in [73, 111]. V. Prassa et al. [106] predict only the Kπ = 9- state, G.G.
Adamian only the Kπ = 10- state below E∗ = 1.5 MeV. In addition F.R. Xu et al.
[73], H.L. Liu et al. [111] and V. Prassa et al. [106] predict a Kπ = 10-
2-quasi-proton - state in the range E∗ = (1.0-1.5) MeV, In [111] the Kπ = 10-
2-quasi-neutron - state is favoured as the isomeric one due to the, according
to the Gallagher rule favored coupling in energy ($\uparrow\downarrow$
configuration) in contrast to the unfavoured coupling of the Kπ = 9- state
($\uparrow\uparrow$ configuration), while the Kπ = 10- 2-quasi-proton - state
is not considered as the isomeric one. Besides the Kπ = 10- 2-quasi-neutron -
state G.G. Adamian et al. also consider a Kπ = 6+ 2-quasi-neutron - state
(configuration $\nu$1/2-[761]$\downarrow$ $\otimes$ $\nu$11/2-[725]$\uparrow$)
as a possible isomeric one. In addition they obtain at E∗ $<$ 1.5 MeV a Kπ =
8+ 2-quasi-neutron - state (confguration not given, but probably
$\nu$7/2+[613]$\uparrow$ $\otimes$ $\nu$9/2+[604]$\downarrow$ as expected in
268Ds) and a Kπ = 6- 2-quasi-proton - state; here also no configuration is
given, but with respect to the calculated Nilsson levels shown in fig. 22 a
possible configuration could be $\pi$11/2+[615]$\uparrow$ $\otimes$
$\nu$1/2-[510]$\uparrow$.

7.3 $\alpha$ decay of K isomers
$\alpha$ decay of K isomers has been reported so far only for three cases
254m1No [33], 270mDs [26, 74, 75] and 266mHs [74, 75].
For 254m1No two $\alpha$ events with individual energies of E = 9369 keV and E
= 9336 keV were observed, while for the transition 254mNo
${}^{\alpha}_{\rightarrow}$ 250gFm a value Eα = 9370$\pm$10 keV was expected.
The lower energy of the second decay may be regarded as a hint that not the
ground state of 250Fm was populated, but a low lying member of the ground-
state rotational band. A branching ratio b${}_{\alpha}\leq$ 1x10-4 and a
partial $\alpha$ decay half-life of $\geq$2750 s were obtained. The
theoretical half-life [76, 77] for a 9370-keV transition is Tα = 3.63 ms,
resulting in a hindrance factor HF $\geq$ 7.6x105, which is a factor
$\geq$2435 or $\geq$26600 higher than hindrance factors expected for $\Delta$L
= 8 or $\Delta$L = 6 transitions (PL/P0 = exp[-2.027L(L+1)Z-1/2A-1/6])[109].
But here we also have to consider that besides structural hindrance according
to the selection rules for $\alpha$ decay for transitions between states of
different parities (e.g. 8- $\rightarrow$ 0+, 2+) only odd values of angular
momentum change are allowed, which introduces strong hindrance between states
of even values of angular momentum difference [113].
270mDs is insofar a specific case of a K isomer as it decays to a large extent
by $\alpha$ emission. As the half-lives of the ground state (0.17 ms) and the
isomeric state (3.81 ms) [78] differ by roughly a factor of 22 a rough
estimate of the $\alpha$ branching can be obtained from the ratio of $\alpha$
\- particles in the energy range of the ground state $\alpha$ decays having
correlation times a factor of five longer ($>$1 ms) than the half-life of the
ground state, which results in a branching bα $\approx$ 0.93 [78], resulting
in a partial $\alpha$ decay half-life of Tα = 4.1 ms. Assuming events with
energies Eα $>$ 12 MeV as decays into the ground state (or a ’low’ member of
the ground-state rotational band) one obtains a branching ( 6 out of 15
decays) of 0.4 and hence a partial $\alpha$ decay half-life of Tα $\approx$
4.1/0.4 ms $\approx$ 10.25 ms. For the $\alpha$ events of the highest energies
with a mean value of 12.136 MeV one obtains a theoretical $\alpha$ decay half-
life of Tα = 3.9x10-7 s resulting in a hindrance factor HF $\approx$ 26300\.
Considering transitions 10-, 9- $\rightarrow$ 0+ one expects hindrance factors
of 4800 and 947 for $\Delta$L = 10 and $\Delta$L = 9 transitions [109].
Evidently the differences between the ’total’ hindrance factors (from the
ratio of the experimental and theoretical $\alpha$ half-lives) and those due
to the angular momentum change are much smaller than in the case of 254m1No.
If one now simply writes the total hindrance factor HF(tot) as the product of
the angular momentum hindrance factor HF($\Delta$L) and the ’nuclear structure
based hindrance’ HF(struct) as HF(tot) = HF($\Delta$L) x HF(struct) one
obtains HF(struct)(270mDs) $<$$<$ HF(struct)(254m1No) (see table 10).

