\mathcal{D}}_{1:t-1})=\mathcal{N}({\bm{\theta}}_{t-1}|\bm{\mu}_{t-1},(\bm{\Upsilon}_{t-1}+\mathbf{W}_{t-1}\mathbf{W}_{t-1}^{\mkern-1.5mu\mathsf{T}})^{-1})$
yields

$\displaystyle
p_{t}({\bm{\theta}}_{t-1}|{\mathcal{D}}_{1:t-1})=\mathcal{N}\left({\bm{\theta}}_{t-1}\middle|\bm{\mu}_{t-1},(1+\alpha)\left(\bm{\Upsilon}_{t-1}+\alpha\eta_{t-1}\mathbf{I}_{P}+\mathbf{W}_{t-1}\mathbf{W}_{t-1}^{\mkern-1.5mu\mathsf{T}}\right)^{-1}\right)$
(94)

implying

$\displaystyle\acute{\bm{\mu}}_{t-1}$ $\displaystyle=\bm{\mu}_{t-1}$ (95)
$\displaystyle\acute{\bm{\Upsilon}}_{t-1}$
$\displaystyle=\frac{1}{1+\alpha}\bm{\Upsilon}_{t-1}+\frac{\alpha\eta_{t-1}}{1+\alpha}\mathbf{I}_{P}$
(96) $\displaystyle\acute{\mathbf{W}}_{t-1}$
$\displaystyle=\frac{1}{\sqrt{1+\alpha}}\mathbf{W}_{t-1}$ (97)

##  Appendix F Spherical LO-FI

Here we describe a restricted version of LO-FI in which the diagonal part of
the precision is isotropic, $\bm{\Upsilon}_{t}=\eta_{t}\mathbf{I}_{P}$. We
denote this class of spherical plus low-rank models by SPL($L$), and refer to
this algorithm as spherical LO-FI, in contrast to the diagonal LO-FI presented
in the main text. Perhaps surprisingly, we find that the spherical restriction
can slightly help predictive performance (see UCI regression results in table
3), which is consistent with the claims in (Tomczak et al., 2020). However,
the gains are not consistent across datasets.

The spherical restriction also allows a more efficient predict step, taking
$O(P)$ instead of $O(PL^{2})$ as in diagonal LO-FI, although in practice the
running times are indistinguishable (see fig. 7). The update step takes
$O(P\tilde{L}^{2})$, matching diagonal LO-FI, although we present an
alternative approximate method in section F.5.2 that takes only $O(PLC)$.

###  F.1 Warmup

To motivate our approach, consider the case of stationary parameters, where
$p({\bm{\theta}}_{t}|{\bm{\theta}}_{t-1})=\delta({\bm{\theta}}_{t}-{\bm{\theta}}_{t-1})$.
Then $\bm{\Sigma}_{t|t-1}=\bm{\Sigma}_{t-1}$ and hence eq. 26 becomes
$\bm{\Sigma}_{t}^{-1}=\bm{\Sigma}_{t-1}^{-1}+\mathbf{H}_{t}^{\mkern-1.5mu\mathsf{T}}\mathbf{R}_{t}^{-1}\mathbf{H}_{t}$.
Hence we can unwind eq. 26 to get

$\displaystyle\bm{\Sigma}_{t}^{-1}=\eta_{0}\mathbf{I}_{P}+\sum_{i=1}^{t}\mathbf{G}_{i}\mathbf{G}_{i}^{\mkern-1.5mu\mathsf{T}}$
(98)

where
$\mathbf{G}_{t}=\mathbf{H}_{t}^{\mkern-1.5mu\mathsf{T}}\mathbf{A}_{t}^{\mkern-1.5mu\mathsf{T}}\in^{P\times
C}$ is the transposed Jacobian of the standardized observation vector
$\mathbf{A}_{t}{\bm{y}}_{t}$. The data-driven part of eq. 98 is a sum of outer
products of gradients, taken over all time steps and (standardized) outcome
dimensions. We seek a low-rank approximation of this sum,

