Multi-ﬁdelity Monte Carlo: a pseudo-marginal approach
Anonymous Author(s)
AfﬁliationAddress email
Abstract
Markov chain Monte Carlo (MCMC) is an established approach for uncertaintyquantiﬁcation and propagation in scientiﬁc applications.
A key challenge in applying MCMC to scientiﬁc domains is computation: the target density of interest isoften a function of expensive computations, such as a high-ﬁdelity physical simulation, an intractable integral, or a slowly-converging iterative algorithm.
Thus, usingan MCMC algorithms with an expensive target density becomes impractical, asthese expensive computations need to be evaluated at each iteration of the algorithm.
In practice, these computations often approximated via a cheaper, low-ﬁdelity computation, leading to bias in the resulting target density.
Multi-ﬁdelity MCMCalgorithms combine likelihoods of varying ﬁdelities in order to obtain an approximate target density with lower computational cost.
In this paper, we describe aclass of asymptotically exact multi-ﬁdelity MCMC algorithms for the setting wherea sequence of likelihoods of increasing ﬁdelity can be computed that approximatesthe high-ﬁdelity likelihood.
We take a pseudo-marginal MCMC approach for multiﬁdelity inference that utilizes a low-ﬁdelity unbiased estimator of the high-ﬁdelitylikelihood constructed via randomized truncation of a telescoping series of thelow-ﬁdelity sequence of models.
Finally, we discuss and evaluate the proposedmulti-ﬁdelity MCMC approach on several applications, including log-Gaussian
Cox process modeling, Bayesian ODE system identiﬁcation, PDE-constrainedoptimization, and Gaussian process regression parameter inference.
Introduction
Simulation and computational modeling play a key role in science, engineering, economics andmany other areas.
When these models are high-quality and accurate, they are important for scientiﬁcdiscovery, design and data-driven decision making.
However, the ability to accurately modelcomplex physical phenomena often comes with a signiﬁcant cost—many models involve expensivecomputations that then need to be evaluated repeatedly in, for instance, a sampling or optimizationalgorithm.
Examples of model classes with expensive computations include intractable integrals orsums, expensive quantum simulations [42], expensive numerical simulations arising from partialdifferential equations (PDEs) [37], and large systems of ordinary equations (ODEs).
In many situations, one has the ability to trade off computational cost against ﬁdelity or accuracy in theresult.
Such a tradeoff might arise from the choice of discretization or the number of basis functionswhen solving a PDE, or the number of quadrature points when estimating an integral.
Sometimes it ispossible to leverage lower-ﬁdelity models to help accelerate high-quality solutions, e.g., by usingmultigrid methods [22] for spatial discretizations.
More generally, multi-ﬁdelity methods combinemultiple models of varying cost and ﬁdelity to accelerate computational algorithms and have beenapplied to solving inverse problems [10, 23, 37], trust region optimization [1, 4, 14, 30, 38], Bayesianoptimization [7, 20, 26, 27, 40, 44], Bayesian quadrature [16, 45], and sequential learning [21, 34].
[Figure]
One critically important tool for scientiﬁc and engineering computation is Markov chain Monte Carlo(MCMC), which is widely used for uncertainty quantiﬁcation, optimization, and integration.
MCMCmethods are recipes for constructing a Markov chain with some desired target distribution as thelimiting distribution.
One uses pseudo-random numbers to simulate transitions of the Markov chainin order to produce samples from the target distribution.
However, MCMC often becomes impracticalfor high-ﬁdelity models, where a single step of the Markov chain may, for instance, involve anumerical simulation that takes hours or days to complete.
Multi-ﬁdelity methods for MCMC focuson constructing Markov chain transition operators that are sometimes able to use inexpensive lowﬁdelity evaluations instead of expensive high-ﬁdelity evaluations.
The goal is to increase the effectivenumber of samples generated by the algorithm, given a constrained computational budget.
A largefocus of the multi-ﬁdelity MCMC literature is on two-stage Metropolis-Hastings (M-H) methods[9, 13], which use a single low-ﬁdelity model for early rejection of a proposed sample, thereby oftenshort-circuiting the evaluation of the expensive, high-ﬁdelity model.
However, there are several limitations of two-stage multi-ﬁdelity Monte Carlo.