Table 10: Hindrance of $\alpha$ decay for transition from the K isomer into the ground-state or low lying members of the ground-state rotational band of the daughter nucleus for 254m1No and 270mDs.  |  254m1No |  270mDs
---|---|---
Tα(calc) | 3.63 ms | 0.39 $\mu$s
Tα(exp) |  $\geq$2750 | 10.25 ms
HF |  $\geq$7.58 x 105 | 26148
HF($\Delta$L) | 28.5 ($\Delta$L = 6) | 932 ($\Delta$L = 9)
| 311.2 ($\Delta$L = 8) | 4260 ($\Delta$L = 10)
HF(struct) | 26581 ($\Delta$L = 6) | 28 ($\Delta$L = 9)
| 2435 ($\Delta$L = 8) | 6 ($\Delta$L = 10)

$\alpha$ decay of 270mDs was investigated theoretically by R.M Clark and D.
Rudolph [114] using the ’superfluid tunneling model’(STM). They were able to
reproduce the ground-state $\alpha$ decay half-lives of all known even-even
nuclei in the range Z = 100 - 118. Applying their model to the $\alpha$ decay
of 270mDs they obtain a partial $\alpha$ decay half-life of 36 ms for a
$\Delta$L = 9 transition, when reducing the pairing gap parameter to sixty
percent of the value used to reproduce the half-lives for the ground-state
$\alpha$ \- decay of the even - even nuclei, which caused an increase of the
half-life by a factor of $\approx$136\. This value is not in severe
disagreement with the experimental partial half-life of 11 ms. For the decay
of 270mDs into the assumed isomer in 266Hs they obtain a partial $\alpha$
decay half-life of 31 ms, which can be compared with the experimental value of
63 ms, assuming a relative intensity of 0.07 (one out of fifteen decays) for
that transition. Thus, the agreement of the results of R.M. Clark and D.
Rudolph with the experimental data is not so bad. It should be noted that
reduction of the pairing gap parameter does not mean simply a reduction of the
pairing, but includes also reduction of the decay probability due to nuclear
structure ’effects’ [114]. Consquently the authors conclude ’we find that the
effects of nuclear structure of the multi-quasiparticle isomer, which is
accounted for in the STM model by a reduction of the pairing gap parameter
must be included’ [114].
In conclusion we have to state that the quite low values of the ’structural’
hindrance factors are in disagreement with ’very high’ hindrance factors
expected for transitions violating by the selection rules, in this case for
$\alpha$ decay with an even value for angular momentum change and a parity
change, i.e. for the transition 10- $\rightarrow$ 0+, 2+. In other words, the
$\alpha$ decay hindrance factors do not support a Kπ = 10- configuration of
the isomer, as it seems preferred by the nuclear structure calculations and
applying the Gallagher rule, but rather suggest a Kπ = 9-. This statement,
however, is quite vague and further detailed investigations are required to
clarify the situation.

Figure 24: Barrier curvature energy for even - even nuclei.

###  7.3 7.4 Stability of K isomers against fission

It is well known that spontaneous fission of nuclei with odd numbers of
protons and/or neutrons is hindered compared to even - even nuclei. This
feature qualitativly can be understood by the conservation of angular momentum
and parity. While a pair of nuclei coupling their angular momenta to L = 0 may
change the nuclear level at crossing points during deformation towards the
fission configuration, thus following the energetically most favourable path
[115], for an unpaired nucleon this is normally not possible as it has to keep
its angular momentum, which leads to an effective increase of the fission
barrier, in literature denoted as ’specalization energy’ and thus to an
increase of the half-life [116]. Quantitatively this increase can be expressed
by a hindrance factor HF = Tsf(exp)/Tee, where Tsf(exp) is the experimental
half-life and Tee the ’unhindered’ half-life, usually taken as the geometric
mean of the half-lives of the neighbouring even-even nuclei. There is no
general rule for estimation of such hindrance factors, since the increase of
the effective fission barrier depends of the nuclear structure of the
considered nuclei and on the increase in energy of the occupied single
particle level at deformation. Neverteless a correlation between the steepness
of increase or decrease in energy at deformation seems evident [102]. In odd-
odd nuclei with an unpaired proton and an unpaired neutron, both nucleons have
to keep their angular momenta, leading to higher hindrance factors than
obtained for odd-mass nuclei. Consequently only very few cases of spontaneous
fission of odd-odd nuclei are known. It should be mentioned that detection of
spontaneous fission is complicated by the fact, that the odd-odd nucleus may
decay by $\beta^{-}$-, $\beta^{+}$\- or electron capture decay into an even-
even nucleus of much lower half-life that undergoes spontaneous fission. So
techniques to discriminate between ’direct’ spontaneous fission and
spontaneous fission after $\beta$\- or electron capture decay have to be
applied.
In K isomers, where pairs of nucleons are broken and the nucleons are excited
into different levels, the situation resembles that in odd-odd nuclei, i.e. a
strong hindrance of spontaneous fission can be expected. So far, only for two
cases, 256mFm [25] and 254m1No (see sect. 6.2) spontaneous fission was
observed. It should be mentioned that here one is faced with similar situation
as in odd-odd nuclei. In some nuclei (253,254Rf, 250No) two fission activities
were observed. In these three cases, however, it was shown, that the isomer
decays by internal transitions into the ground-state, which then undergoes
spontaneous fission.
To obtain information about fission hindrance of K isomers F.P. Heßberger et
al. [33] performed for the case of 254m1,m2No some basic calculations to
estimate fission halflives of these isomers. The calculations were based on
the empirical description of fission half-lives suggested by V.E. Viola and
B.D. Wilkins [117]. Similar to description of $\alpha$ decay spontaneous
fission was estimated as a tunneling through a one-dimensional fission
barrier. The fission half-life thus was expressed as