$\displaystyle\mathbf{W}_{t}\mathbf{W}_{t}^{\mkern-1.5mu\mathsf{T}}\approx\sum_{i=1}^{t}\mathbf{G}_{i}\mathbf{G}_{i}^{\mkern-1.5mu\mathsf{T}}$
(99)

with $\mathbf{W}_{t}\in^{P\times L}$. LO-FI’s update step uses incremental SVD
after each observation to maintain $\mathbf{W}_{t}$ as an approximation of the
top $L$ non-normalized singular vectors of
$[\mathbf{G}_{1},\dots,\mathbf{G}_{t}]$. Section F.5 describes two alternative
versions of incremental SVD, one matching that of diagonal LO-FI (section
F.5.1) and the other using a more efficient projection approximation (section
F.5.2). In both cases we will have
$\mathbf{W}_{t}=\mathbf{U}_{t}\bm{\Lambda}_{t}$, where
$\bm{\Lambda}_{t}=\mathrm{diag}(\bm{\lambda}_{t})$ is a diagonal $L\times L$
matrix, and
$\mathbf{U}_{t}^{\mkern-1.5mu\mathsf{T}}\mathbf{U}_{t}=\mathbf{I}_{L}$.
Therefore the approximate posterior is written as

$\displaystyle
p({\bm{\theta}}_{t}|{\mathcal{D}}_{1:t})=\mathcal{N}\left({\bm{\theta}}_{t}\middle|\bm{\mu}_{t},\left(\eta_{t}\mathbf{I}_{P}+\mathbf{U}_{t}\bm{\Lambda}_{t}^{2}\mathbf{U}_{t}^{\mkern-1.5mu\mathsf{T}}\right)^{-1}\right)$
(100)

Unlike in diagonal LO-FI, the spherical part of the precision is data-
independent. This is because any data-driven update, like eq. 41, would make
it nonspherical. Therefore $\eta$ evolves only due to the dynamics in our
generative model, eq. 14.

###  F.2 Steady-state assumption

We find it helpful to make the steady-state assumption that
$\mathbb{V}\left[{{\bm{\theta}}_{t}}\right]=\mathbb{V}\left[{{\bm{\theta}}_{0}}\right]$
for all $t$, which is the same as the “variance preserving” OU process used in
diffusion probabilistic models (Song et al., 2021; Ho et al., 2020). Because
$\mathbb{V}\left[{{\bm{\theta}}_{0}}\right]=\eta_{0}^{-1}\mathbf{I}_{P}$, and
because $\mathbb{V}\left[{{\bm{\theta}}_{t}}\right]$ and $\eta_{t}^{-1}$ both
evolve according to eq. 14,
$\eta_{t}^{-1}=\gamma_{t}^{2}\eta_{t-1}^{-1}+q_{t}$, we have by induction that
$\mathbb{V}\left[{{\bm{\theta}}_{t}}\right]=\eta_{t}^{-1}\mathbf{I}_{P}$ for
all $t$. Therefore the steady-state assumption is equivalent to
$\eta_{t}=\eta_{0}$ and implies the following constraint for all $t$:

$\displaystyle\gamma_{t}^{2}+q_{t}\eta_{t-1}=1$ (101)

###  F.3 Notation

We use $\overline{\square}$ and $\tilde{\square}$ to denote objects whose
“focal” dimension is grown from $L$ to $P$ and $\tilde{L}$, respectively. For
example, $\mathbf{U}_{t}$ has size $P\times L$ while
$\overline{\mathbf{U}}_{t}$ has size $P\times P$ (see eqs. 104 and 105), and
$\bm{\Lambda}_{t}$ has size $L\times L$ while $\tilde{\bm{\Lambda}}_{t}$ has
size $P\times\tilde{L}$ (with $\tilde{L}$ nonzero entries; see eqs. 39 and
121).