First, in manyapplications, a hierarchy of cheaper, low-ﬁdelity models is available; for instance, in the case ofintegration, k -point quadrature estimates form a hierarchy of low-ﬁdelity models, and in the case of a
PDE, varying the discretization.
Thus, the two-stage approach does not fully utilize the availabilityof a hierarchy of ﬁdelities and may be more suitable for settings where the high- and low-ﬁdelitymodels are not hierarchically related, e.g., semi-empirical methods vs. Hartree-Fock in computationalchemistry.
In addition, in such applications, there is often a limiting model of interest, such as acontinuous function that the low-ﬁdelity discretizations approximate.
Two-stage MCMC does notasymptotically sample from this limiting target density and will at best sample from an approximationof the biased, high-ﬁdelity posterior.
Finally, the two-stage method is unnatural to generalize to moresophisticated MCMC algorithms such as slice samplers and Hamiltonian Monte Carlo (HMC).
We propose a class of multi-ﬁdelity MCMC methods designed for applications with a hierarchy oflow-ﬁdelity models available.
More speciﬁcally, we assume access to a sequence of low-ﬁdelitymodels that converge to a “perfect-ﬁdelity” model in the limit.
Within an MCMC algorithm, wecan approximate the perfect-ﬁdelity target density with an unbiased estimator constructed from arandomized truncation of the inﬁnite telescoping series of low-ﬁdelity target densities.
This class ofmulti-ﬁdelity MCMC is an example of a pseudo-marginal MCMC algorithm—the unbiased estimatoressentially guarantees that the algorithm is asymptotically exact in that it eventually generates samplesfrom the perfect-ﬁdelity target distribution.
Our approach introduces the ﬁdelity of a model as anauxiliary random variable that is evolved separately from the target variable according to its ownconditional target distribution; this technique can be used in conjunction with any suitable MCMCupdate that leaves the conditional update for the target variable of interest invariant, such as MH,slice sampling, elliptical slice sampling, or Hamiltonian Monte Carlo.
We apply pseudo-marginalmulti-ﬁdelity MCMC to several problems, including a log-Gaussian Cox process, Bayesian ODEsystem identiﬁcation, PDE-constrained optimization, and Gaussian process parameter inference.
Related work.
Multi-ﬁdelity MCMC methods are typically applied in a two-stage scenario wherethe goal is to reduce the computational cost of using a single expensive high-ﬁdelity model by using acheap low-ﬁdelity model as a low-pass ﬁlter for a delayed acceptance/rejection algorithm [9, 10, 13];see Peherstorfer et al.
[35] for a survey.
Higdon et al.
[23] propose coupling a high-ﬁdelity Markovchain with a low-ﬁdelity Markov chain via a product chain.
In constrast, our approach aims to sample from a “perfect-ﬁdelity" target density while reducing computational cost; two-stage MCMCalgorithms result in biased estimates with respect to this target density.
A related class of methods ismultilevel Monte Carlo [12, 17, 18, 43], which uses a hierarchy of multi-ﬁdelity models for Monte
Carlo estimation by expressing the expectation of a high-ﬁdelity model as a telescoping sum oflow-ﬁdelity models.
Dodwell et al.
[12] use the M-H algorithm to form the multilevel Monte Carloestimates, simulating from a separate Markov chain for each level of the telescoping sum.
In practicemultilevel Monte carlo requires choosing a ﬁnite number of ﬁdelities, inducing bias in the estimatorwith respect to the (limiting) perfect-ﬁdelity model.
In contrast, our method uses a randomizedﬁdelity within a single Markov chain with the perfect-ﬁdelity model as the target.
Our approach applies pseudo-marginal MCMC to multi-ﬁdelity problems.
There is a rich literaturedeveloping pseudo-marginal MCMC methods [2, 6] for so-called “doubly-intractable” likelihoods,likelihoods when the likelihood is intractable to evaluate.
There are several approaches in the pseudomarginal MCMC literature are particular relevant to our work.
The ﬁrst is are PM-MCMC approachesintroduced by Lyne et al.
[29], which describes a class of pseudo-marginal M-H methods that use
Russian roulette estimators to obtain unbiased estimators of the likelihood.
However, this methodsamples the variable of interest jointly with the auxillary randomness, which can lead to sticking.