(7.4.1) Tsf = ln2 x (nP)-1

where ’n’ denotes the frequency of barrier assaults, ’P’ the barrier
penetration probability.
The fission barrier was approximated by an inverted parabola, and the barrier
transmission probability was calculated using the Hill-Wheeler approximation
[118], resulting in

(7.4.2) P = [1+exp(2$\pi$/$\hbar\omega_{f}$)x(Bf-E))]-1

with $\hbar\omega_{f}$ representing the barrier curvature energy, Bf the
fission barrier and E the excitation energy above the ground state.
The fission half-life for the ground state (E∗ = 0) can thus be written as

(7.4.3) lg(Tsf/years) = (2.73/$\hbar\omega_{f}$ x Bsf) - 28.04

It should be noted, however, that is approximation is quite crude, as in the
region of the actinides the fission barrier is known to be double humped [119]
as sketched in fig. 24 in the inset. On the basis of the above relation for
the fission half-lives F.P. Heßberger et al. [120] analyzed the barrier
curvature energies for the known even-even nuclei using the fission barrier as
the sum Bsf = B${}^{LD}_{sf}$ \- $\delta$Ugs with the liquid drop fission
barrier B${}^{LD}_{sf}$ as suggested by M. Dahlinger et al. [121] and
$\delta$Ugs being the ground-state shell correction energy as presented by P.
Möller and J.R. Nix [122]. On this basis F.P. Heßberger et al. [120] obtained
quite smooth behavior of $\hbar\omega_{f}$ around 0.4 MeV as function of the
fissility parameter x†††††† x = Z2 / (49.22 x A (1-0.3803 x I2 \- 20.489 x
I4)) and I = (A-2Z)/A for the isotopes of uranium to californium (region 1),
$\hbar\omega_{f}$ around 0.8 MeV, for the isotopes of nobelium, rutherfordium
and seaborgium, where the outer fission barrier was predicted to lie in energy
below the ground state [123] (region 3) and a strong variation with x for the
fermium and nobelium isotopes (region 2), which was explained by quite broad
barriers in region 1, narrow barriers in region 3 and a ’transition region’
(region 2). On this basis F.P. Heßberger et al. [33] calculated, however using
liquid drops fissions barriers according to A. Sierk [124] and ground state
shell correction energies from R. Smolanczuk and A. Sobiczewski [127], fission
half-lives for 254m1,254m2No of 1 s (254m1No) and $<$0.5 $\mu$s (254m2No) and
hence hindrance factors of $\approx$1375 (254m1No) and $>$3.3 x 106 (254m2No).
The hindrance factor HF = 1375 seems quite low, but one has to consider that
it is very sensitive to the value of the barrier curvature $\hbar\omega_{f}$
and the effective fission barrier (Bsf-E∗). In [33] quite conservative values
of (Bsf-E∗) = 5.643 MeV and $\hbar\omega_{f}$ = 0.75 MeV were used; using a
slighty higher value $\hbar\omega_{f}$ = 0.8 MeV and (Bsf-E∗) = 5.47 MeV using
the fission barrier from [126], one obtains Tsf = 0.013 s, hence a two orders
of magnitude lower value (see table 11).
In fig. 24 the results for the barrier curvature energies $\hbar\omega_{f}$
according to relation 7.4.3. using, however, fission barriers as published by
P. Möller et al. [126] (full squares) for the elements thorium to hassium are
presented as function of the fissility parameter x; for comparison results for
the isotopes of the elements rutherfordium, seaborgium, and hassium using the
dynamical fission barriers B${}^{dyn}_{f}$ as published by R. Smolanczuk et
al. [127] are shown as open squares in fig. 24. Evidently an increase of the
barrier curvature energies at increasing fissility is observed, with partly
strong variations within an isotope line, which is seen a signature for a
change of the shape of the fission barrier. The values using fission barriers
of [127] are higher than those using fission barriers of [126] since [127]
predict higher fission barriers.
For the case of 254m1No one obtaines a fission half-live of Tsf = 0.0339 s
using the value of $\hbar\omega_{f}$ for the ground state. Respecting,
however, that the excitation energy of the isomer may be already above the
height of the outer barrier, thus taking the barrier curvature obtained for
250No, which is expected to have a narrow single-humped barrier [123] one
obtains Tsf = 0.48 $\mu$s, and hence hindrance factors of HF = 40560 and HF =
2.86$\times$109.
Recently the idea of parametrizing the barrier curvature energy was resumed by
J. Khuyagbaatar [128]. He obtained somewhat higher values of $\hbar\omega_{f}$
and a fission half-life of 9.4$\times$10-4 for 254m1No. The results are
summarized in table 11; evidently the hindrance factor reported in [33] seems
unrealistic low, seemingly using a too low value for the barrier curvature
energy. But it also seen that the hindrance factors resulting from the other
analyses also straggle in a wide range of more than four orders of magnitude,
which can be ascribed due to the rather simple form of calculating the fission
half-lives.
In the case of 256Fm Khuyagbaatar report a calculated unhindert fission half-
life of 2.2$\times$10-7 s, while the analysis presented here delivers a value
Tsf(calc) = 3.6$\times$10-8 s and hence hindrance factors of 3.6$\times$103
[128] and 2.2$\times$105.
A comparison for 254m1No is shown in table 11. Evidently the cases 3 and 4
exhibit hindrance factors of $>$1$\times$106 which is higher than hindrance
factor for odd-mass nuclei which are typically (except a few cases)
$<$$\times$106, i.e. one can conclude that fission of 254m1No is more hindered
than typically for nuclei with one unpaired nucleon. A comparison with odd-odd
nuclei is hardly possible, as there are only two cases of odd-odd nuclei where
direct fission is reported, 262Db (a less certain case) and 260Md. In the case
of 262Db a quite low hindrance factor (HF$\approx$3250) is obtained, while for
260Md a quite high value (HF$\approx$9.2$\times$108) is obtained (see [102]).
Thus one may conclude that fission hindrance of 2- quasiparticle - K isomers
(with two unpaired nucleons) indeed rather resembles the case of odd-odd
nuclei with each an unpaired proton and neutron. At present state, such a
conclusion is indeed still somewhat speculative, more information of
spontaneous fission of K isomers and odd-odd nuclei is required as well as
efforts from theory to calculate fission half-lives of K isomers is required.