###  F.4 Predict step for the parameters

The predict step for the parameters,
$p({\bm{\theta}}_{t}|{\mathcal{D}}_{1:t-1})=\mathcal{N}({\bm{\theta}}_{t-1}|\bm{\mu}_{t|t-1},\bm{\Sigma}_{t|t-1})$,
just requires pushing the previous posterior through the linear-Gaussian
dynamics model in eq. 14:

$\displaystyle\bm{\mu}_{t|t-1}$ $\displaystyle=\gamma_{t}\bm{\mu}_{t-1}$ (102)
$\displaystyle\bm{\Sigma}_{t|t-1}$
$\displaystyle=\gamma_{t}^{2}\bm{\Sigma}_{t-1}+q_{t}\mathbf{I}_{P}$ (103)

To efficiently compute $\bm{\Sigma}_{t-1}$, let $\overline{\mathbf{U}}_{t-1}$
be an orthonormal matrix extending $\mathbf{U}_{t-1}$ from $P\times L$ to
$P\times P$, and let $\overline{\bm{\lambda}}_{t-1}\in^{P}$ be a vector
extending $\bm{\lambda}_{t-1}$ with zeros:

$\displaystyle\overline{\mathbf{U}}_{t-1}\overline{\mathbf{U}}_{t-1}^{\mkern-1.5mu\mathsf{T}}$
$\displaystyle=\mathbf{I}_{P}$ (104)
$\displaystyle\overline{\mathbf{U}}_{t-1}[:,1{:}L]$
$\displaystyle=\mathbf{U}_{t-1}$ (105)
$\displaystyle\overline{\bm{\lambda}}_{t-1}[1{:}L]$
$\displaystyle=\bm{\lambda}_{t-1}$ (106)
$\displaystyle\overline{\bm{\lambda}}_{t-1}[(L+1){:}P]$ $\displaystyle=\bm{0}$
(107)

Then we can diagonalize using $\overline{\mathbf{U}}_{t-1}$:

$\displaystyle\bm{\Sigma}_{t-1}$
$\displaystyle=\left(\eta_{t-1}\mathbf{I}_{P}+\mathbf{U}_{t-1}\bm{\Lambda}_{t-1}^{2}\mathbf{U}_{t-1}^{{\mkern-1.5mu\mathsf{T}}}\right)^{-1}$
(108)
$\displaystyle=\left(\overline{\mathbf{U}}_{t-1}\mathrm{diag}\left(\eta_{t-1}+\overline{\bm{\lambda}}_{t-1}^{2}\right)\overline{\mathbf{U}}_{t-1}^{{\mkern-1.5mu\mathsf{T}}}\right)^{-1}$
(109)
$\displaystyle=\overline{\mathbf{U}}_{t-1}\mathrm{diag}\left(\eta_{t-1}+\overline{\bm{\lambda}}_{t-1}^{2}\right)^{-1}\overline{\mathbf{U}}_{t-1}^{{\mkern-1.5mu\mathsf{T}}}$
(110)

Substituting into eq. 103 gives an efficient expression for the precision:

$\displaystyle\bm{\Sigma}_{t|t-1}^{-1}$
$\displaystyle=\left(\gamma_{t}^{2}\overline{\mathbf{U}}_{t-1}\mathrm{diag}\left(\eta_{t-1}+\overline{\bm{\lambda}}_{t-1}^{2}\right)^{-1}\overline{\mathbf{U}}_{t-1}^{{\mkern-1.5mu\mathsf{T}}}+q_{t}\mathbf{I}_{P}\right)^{-1}$
(111)
$\displaystyle=\overline{\mathbf{U}}_{t-1}\mathrm{diag}\left(\frac{\eta_{t-1}+\overline{\bm{\lambda}}_{t-1}^{2}}{\gamma_{t}^{2}+q_{t}\eta_{t-1}+q_{t}\overline{\bm{\lambda}}_{t-1}^{2}}\right)\overline{\mathbf{U}}_{t-1}^{{\mkern-1.5mu\mathsf{T}}}$
(112)
$\displaystyle=\frac{\eta_{t-1}}{\gamma_{t}^{2}+q_{t}\eta_{t-1}}\mathbf{I}_{P}+\mathbf{U}_{t-1}\mathrm{diag}\left(\frac{\gamma_{t}^{2}\bm{\lambda}_{t-1}^{2}}{(\gamma_{t}^{2}+q_{t}\eta_{t-1})(\gamma_{t}^{2}+q_{t}\eta_{t-1}+q_{t}\bm{\lambda}_{t-1}^{2})}\right)\mathbf{U}_{t-1}^{{\mkern-1.5mu\mathsf{T}}}$
(113)