Alternatively, several methods have considered sampling the randomness separately.
The idea ofclamping random numbers is explored in depth by Andrieu et al.
[3] and Murray and Graham [32];the latter applies to this pseudo-marginal slice sampling.
In particular, our approach applies theseideas for the speciﬁc setting of multi-ﬁdelity models, where the ﬁdelity is treated as an auxillaryvariable.
Finally, while our approach applies to doubly-intractable problems, we are also motivatedby a larger class of multi-ﬁdelity problems studied in the computational sciences that may not evenbe inference problems, such as quantum simulations and PDE-constrained optimization.
Multi-ﬁdelity MCMC
Monte Carlo methods approximate integrals and sums that can be expressed an expectation:
[Equation]
where θ ( t ) ∼ π, (1) and where π : Θ → R + is the target density , h : Θ → R is a function of interest, and θ ( t ) aresamples from π .
Markov chain Monte Carlo methods are then used to generate samples θ ( t ) from πby simulating from a Markov chain with target π .
In many settings, pointwise evaluations of the target function π ( θ ) are expensive or even intractable;from here on we will assume that the goal is to compute statistics with respect to a perfect-ﬁdelitytarget density π ∞ ( θ ) of interest.
In practice, the estimate in Equation (1) is instead estimated withrespect to a cheaper, low-ﬁdelity density π k ( θ ) , where k ∈ N := { 1 , 2 , . . . } .
In particular, weconsider settings where there is a sequence of low-ﬁdelity densities available that converge to thetarget, i.e., π k ( θ ) k →∞ −→ π ∞ ( θ ) , where we assume that as k increases, the model becomes higher inﬁdelity (with respect to π ∞ ) but more costly to evaluate, increasing in expense super-linearly with k .
For instance, π ∞ could represent a target density that depends on an intractable integral, the solutionof a PDE, the solution of a large system of ODEs, the solution of a large system of linear equations, orthe minimum of a non-convex function.
Thus, a typical evaluation of π ∞ requires an approximationat a ﬁdelity k with a tolerable level of bias for a given computational budget.
Here increasing k couldcorrespond to ﬁner discretizations of differential equations, increasing numbers of quadrature points,or performing a larger number of iterations in a linear solver or optimization routine.
In the multi-ﬁdelity setting, the goal is to combine multiple ﬁdelity models within an MCMCalgorithm to reduce the computational cost of estimating Equation (1).
In this paper, we describea class of MCMC algorithms that leverage the sequence of low-ﬁdelity models π k .
Our strategyfor multi-ﬁdelity MCMC (MF-MCMC) will be to construct an unbiased estimator of π ∞ ( θ ) usingrandom choices of the ﬁdelity K and then to include K in the Markov chain as an auxiliary variable.
By carefully constructing such a Markov chain, it will be possible to asymptotically estimate thefunctional in Equation (1) as though the samples were taken from the perfect-ﬁdelity model; eachstep of the Markov chain will nevertheless only require a ﬁnite amount of computation.
Finally, our approach allows us to essentially plug in any valid MCMC algorithm, and we apply this strategy todevelop multi-ﬁdelity variants of a number of MCMC algorithms, such as M-H and slice sampling.
2.1 Pseudo-marginal MCMC for the multi-ﬁdelity setting
Pseudo-marginal MCMC [2, 6] is a class of auxillary-variable MCMC algorithms that replaces thetarget density
π ( θ ) with an estimator ˆ π ( θ ) that is a function of a random variable.
If the estimator isnonnegative and unbiased, i.e., for all θ ∈ Θ , ˆ π ( θ ) ≥ 0 and E [ˆ π ( θ )]
= π ( θ ) , then MCMC transitionsthat use the estimator still have π ( θ ) as their invariant distribution.
This property is sometimes referredto as “exact-approximate” MCMC as the transitions are approximate but the limiting distributionis exact.
Estimators can be constructed from a variety of methods, including particle ﬁltering [2];our approach will use randomized series truncations, which has been consider in pseudo-marginal
MCMC methods such as Lyne et al.
[29] and Georgoulas et al.
We now apply the pseudo-marginal approach to the multi-ﬁdelity inference setting.
Here the targetdensity estimator arises from a random choice of the ﬁdelity K ∈ N that is governed by a distributionµ on N .