Table 11: Comparison of fission hindrances for 254m1No for different barrier curvature energies $\hbar\omega_{f}$ values. Tsf = 1375 s for all cases. reference | Tsf(calc) | HF
---|---|---
[33] | 1 s | 1375
[33], $\hbar\omega_{f}$=0.8 MeV | 0.013 s | 1.06$\times$105
this work, $\hbar\omega_{f}$=0.783 MeV | 0.034 s | 4.1$\times$104
this work, $\hbar\omega_{f}$=1.05 MeV | 0.048 $\mu$s | 2.9$\times$109
[128] | 0.94 ms s | 1.4$\times$106

Finally one other item should be mentioned; besides $\hbar\omega_{f}$ fission
half-lives calculated using the above mentioned method are also quite
sensitive to the fission barrier. To show the situation fission half-lives of
256Rf for the ground-state and at an excitation energy of 1 MeV were
calculated using once the fission barriers from [126] and the analyzed barrier
curvatures and once the dynamical fission barriers from [127]. Using in both
cases the values of $\hbar\omega_{f}$ extracted for the ground-state fission
half-lives of Tsf(E∗=1 MeV) = 7.6 $\mu$s (fission barriers from [126]) and
Tsf(E∗= 1 MeV) = 30 $\mu$s (fission barriers from [127]) and hence half-life
ratios Tsf(gs)/Tsf(E∗=1 MeV) = 860 and Tsf(gs)/Tsf(E∗=1 MeV) = 210 are
obtained; although these ratios vary within a factor of four, this difference
is not of significant relevance for qualitative discussion on fission
hindrance of K isomers.