This implies the updates

$\displaystyle\eta_{t}$
$\displaystyle=\frac{\eta_{t-1}}{\gamma_{t}^{2}+q_{t}\eta_{t-1}}$ (114)
$\displaystyle\bm{\lambda}_{t|t-1}^{2}$
$\displaystyle=\frac{\gamma_{t}^{2}\bm{\lambda}_{t-1}^{2}}{(\gamma_{t}^{2}+q_{t}\eta_{t-1})(\gamma_{t}^{2}+q_{t}\eta_{t-1}+q_{t}\bm{\lambda}_{t-1}^{2})}$
(115) $\displaystyle\mathbf{U}_{t|t-1}$ $\displaystyle=\mathbf{U}_{t-1}$ (116)

Under the steady-state assumption, eq. 101, these reduce to

$\displaystyle\eta_{t}$ $\displaystyle=\eta_{t-1}$ (117)
$\displaystyle\bm{\lambda}_{t|t-1}^{2}$
$\displaystyle=\frac{\gamma_{t}^{2}\bm{\lambda}_{t-1}^{2}}{1+q_{t}\bm{\lambda}_{t-1}^{2}}$
(118) $\displaystyle\mathbf{U}_{t|t-1}$ $\displaystyle=\mathbf{U}_{t-1}$ (119)

See algorithm 6 for the pseudocode.

1def
$\text{predict}(\bm{\mu}_{t-1},\bm{\lambda}_{t-1},\mathbf{U}_{t-1},\eta_{t-1},{\bm{x}}_{t},\gamma_{t},q_{t})$:

2 $\bm{\mu}_{t|t-1}=\gamma\bm{\mu}_{t-1}$

3
$\bm{\lambda}_{t|t-1}=\sqrt{\frac{\gamma_{t}^{2}\bm{\lambda}_{t-1}^{2}}{(\gamma_{t}^{2}+q_{t}\eta_{t-1})(\gamma_{t}^{2}+q_{t}\eta_{t-1}+q_{t}\bm{\lambda}_{t-1}^{2})}}$
// componentwise

4 $\mathbf{U}_{t|t-1}=\mathbf{U}_{t-1}$

5 $\eta_{t}=\frac{\eta_{t-1}}{\gamma_{t}^{2}+q_{t}\eta_{t-1}}$

6 $\hat{{\bm{y}}}_{t}=h({\bm{x}}_{t},\bm{\mu}_{t|t-1})$

Return
$(\hat{{\bm{y}}}_{t},\bm{\mu}_{t|t-1},\bm{\lambda}_{t|t-1},\mathbf{U}_{t|t-1},\eta_{t})$

Algorithm 6 LO-FI predict step (spherical).

###  F.5 Update step

Algorithm 7 shows the pseudocode for spherical LO-FI’s update step. The mean
update is the same as for diagonal LO-FI, eq. 38. Substituting the spherical
part of the precision, $\bm{\Upsilon}_{t|t-1}=\eta_{t}\mathbf{I}_{P}$, yields

$\bm{\mu}_{t}=\bm{\mu}_{t|t-1}+\eta_{t}^{-1}\left(\mathbf{I}_{P}-\tilde{\mathbf{W}}_{t}\left(\eta_{t}\mathbf{I}_{\tilde{L}}+\tilde{\mathbf{W}}_{t}^{{\mkern-1.5mu\mathsf{T}}}\tilde{\mathbf{W}}_{t}\right)^{-1}\tilde{\mathbf{W}}_{t}^{{\mkern-1.5mu\mathsf{T}}}\right)\mathbf{H}_{t}^{{\mkern-1.5mu\mathsf{T}}}\mathbf{R}_{t}^{-1}{\bm{e}}_{t}$
(120)