We denote the estimator using ˆ π K ( θ ) to make the dependence on the random ﬁdelity Kexplicit.
The estimator is constructed such that it is unbiased with respect to µ , i.e.,
[Equation]
The distribution µ is also constructed by the user: ideally, the estimator ˆ π K ( θ ) will prefer smallervalues of K while having sufﬁciently low variance as to allow the Markov chain to mix effectively.
Thus the simulations can be run at inexpensive low-ﬁdelities, while the estimates will be as though itthe perfect-ﬁdelity model were being used.
The standard pseudo-marginal MCMC approach is to construct a Markov chain with that has thefollowing joint density as its stationary distribution:
[Equation]
Observe that while Equation (3) does not depend on the perfect-ﬁdelity target density π ∞ , it returnsthe desired marginal π ∞ via Equation (2).
As a concrete example, a pseudo-marginal M-H algorithmgenerates a new state θ (cid:48) and ﬁdelity K (cid:48) jointly using q ( θ (cid:48) ; θ ) as the proposal for θ (cid:48) , q ( K (cid:48) ; K ) =
µ ( K (cid:48) )
as the proposal distribution for the ﬁdelity, and accepts/rejects the state according to
[Equation]
where the equality holds since the distribution terms for K and K (cid:48) cancel.
Note that the righthandside of Equation (4) is the standard M-H ratio but that the target π is replaced with the estimator ˆ π K .
However, the standard pseudo-marginal MCMC approach can “get stuck” when the estimator is noisyand fail to accept new states.
Thus, we apply the approach in Murray and Graham [32] that augmentsthe Markov chain to include the randomness of the estimator; here the randomness of the estimatorarises from the ﬁdelity K .
Concretely, we construct a Markov chain that simulates from Equation (3)by alternating sampling between the conditional target densities π ( K | θ ) and π ( θ | K ) (steps 5 and6 of Algorithm 1, respectively).
We refer to this strategy as multi-ﬁdelity MCMC (MF-MCMC),since by conditioning on K
= k , the update for the state θ becomes a standard deterministic updateapplied to a low-ﬁdelity model
ˆ π k ( θ ) , and any appropriate MCMC update can be used here, makingit straightforward to use complex MCMC methods, such as slice sampling and HMC.
Similarly, anysuitable MCMC update for the ﬁdelity K can be used using the conditional target π
( K | θ ) .
Many techniques can be used to construct an unbiased estimator of π ∞ with randomness K ; wedescribe a general approach in the next section.
However, it is generally difﬁcult to guarantee theestimator is nonnegative, as required by pseudo-marginal MCMC.
One technique considered by
Lin et al.
[28] and Lyne et al.
[29] is to instead sample from the target distribution induced by theabsolute value of the estimator and applying a sign-correction to the ﬁnal Monte Carlo estimate in
Equation (1), an approach borrowed from the quantum Monte Carlo literature where it is necessaryfor modeling fermionic particles.
This approach has been applied to the M-H algorithm, but we notethat this general approach can be applied much more broadly, as we do in this work.
In problems where the estimator may be negative, we sample from the conditional target distributionsusing the absolute value of the estimator | ˆ π K ( θ ) | , which we denote by ˜ π
( K | θ ) ∝ µ ( K )
| ˆ π k ( θ ) | and˜ π ( θ | K = k ) ∝ | π k ( θ ) | .
The estimate in Equation (1) is then corrected using the signs σ ( θ, k ) ,
[Equation]
where { ( K ( t ) , θ ( t ) ) }
Tt =1 are the sampled values from the joint distribution ˜
π ( θ, K ) ∝ | ˆ π K ( θ )
| µ ( K ) .
Importantly, the sign-corrected estimate still leads to the asymptotically unbiased estimate of functional of interest.
Let σ ( θ, k ) denote the sign of the estimator such that ˆ π k ( θ )
=
σ ( θ, k ) | ˆ π k ( θ ) | .
Theestimator ˆ I T in Equation (5) is formed using a Monte Carlo estimate of the functional after expandingit into its joint distribution, i.e.,
[Equation]
The full multi-ﬁdelity MCMC algorithm with sign correction summarized in Algorithm 1.
We notethat while the Markov chain no longer converges to the target π ∞ but instead an absolute valuetarget density, the ﬁnal estimate after sign-correction – which is the downstream goal of interest – isunbiased due to Equation (6).