##  8 8\. Outlook

The occurence of K isomers in strongly prolate deformed nuclei is a physically
interesting phenomenom. Thanks to improved experimental techniques
investigation of K isomers became a key aspect in spectroscopy of transfermium
nuclei during the past two decades. Although about thirty new K isomers were
discovered in that region of nuclei, information on the structure and decay
properties is scarce for most of the cases due to low production rates for
most of the isomers which do not allow for detailed $\gamma$ spectroscopic
investigations.
A specific feature pointed out quite often is the longer half-life of some
isomers compared to the ground - state, which was also discussed in context
with the stability of ’superheavy’ nuclei in vicinity of the closed proton and
neutron shells expected at Z = 114, 120 or 126 and N = 172 or 184. Here one
has to keep in mind, however, that K isomerism is a phenomenom in strongly
deformed nuclei, while the nuclei around the closed proton an neutron shells
are spherical. So K isomers are not expected in the region of spherical
superheavy elements.
Stability and decay properties of K isomers (also with respect to those of the
ground-state) are strongly related to the properties of the radioactive decay
modes, internal transitions, $\alpha$ decay, spontaneous fission, relevant for
the decay of the isomers.
The ratios (partial half-lives) for internal transitions are essentially
defined by the K hindrance, those for $\alpha$ decay by the Q-value, the
’angular momentum hindrance’ and by hindrance due to changes in nuclear
structure and the parity. Spontaneous fission half-lives are essentially
defined by hight and shape of the fission barrier and a fission hindrance due
to unpaired nucleons. It is thus the interplay between probabilities of these
three decay modes which finally defines the decay of the K isomer. As long as
partial half-lives for $\alpha$ decay and spontaneous fission are
significantly larger than those for internal transitions, the latter will
dominate and $\alpha$ decay and spontaneous fission will be exotic decay modes
with low intensities as in the case of 254m1No. Half-lives of the K isomers
are definitely shorter than those of the ground-states in such cases. The
situation changes, when partial half-lives for $\alpha$ decay or spontaneous
fission come into the same order of magnitude as the half-lives for internal
transitions. In those cases $\alpha$ decay or spontaneous fission can become
the essential decay modes of K isomers and also half-lives of the isomeric
state may become longer than that of the ground state due to strong hindrance
of the isomeric decays while $\alpha$ decay or spontaneous fission of the
ground-state is unhindered or at least less hindered. An example is the decay
of 270,270mDs by $\alpha$ emission, where $\alpha$ decay of the isomeric state
is strongly hindered, and thus half-life is longer than that of the ground-
state despite of a higher Qα \- value of the isomer. Another phenomenom is the
observation of two fission activities for 250No and 254Rf (254Rf, T1/2 = 247
$\mu$s), with the shorter one attributed to the ground-state and the longer
one attributed to the isomeric state. Here it was shown that direct fission of
the isomer is at best a decay mode of very low intensity due to the strong
hindrance caused by the unpaired nucleons despite a significantly ($\leq$1
MeV) lower fission barrier of the isomer. The longer-lived fission activity
attributed to the isomer rather represents spontaneous fission of the ground-
state delayed by internal transitions. Thus we can conclude, although details
of the decays are theoretically not well understood, that decay of the latter
K isomers can be understood on the basis of the more or less well known
properties of internal transitions, $\alpha$ decay and spontaneous fission and
certainly do not contain some kind of ’new physics’.
Although properties of K isomers in the transfermium region will not have
direct impact on the expected properties of ’spherical’ superheavy nuclei,
study of their properties will deliver valuable information on energies and
ordering of single particle levels (Nilsson levels) which certainly will have
feedback on the prediction and location of spherical proton and neutron shells
in the range Z = 114-126 and N = 172-184.

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[a]Julien Frison

# Bayesian Inference for Contemporary Lattice Quantum Field Theory

###### Abstract

Bayesian inference provides a rigorous framework to encapsulate our knowledge
and uncertainty regarding various physical quantities in a well-defined and
self-contained manner. Utilising modern tools, such Bayesian models can be
constructed with a remarkable flexibility, leaving us totally free to
carefully choose which assumption should be strictly enforced and which should
on the contrary be relaxed. The practical evaluation of these assumptions,
together with the data-driven selection or averaging of models, also appears
in a very natural way. In this presentation, I discuss its application in the
context of lattice QCD and its common statistical problems. As a concrete
illustration, I present a few parametric and non-parametric hierarchical
models applied to actual correlator data, from single exponential fits to
spectral functions.

##  1 Introduction

Probabilistic programming languages (PPL) have made huge progress in the last
decade, in large part thanks to new Monte-Carlo samplers [1] derived from
those originally developped for Lattice QCD [2]. Combined with modern
computing resources and Machine Learning libraries [3], it opens the door to
the automatic evaluation of Bayesian Networks of arbitrary complexity. PPL
usually implement various methods, and in particular it also connects to other
Machine Learning techniques such as VAE [4] and normalising flows [5], which
are often justified theoretically as a good approximation of a Bayesian
inference.

To this day, this has been widely applied to epidemiology [6], finance,
geology, pharmaceutical industry, marketing, social sciences and many other
fields. However, applications to LQCD are scarce, and to our knowledge mostly
limited to [7, 8].

Instead, LQCD is mostly based on a combination of traditional statistical
methods developped in the 70’s or earlier. This includes using the Central
Limit Theorem on Markov Chains [9], Generalised Least Square [10], Delta
method [11], Jacknife [12] and Bootstrapping [13]. The main exception in this
picture is the successful adoption of (pseudo-)Bayesian Model Averaging [14]
by a large share of the community.