####  F.5.1 Precision update: SVD version

Our primary proposed update step for spherical LO-FI is essentially the same
as that for diagonal LO-FI. We define $\tilde{\mathbf{W}}_{t}$ as in eq. 31,
calculate its SVD as in eq. 39, and keep the top $L$ singular values and
vectors (mirroring eq. 40):

$\displaystyle\bm{\lambda}_{t|t-1}$
$\displaystyle=\tilde{\bm{\lambda}}_{t|t-1}[1{:}L]$ (121)
$\displaystyle\mathbf{U}_{t|t-1}$
$\displaystyle=\tilde{\mathbf{U}}_{t|t-1}[:,1{:}L]$ (122)

To keep the diagonal part of the precision spherical, we do not update it in
response to data (cf. eq. 41).

1def
$\text{update}(\bm{\mu}_{t|t-1},\bm{\lambda}_{t|t-1},\mathbf{U}_{t|t-1},\eta_{t},{\bm{x}}_{t},{\bm{y}}_{t},\hat{{\bm{y}}}_{t},\mathbb{V}[{\bm{y}}|\cdot],L)$:

2 ${\bm{e}}_{t}={\bm{y}}_{t}-\hat{{\bm{y}}}_{t}$

3 $\mathbf{R}_{t}=\mathbb{V}[{\bm{y}}|\hat{{\bm{y}}_{t}}]$

4 $\mathbf{A}_{t}^{\mkern-1.5mu\mathsf{T}}=\mathrm{chol}(\mathbf{R}_{t}^{-1})$

5
$\tilde{\mathbf{W}}_{t}=\left[\begin{array}[]{cc}\mathbf{U}_{t|t-1}\mathrm{diag}(\bm{\lambda}_{t|t-1})&\mathbf{H}_{t}^{{\mkern-1.5mu\mathsf{T}}}\mathbf{A}_{t}^{{\mkern-1.5mu\mathsf{T}}}\end{array}\right]$

6
$\bm{\mu}_{t}=\bm{\mu}_{t|t-1}+\eta_{t}^{-1}\left(\mathbf{I}_{P}-\tilde{\mathbf{W}}_{t}\left(\eta_{t}\mathbf{I}_{\tilde{L}}+\tilde{\mathbf{W}}_{t}^{\mkern-1.5mu\mathsf{T}}\tilde{\mathbf{W}}_{t}\right)^{-1}\tilde{\mathbf{W}}_{t}^{\mkern-1.5mu\mathsf{T}}\right)\mathbf{H}_{t}^{\mkern-1.5mu\mathsf{T}}\mathbf{R}_{t}^{-1}{\bm{e}}_{t}$

7 if Full-SVD then

8
$(\tilde{\bm{\lambda}}_{t},\tilde{\mathbf{U}}_{t})=\text{SVD}(\tilde{\mathbf{W}}_{t})$

9
$(\bm{\lambda}_{t},\mathbf{U}_{t})=\text{top-L}(\tilde{\mathbf{U}}_{t},\tilde{\bm{\lambda}}_{t})$

10else

11 $(\bm{\lambda}_{t},\mathbf{U}_{t})=\text{SVD-
orth}(\bm{\lambda}_{t|t-1},\mathbf{U}_{t|t-1},\mathbf{H}_{t},\mathbf{A}_{t})$

12

Return $(\bm{\mu}_{t},\bm{\lambda}_{t},\mathbf{U}_{t})$

Algorithm 7 LO-FI update step (spherical).

####  F.5.2 Precision update: Orthogonal projection version

Computing the SVD takes $O(P\tilde{L}^{2})$ time, which may be expensive. We
now present an alternative that takes $O(PLC)$ time, but which is less
accurate. The approach is based on the ORFit method (Min et al., 2022), which
uses orthogonal projections to make the SVD fast to compute.