While this may seem limiting if one is interested in the posterior itself,useful unbiased posterior summaries may be still be obtained via the functional, such as the posteriormean, variance, quantiles, and histograms that may be used to visualize marginal distributions.
Unbiased low-ﬁdelity estimators via randomized truncations
In this section, we discuss how to construct an unbiased estimator of π ∞ ( θ ) , given a sequence oflow-ﬁdelity likelihoods with the property π k ( θ ) → π ∞ ( θ ) as k → ∞ .
This estimator has the propertythat it requires a ﬁnite amount of computation with probability one, and it also has a tunable amountof expected computation per estimate, i.e., it uses low-ﬁdelity density evaluations to estimate theperfect-ﬁdelity target density.
The central idea of this estimator has been used for decades, goingback to John von Neumann and Stanislaw Ulam.
More recently it has found use in related work suchas Lyne et al.
[29], Beatson and Adams [5], Jacob et al.
[25], and Glynn and Rhee [19].
First notethat we can express the perfect-ﬁdelity model as a telescoping sum: let π 0 ( θ ) = 0 and write
[Equation]
The estimator ˆ π K is then constructed by taking a random truncation K ∼ µ of the inﬁnite telescopingseries.
The sampled terms in the sum are then reweighted to ensure the estimator remains unbiased:
[Equation]
Two approaches are commonly used to ensure that the resulting estimator is unbiased: weightedsingle-term estimators and Russian roulette estimators.
The single-term estimator [29] is constructedby importance sampling a term from the series in Equation (7): the truncation level is drawn as
K ∼ µ , and the K th term is used to form the estimate ˆ π K ( θ )
= µ ( K ) − 1 ( π K ( θ ) − π K − 1 ( θ )) .
Thus, the weight in Equation (8) is W k,K = µ ( K ) − 1 1 ( K = k ) .
In the Russian rouletteestimator, the remaining terms in the estimator are reweighted by their survival probabilities,i.e., W k,K =
(1 − (cid:80)
k − 1 k (cid:48)
=1 µ ( k (cid:48) )) − 1 1 ( K ≥ k ) .
The distribution µ controls the number of terms inthe estimator, and a good proposal distribution should match the tails of the sequence of low-ﬁdelitydensities [5, 29, 36].
The ability to use cheaper models is a key feature of multi-ﬁdelity inference, and the low-ﬁdelityestimator provides a means to reduce the computational cost of multi-ﬁdelity Monte Carlo.
However,these estimators are an example of a class of methods that explores a compute-variance tradeoff:computationally cheaper estimates leads to high variability.
The resulting increase in variance slowsdown the convergence of the MCMC procedure and could lead to an overall less efﬁcient method dueto a reduced effective sample size.
1: Input : Initial state θ and ﬁdelity K , truncation distribution µ 2: for t = 1 , . .
, T do 3:
Given current K and θ , form estimator ˆ π K ( θ )
=
(cid:80)
Kk
=1 w k,K
( π k ( θ ) − π k − 1 ( θ )) 4:
Save sign σ ( θ, K ) =
sign (ˆ π K ( θ )) 5:
Update ﬁdelity K leaving invariant the target conditional ˜ π
( K | θ ) ∝ µ ( K ) | ˆ π K ( θ ) | 6:
Update state θ leaving invariant the target conditional ˜ π
( θ | K = k ) ∝ | ˆ π k ( θ ) | 7: end for 8: Output : Samples { ( θ ( t ) , K ( t ) ) } and estimate ˆ I T = (cid:16)(cid:80)
Tt =1 σ ( t ) h
( θ ( t ) )
(cid:17) / (cid:16)(cid:80)
Tt =1 σ ( t ) (cid:17)
Summary of the multi-ﬁdelity MCMC recipe
Here we summarize the recipe for constructing a multi-ﬁdelity Markov chain Monte Carlo algorithm.
First, identify a sequence of increasing-ﬁdelity target densities with the property that their limit isthe desired “perfect-ﬁdelity” density.
Low-ﬁdelity densities should be cheap with the cost rapidlyincreasing within the sequence.