The objective of this work is to learn physical parameters or functions, in a
numerically efficient way, which extracts as much information as possible from
our data, built on a theoretically well-defined probabilistic interpretation.
This should additionally provide a unified and consistent framework describing
the whole process from the data to the final result, combining as much as
possible the strengths of all the different methods currently used in the LQCD
state-of-the-art. As a result, one can get rid of unnecessary assumptions and
gain flexibility on model building. Eventually, an important feature of this
Bayesian point-of-view is the ability to provide interpretable metrics to test
afterwards which assumptions should be used or not.

In Sec. 2.1 we will summarise the state-of-the-art and discuss how one can
expect this to improve from Bayesian inference. Sec. 3 will then introduce the
formalism and tools at our disposal. Eventually, the main discussion in Sec. 4
will focus on giving an overview of some models illustrating the diversity of
problems which can be unified into a single framework.

##  2 Current methods and beyond

###  2.1 Current methods

State-of-the-art LQCD analyses are mostly organised in three somewhat
independent layers, which can sometimes conflict with each other.

Its foundation usually consist in a resampling, either block[15] bootstrapping
or jacknife. Some practioners prefer using the $\Gamma$ method [16], and the
best choice depends on the problem considered. While bootstrapping is an exact
non-parametric method for non-linear functions of non-Gaussian data, the
$\Gamma$ method has a demonstrably faster convergence for strongly auto-
correlated data.

The next layer consists in a $\chi^{2}$ fit, which is essentially a maximum
likelihood estimator with a Gaussian likelihood. One practical difficulty is
the need for the covariance to be known with sufficiently good precision
before fitting, and the lack of a clear definition for what sufficiently good
means. Another difficulty is that (non-regularised) non-linear fits are often
unstable for a moderately large number of parameters. Convergence is usually
not theoretically guaranteed, and it is a fortiori not guaranteed to converge
to something meaningful given only a finite amount of data.

The last piece is the Akaike Information Criterion[17], to give a more
rigorous meaning to the results of the previous layers. However, this requires
prior knowledge of the correlated $\chi^{2}$, which is often not reliable.
Additionally, in principle it requires the guarantee that one model
parametrises the truth. Finally, evaluating many models over many boostrap
samples tends to be a computationally intensive task, even when only a few
models contribute significantly to the results.

###  2.2 A fully Bayesian solution

In this work we replace all of this with a single method which is fully
Bayesian from the start to the end. It natively provides distributions and
confidence intervals for every single object, with an automatic propagation of
all errors, instead of wrapping a posteriori a point estimate inside an extra
layer. Every assumption is encoded into the choice of model, while the same
framework is used for arbitrarily complicated models or choices of models
covering different regimes. These models can be evaluated with an HMC, which
might take time to converge in practice but is theoretically guaranteed to
converge into a well-defined posterior distribution. Eventually, widely
applicable criteria[18, 19] exist to evaluate the distance from the model to
the truth, making model averaging and selection more reliable.

##  3 Bayesian formalism

###  3.1 The Bayes formula

Our inference is based on an “inversion” of probabilities with the Bayes
formula

$P\left(a\mid y,M\right)=\frac{P\left(y\mid a,M\right)P\left(a\mid
M\right)}{P\left(y\mid M\right)}.$ (1)

In the usual terminology, the left hand side is called the posterior
distribution and expresses the probability distribution of a set of parameters
$a$ in the model $M$ when a data sample $y$ is known. This distribution is our
final result, from which one can extract quantiles and other estimators to
form our intervals of confidence. In the right hand side, the numerator is
made of a likelihood multiplied by a prior, both being chosen by the modeller.
The denominator, called marginal likelihood or evidence, is usually ignored.

Starting from these elements, one can also define the Posterior Predictive
Distribution, which expresses a Bayesian model as a generative machine-
learning model:

$P\left(y^{\prime}\mid y,M\right)=\int P\left(y^{\prime}\mid
a,M\right)P\left(a\mid y,M\right)\mathrm{d}a.$ (2)

###  3.2 Applying the HMC

A close formula is usually not needed for the posterior, and PPL are
specifically made to deal with situations where such a manual derivation is
not possible. Configurations of the parameters $a$ can be sampled from an HMC-
based algorithm, similarly to how lattice configurations are drawn. Pushing
the analogy, the negative log-likelihood (for instance a $\chi^{2}$ in a fit)
corresponds to the choice of a lattice action, while the marginal distribution
corresponds to the partition function. A major difference is that here we do
not need to manually provide an expression for the HMC forces.

###  3.3 Software

Many languages and libraries exist for Probabilistic Programming. We chose to
base this work on PyMC[20], with its vectorisation and automatic
differentiation included through PyTensor. With this tools at our disposal,
the task for the user is usually limited to writing models in the form of a
mathematical expression. These are not restricted to comply with any specific
approximation, but only to provide a tree which compiles.