To explain the method, we start by considering the special case of a
linearized scalar output model of the form

$\displaystyle\mathcal{N}(y_{t}|h({\bm{x}}_{t},\bm{\mu}_{t|t-1})+{\bm{g}}_{t}^{\mkern-1.5mu\mathsf{T}}({\bm{\theta}}_{t}-\bm{\mu}_{t|t-1}),R)$
(123)

where
${\bm{g}}_{t}=\nabla_{{\bm{\theta}}}h({\bm{x}}_{t},{\bm{\theta}})_{\bm{\mu}_{t|t-1}}=\mathbf{H}_{t}^{\mkern-1.5mu\mathsf{T}}$
is the gradient. So $\tilde{\mathbf{W}}_{t}$ becomes a $P\times(L+1)$ matrix,
given by
$\tilde{\mathbf{W}}_{t}=\left[\begin{array}[]{cc}\mathbf{U}_{t-1}\bm{\Lambda}_{t-1}&{\bm{g}}_{t}\end{array}\right]$.
There is no closed-form method for computing the SVD of this new matrix,
because the new gradient will generally be oblique to the existing vectors.
The ORFit method (Min et al., 2022) makes the problem tractable by replacing
the gradient ${\bm{g}}_{t}$ by its projection onto the subspace orthogonal to
the current basis set. That is, it replaces ${\bm{g}}_{t}$ with

$\displaystyle{\bm{v}}_{t}$
$\displaystyle=\left(\mathbf{I}_{P}-\mathbf{U}_{t|t-1}\mathbf{U}_{t|t-1}^{{\mkern-1.5mu\mathsf{T}}}\right){\bm{g}}_{t}$
(124)

Computing the SVD of
$\ring{\mathbf{W}}_{t}=\left[\begin{array}[]{cc}\mathbf{U}_{t|t-1}\bm{\Lambda}_{t|t-1}&{\bm{v}}_{t}\end{array}\right]$
is trivial because its columns are orthogonal. First let
$\bm{\lambda}_{t}=\bm{\lambda}_{t|t-1}$ and
$\mathbf{U}_{t}=\mathbf{U}_{t|t-1}$. Now compute $v=||{\bm{v}}_{t}||$ and let
$k=\operatornamewithlimits{argmin}_{j^{\prime}}\lambda_{t-1}[j^{\prime}]$. If
$v>\lambda_{t}[k]$, then we replace $\lambda_{t}[k]$ with $v$, and
$\mathbf{U}_{t}[:,k]$ with ${\bm{v}}_{t}/v$. That is, we discard an old basis
vector if the new observation is more informative, in the sense of Fisher
information with respect to the linearized observation model.

We can generalize to handle $C$-dimensional outputs, to efficiently compute a
truncated rank-$L$ SVD of $\tilde{\mathbf{W}}_{t}$ in eq. 31, by incrementally
applying the above procedure to each column of the generalized matrix of
gradients,
$\mathbf{H}_{t}^{{\mkern-1.5mu\mathsf{T}}}\mathbf{A}_{t}^{{\mkern-1.5mu\mathsf{T}}}$.
To reduce the dependence on the order of projection, we visit the columns in a
random order. We denote this operation by

$\displaystyle(\mathbf{U}_{t},\bm{\Lambda}_{t})=\text{SVD-
orth}(\mathbf{U}_{t|t-1},\bm{\Lambda}_{t|t-1},\mathbf{H}_{t},\mathbf{A}_{t},L).$
(125)

See algorithm 8 for the pseudocode. This takes $O(PLC)$ time.

1def $\text{SVD-orth}(\bm{\lambda},\mathbf{U},\mathbf{H},\mathbf{A})$:

2 Sample $\bm{\pi}\in\text{perm}(C)$

3 for $j\in\bm{\pi}$ do

4
${\bm{v}}_{j}=\left(\mathbf{I}_{P}-\mathbf{U}\mathbf{U}^{{\mkern-1.5mu\mathsf{T}}}\right)\mathbf{H}^{{\mkern-1.5mu\mathsf{T}}}\left[\mathbf{A}^{{\mkern-1.5mu\mathsf{T}}}\right]_{\cdot
j}$

5 $v_{j}=\|{\bm{v}}_{j}\|$

6 $k=\operatornamewithlimits{argmin}\bm{\lambda}$

7 if $v_{j}>\lambda_{k}$ then

8 $\mathbf{U}[:,k]=\frac{{\bm{v}}_{j}}{v_{j}}$

9 $\lambda_{k}=v_{j}$

10

Return $(\bm{\lambda},\mathbf{U})$

Algorithm 8 Incremental SVD using orthogonal projection.