In the context of Bayesian inference, it may be appropriate to focusthe multi-ﬁdelity aspects on the likelihood term and construct the target densities via, e.g., π k
( θ ; D )
∝π
( θ ) L k ( θ ; D ) , where π 0 is the prior, L k is a low-ﬁdelity likelihood, and D are the observations.
This likelihood-based version is what we use in several of the experiments.
Next, introduce a truncation distribution µ on N .
This truncation distribution should be chosen tobalance between expected cost and variance of the resulting estimator; our overall goal is to mostlyuse cheap low-ﬁdelity densities, but high-variance estimators will presumably damage the mixingtime and/or the asymptotic variance.
Initialize the Markov chain with a reasonable choice for θ and a draw of K from the distribution µ .
Each step of the Markov chain simulation consists of an update to θ given K and an update of Kgiven θ .
The update of θ given K can be performed using any standard MCMC algorithm, e.g., M-H,slice sampling, or HMC, applied to the low-ﬁdelity estimator.
It is important to use the absolutevalue of the estimator and keep track of its sign.
The update of K given θ is also ﬂexible, but it isreasonable to construct the update so that only a few K are considered in each step, as each of thoseﬁdelities will need to be evaluated.
By default, we consider a simple random walk on the positiveintegers for our experiments.
After running a sufﬁcient number of steps of the Markov chain, use thesign corrected-estimator in Equation (5) to compute the expectation of the function h ( θ ) .
Experiments
In all experiments, we use a random-walk M-H update to sample from the conditional K | θ , andtruncation distribution µ ( K ) =
geometric
( K ; γ 0 ) .
Additional experimental details are in Appendix F.
5.1 Toy conjugate Gaussian models
We ﬁrst examine a toy example that does not involve any high- or low-ﬁdelity computation (i.e., thecomputational cost does not increase with k ): consider a high-ﬁdelity model L ∞ ( θ ) =
N ( x ; θ, σ ∞ )and a low-ﬁdelity model L k ( θ ) =
N ( x ; θ, σ k ) , where σ 2 k → σ 2 ∞ .
Here we we consider the sequenceσ 2 k
= 1 + 2 /k 2 and σ 2 ∞ = 1 .
In Figure 2 we compare the performance of single-ﬁdelity andmulti-ﬁdelity M-H and slice sampling.
We also compare to the two-stage M-H algorithm summarizedin Appendix C.4; here we consider 2 two-stage setups of k = { 1000 , 10 } and k = { 100 , 5 } .
Thehistograms show the bias of each method after simulating 10,000 samples, and the solid gray curvedenotes the exact posterior density.
The number of cost-adjusted likelihoods was computed bymultiplying each likelihood evaluation by the ﬁdelity.
Here we see that the multi-ﬁdelity methodsconverge to the same value as the high-ﬁdelity but in fewer cost-adjusted likelihood evaluations.
5.2 Log-Gaussian Cox processes
We examine an application of MF-MCMC to the log Gaussian Cox process (LGCP) model [31], wherethe perfect-ﬁdelity model is a function of an integral and the lower-ﬁdelity sequence of models arisesfrom k -point quadrature estimates.
Let log f ∼ GP(0 , k θ ) , where k ( x, x (cid:48) )
=
exp (cid:0) − 12 (cid:96) 2 (cid:107) x − x (cid:48) (cid:107) 22 (cid:1) ,
[Figure]
where (cid:96) is a lengthscale hyperparameter.
Consider an inhomogenous Poisson process on X ⊆ R Dwith intensity λ ( x ) =
e f ( x ) .
Given a random set of points { X 1 , . . .
, X N } , where X n ∈ X ,
[Equation]
Typically, inference in the LGCP uses a grid-based approximation of Equation (9), where the pointsare binned into counts and each bin has a Poisson likelihood [11, 33, 41]; this approximation resultsin a biased posterior.
Because the likelihood depends on a high-dimensional latent Gaussian vector,we perform inference for f using the elliptical slice sampling (ESS) algorithm; see Appendix C.3 fora summary of the method.
We approximate the integral in Equation (9) with a trapezoidal quadraturerule I k , where the number of quadrature points is a linear function of k .
We apply multi-ﬁdelity and single-ﬁdelity ESS algorithms to a coal mining disasters data set Carlinet al.
The data contain the dates of 191 coal mine explosions that killed ten or more men in
Britain between 15 March 1851 and 22 March 1962.