Some analytical computations from the user can be useful, for marginalisation
in particular, but only if a speedup is needed. Models can either be
parametric (like traditional $\chi^{2}$ fits) or non-parametric (Bayesian
Bootstrap, Gaussian Processes, Bayesian Neural Networks, …), and parametric
models do not need to have a positive number of degrees of freedom. Finally,
it always provides an estimate of the Kullback-Leibler divergence, a goodness-
of-fit indicator which (unlike frequentist arguments or the Akaike IC) is not
conditioned on the model being true. So in principle the distance from the
model to the truth can be checked from a strongly data-driven point-of-view.

We will show some graphical representations of Bayesian Networks in the
commonly used plate notation, based on the Graphviz library.

##  4 A few models

###  4.1 Data

All the models presented here are tested on some arbitrary correlator data for
illustration. Those come from the CLS ensemble H101, with heavy valence
quarks. This data is provided on [21] alongside with the code to reproduce our
results. This is not the only type of data on which these methods would apply,
and we could perfectly have decided to perform for instance chiral and
continuum fits, or combine lattice and experimental data. Instead, we chose to
show the diversity of models that one can build even from the very same data.

###  4.2 Bayesian Bootstrap

As this is usually the first step in a LQCD analysis, reproducing the
bootstrap is interesting. It has been noticed very early[22] that the
classical boostrap is closely related to a Bayesian model. This is actually
one of the simplest example of non-parametric model, whose graph is show in
Fig. 1(a). It is an extreme came of Dirichlet Process Mixture, where the base
functions are Dirac deltas. In Fig. 1(b) we show a comparison with a classical
bootstrap.

(a) Bayesian Bootstrap in plate notation

(b) Effective mass of a meson

Figure 1: Left: Bayesian Bootstrap model for the observed data $Y_{obs}$ made
of $N$ gauge configurations and $T$ time slices. The parameters $w$ to be
inferred are associated with a Dirichlet prior so that they act as
probabilities summing to one. Right: We show the estimate of the classical
bootstrap together with the Posterior Predictive Distribution of the Bayesian
bootstrap. The two methods are very similar but the Bayesian bootstrap is
expected to have a better convergence.

###  4.3 Infering a covariance

Performing a Generalised Least Square is a notoriously delicate task in LQCD.
Indeed, the true covariance is not directly accessible and the sample
covariance is a noisy estimate which can be ill-conditioned. In the limit when
the fit dimension becomes larger than the number of independent
configurations, the sample covariance is even guaranteed to be singular and
the $\chi^{2}$ can never be built. However, this error is usually neglected in
state-of-the-art analyses.

Regularisation (eg. Ledoit-Wolf) is a partial solution to this issue, but it
does not propagate uncertainties to the final result. Less importantly, it
does not allow a specific choice of model to guide the covariance posterior.

On the other hand, this is handled very naturally in a Bayesian framework, by
promoting the covariance into a model parameter as represented in Fig. 2(a).
Non-positive matrices are simply not allowed by the prior. Near-singular
matrices are suppressed by the large $\chi^{2}$. And finally a Wishart prior
(the conjugate for a Gaussian likelihood) provides shrinkage around its scale
matrix. Results are show in Fig. 2(b) with non-informative priors.

(a) Exponential model with unknown covariance. An uninformative LKJ prior is
used instead of a Wishart prior for practical reasons. One should in principle
check that the result here is independent of the choice of prior.

(b) HMC result for the correlation and comparison with empirical correlation

Figure 2: Left: Representation of a single-exponential model with unknown
covariance for $N=20$ bins and $T=28$ time slices. Top-right: we show the mean
and standard deviation of the posterior distribution of the correlation
matrix, then its inverse. Bottom-right: As a comparison we show the empirical
correlation we were starting with, its deviation from the mean of our HMC
results, and its ill-conditioned pseudo-inverse.

###  4.4 Multi-exponential model with correlations

Multi-exponential models with time correlations plus auto-correlations had
already been presented in [7] where we discuss a marginalisation strategy to
facilitate the sampling. Auto-correlations are implemented through an auto-
regressive process, bearing some similarities with the $\Gamma$ method.
Choosing as a first step a simplified VAR model in which the auto-correlation
modes are independent of time, the auto-regression reads

$y_{\tau}(t)=\rho_{0}+\sum_{i=1}^{r}\rho_{i}y_{\tau-i}(t)+\xi_{\tau}(t),\qquad\langle\xi_{\tau}(t)\xi_{\tau}(t^{\prime})\rangle\not=0.$
(3)

An example is shown in Fig. 3(a) and its results in Fig. 3(b).

(a) 3-exponential model with both autocorrelation (through the lag parameters
$\rho$) and time correlation (through the hyper-parameter $\nu$ of the
marginalised Wishart prior)

(b) HMC trace, where the value of each parameter is shown as a function of the
configuration number

Figure 3: Left: Model used where $Y_{obs}$ is the observed correlator on
$T=30$ time slices for $N=1009$ configurations. Middle: Posterior ditribution
of each free parameter (one colour per component when it is a vector). Good
signal is obtained on each parameter, and the three states are well-separated.
Right: MCMC trace corresponding to the posterior, where the x-axis runs over
Monte-Carlo samples and the y-axis represents the value taken by the
parameter.