###  F.6 Inflation

Inflation operates identically in spherical and diagonal LO-FI, up to a change
in notation. Because spherical LO-FI represents the low-rank part of the
precision as $\mathbf{U}_{t}\bm{\Lambda}_{t}$ instead of $\mathbf{W}_{t}$, the
update to $\mathbf{W}_{t-1}$ (rescaling by $1/\sqrt{1+\alpha}$ as in eqs. 87,
92 and 97) becomes a rescaling of $\bm{\Lambda}_{t-1}$, with
$\mathbf{U}_{t-1}$ unchanged. Likewise, because spherical LO-FI represents the
diagonal part of the precision as $\eta_{t}\mathbf{I}_{P}$ instead of
$\bm{\Upsilon}_{t}$, the update to $\bm{\Upsilon}_{t-1}$ becomes an update to
$\eta_{t-1}$. This update simplifies to $\acute{\eta}_{t-1}=\eta_{t-1}$ for
Bayesian and hybrid inflation (see eqs. 86 and 96 with
$\bm{\Upsilon}_{t-1}=\eta_{t-1}\mathbf{I}_{P}$). This simplification arises
because, in spherical LO-FI, the latent predictive prior exactly coincides
with the spherical part of the precision; therefore discounting the likelihood
and not the prior amounts to deflating $\bm{\Lambda}_{t-1}$ and leaving
$\eta_{t-1}$ unchanged. Under simple inflation, $\bm{\Lambda}_{t-1}$ and
$\eta_{t-1}$ are both deflated. To implement inflation, the parameters
computed here
($\acute{\bm{\mu}}_{t-1},\acute{\eta}_{t-1},\acute{\bm{\Lambda}}_{t-1}$) are
substituted for the posterior parameters
(${\bm{\mu}}_{t-1},{\eta}_{t-1},{\bm{\Lambda}}_{t-1}$) in the predict step
(section F.4).

Bayesian inflation:

$\displaystyle\acute{\bm{\mu}}_{t-1}$
$\displaystyle=\bm{\mu}_{t-1}+\frac{\alpha\eta_{t-1}}{1+\alpha}\left(\acute{\eta}_{t-1}\mathbf{I}_{P}+\acute{\mathbf{U}}_{t-1}\acute{\bm{\Lambda}}_{t-1}^{2}\acute{\mathbf{U}}_{t-1}^{\mkern-1.5mu\mathsf{T}}\right)^{-1}(\Gamma_{t-1}\bm{\mu}_{0}-\bm{\mu}_{t-1})$
(126) $\displaystyle\acute{\eta}_{t-1}$ $\displaystyle=\eta_{t-1}$ (127)
$\displaystyle\acute{\bm{\Lambda}}_{t-1}$
$\displaystyle=\frac{1}{\sqrt{1+\alpha}}\bm{\Lambda}_{t-1}$ (128)

Simple inflation:

$\displaystyle\acute{\bm{\mu}}_{t-1}$ $\displaystyle=\bm{\mu}_{t-1}$ (129)
$\displaystyle\acute{\eta}_{t-1}$
$\displaystyle=\frac{1}{1+\alpha}\bm{\eta}_{t-1}$ (130)
$\displaystyle\acute{\bm{\Lambda}}_{t-1}$
$\displaystyle=\frac{1}{\sqrt{1+\alpha}}\bm{\Lambda}_{t-1}$ (131)

Hybrid inflation:

$\displaystyle\acute{\bm{\mu}}_{t-1}$ $\displaystyle=\bm{\mu}_{t-1}$ (132)
$\displaystyle\acute{\eta}_{t-1}$ $\displaystyle=\eta_{t-1}$ (133)
$\displaystyle\acute{\bm{\Lambda}}_{t-1}$
$\displaystyle=\frac{1}{\sqrt{1+\alpha}}\bm{\Lambda}_{t-1}$ (134)

In all three cases, $\acute{\mathbf{U}}_{t-1}=\mathbf{U}_{t-1}$.