###  4.5 Model averaging

A fully Bayesian model provides many ways to average models, including some
presented in [7]. A particularly interesting one is the construction of a
supermodel which contains all the base models. This can be obtained by adding
a discrete latent variable as in the model of Fig. 4, whose results are shown
in Fig. 5. This variable can trivially be marginalised if preferred.

This is a genuine BMA, based on marginal distributions, and avoids many
discussions on the choice of an Information Criterion and its asymptotic
validity. It also allows to sample all models at the same time and spend more
time on sampling the most relevant models, which means the computing cost
scales better than Akaike-based methods for large model sets.

Figure 4: Model used for the Bayesian Model Average, when $t_{min}$ is not
marginalised. The auxiliary parameters $\mu_{aux}$ describe a cut, while
multi-exponential $\mu_{model}$ is only used for times in the plateau
determined by $t_{min}$. This allows to fix the time extent of the observed
data $Y_{obs}$ and therefore make meaningful comparisons where the posterior
distribution for each sub-model (fixed $t_{min}$) is conditioned with the same
data.

(a) Posterior distribution and trace of $(E,t_{min})$. As we chose the non-
marginalised model, it is important to check that $t_{min}$ moves
sufficiently.

(b) Posterior distribution of $E$ conditioned with a given $t_{min}$, as a
violin plot

Figure 5: The bell-shaped posterior distribution of $E$ on the left is
directly obtained as a combination of all values of $t_{min}$, together with
the posterior probability of each $t_{min}$. One can however look at subsets
of the trace to construct conditional distributions as shown on the right, and
check that all subsets with high $t_{min}$ probability give compatible
results. This also makes it obvious that this method spends more time sampling
the most relevant $t_{min}$ sectors so that the result is smoother there.

###  4.6 Spectral function

As an alternative to multi-exponential inferences such as Sec. 4.4, one can
extract the whole spectral function. A model called BR has already been
presented by [8]. In Fig. 6 we propose a more efficient model based on
(Hilbert Space) Gaussian Processes. It keeps the positivity constraint of BR
while adding some automatically-adjusted smoothing radius. These models are
essentially an infinite sum of exponentials, but are more stable than multi-
exponential models because no free parameter appear in the exponential:
instead the exponentials are applied by a linear operator

${\cal L}\left[f\right](t)=\int K(t,\omega)f(\omega),\qquad
K(t,\omega)=\exp\left(-\omega t\right).$ (4)

As infinitely many energies are included (up to a large cutoff), such models
can typically be fitted on the whole correlator without having to discuss the
choice of a fit interval nor a truncation of the spectrum. Another consequence
is that the ground truth is always included in the space covered by the model:
this has to be compared to the situation of multi-exponential $\chi^{2}$ fits,
where the ground truth is almost-surely not included in a finite set of
truncated models and which tend to become numerically unstable as we approach
the ground truth.

If applied on lattice data in a single finite volume, this spectral function
will not have much physical meaning, but it already provides an interesting
black box to extract reliably a large number of eigenstates (defined for
instance as the modes of the spectral function, sample-per-sample).

(a) $\eta$ and $\ell$ are the parameters of the Matérn kernel for the GP
prior, which are approximated with a HSGP [23]. The correlator is
reconstructed with Simpson integration using the Laplace kernel $K$ and the
actual correlator is observed on $N=403$ configurations and $T=45$ time
slices.

(b) We show the Posterior Predictive Distribution of the spectral function,
where three finite-volume states are very clearly visible and can be measured
to high-precision by locating the maxima of the peaks for each MCMC sample. We
chose the median as central value.

Figure 6: We choose to apply the GP prior to the latent variable $log(f)$ so
that the spectral function $f$ is by construction positive everwhere. Using
constrains from both positivity and smoothness we obtain very precise results
for the spectral function.

##  5 Conclusion

We presented a well-defined bayesian framework to provide rigorous LQCD
results through the whole chain of analysis. Its practical applicability has
been demonstrated on a large range of models, covering most of the challenges
routinely encountered in LQCD. This brings together a large number of
seemingly unrelated issues, which have traditionally been tackled separately
by the community: resampling and non-gaussianity, $\chi^{2}$ fits,
regularisation of the GLS covariance, auto-correlations, model averaging,
inverse problems, …

We hope this could constitute a starting point for the community to explore
more complicated models and take advantage of these connections. We invite the
reader to have a look at the source code in [21]. Non-parametric models in
particular should be explored further, as the example of the spectral
functions shows that a paradigm shift could happen. This would be consistent
with the general trend of machine-learning to move towards universal
approximators.

Finally, we focused our discussion on the use of the HMC algorithm, but
variational methods would deserve deeper exploration as well, from SVI to
neural normalising flows. Each PPL usually allows several such alternatives to
be used interchangeably.

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