Figure 3 (left) shows the estimated meanintensity and standard deviation on coal mining disasters data between one run of multi-ﬁdelity ESSand two single-ﬁdelity ESS runs on a high-ﬁdelity ( k = 1000 quadrature points) and low-ﬁdelity( k = 10 quadrature points) setting.
In this plot, the high- and multi-ﬁdelity means and standarddeviation estimates match well, and the bias in the low-ﬁdelity estimate is quite noticable.
We alsocomputed the cost-adjusted number of likelihood evaluations performed in each iteration of MF-ESSand SF-ESS.
Figure 3 (right) shows the average estimated mean intensity at the time step t =
1880on the three models plotted against the average cost-adjusted number of likelihood evaluations periteration.
Here we can see observe that the multi-ﬁdelity and high-ﬁdelity estimates are quite closeafter many iterations of sampling, but that the multi-ﬁdelity estimate converges with less computation.
[Figure]
5.3 Bayesian ODE system identiﬁcation
We now apply the MF-MCMC approach to Bayesian system identiﬁcation.
Consider the classical
Lotka-Volterra ODE problem.
Let u ( t ) ≥ 0 represent the population size of prey species at time t ,and v ( t ) ≥ 0 represent the population size of the predator species.
The dynamics of the two speciesgiven parameters α, β, γ, δ ≥ 0 are given by a pair of ﬁrst-order ODEs:
[Equation]
System identiﬁcation solves the inverse problem by identifying the parameters of the ODE systemθ =
( α, β, γ, δ ) .
Taking a Bayesian approach, we specify a noise model for the observed data andpriors on the parameters, and we use MCMC to infer a distribution over the solution.
For simplicity,we assume that the initial conditions are known and ﬁx σ = 0 .
Deﬁne z n := ( u ( t n ) , v ( t n )) and let z 1 ( θ ) , . . .
, z N ( θ ) be the solutions to the Lotka-Volterra differential equations given θ at times t 1 , . . .
, t N given the initial conditions and the system parametersθ =
( α, β, γ, δ ) .
Suppose we have observations arising from log( y n )
= log( z n ) + (cid:15) n , where(cid:15) ∼ N (0 , σ 2 I ) .
The low-ﬁdelity likelihood is a function of a numerical solution of the ODEusing a time step of size dt (Equation (F3)).
We compared the performance of a multi-ﬁdelityelliptical slice sampler to two single-ﬁdelity (HF and LF) ellipitical slice samplers with step sizedt = 1 × 10 − 5 , 1 × 10 − 4 .
For the ODE solver, we considered both an Euler solver and an 4th-order
Runge-Kutta solver.
Results for the Euler solver are in Figure 4, and results for the Runge-Kuttasovler are in Figure F3.
In the posterior mean estimates of the parameters, the multi-ﬁdelity slicesampler lead to better estimates in less time than the single-ﬁdelity samplers.
5.4 PDE-constrained optimization
We now consider global optimization of a PDE-constrained objective, where an expensive physicalsimulation is run repeatedly in an outer loop problem.
A common approach for global optimizationis simulated annealing, which has been applied to constrained global optimization [39].
Consider a
[Figure]
model for heat ﬂow in a thin rod of length L with spatial coordinates x ∈ [0 , L ] .
Let u ( x, t ) representthe temperature in the rod at position x and time t , and let ¯ u represent a desired target temperature.
The goal is to minimize a loss function subject to u satisfying a linear heat equation.
This objectivealong with an initial condition and homogenous boundary conditions can be summarized as:minimize u
[Equation]
subject to ∂u
[Equation]
where α, β > 0 are the system parameters.
The goal is to ﬁnd θ =
( α, β ) that minimizes the objectiveand satisﬁes the constraints.
To solve the PDE, we discretize the domain into a grid of size ∆ x andrepresent the second derivative using the central difference formula.
This gives us a system of ODEsthat we solve numerically using a Tsitouras 5/4 Runge-Kutta method, setting ∆ t = 0 .
4∆ x 2 so as tosatisfy a CFL stability condition.
Here the ﬁdelity of the problem is given by the size of the spatialdiscretization ∆ x , which in turn controls the discretization of ∆ t .
We compared against two single-ﬁdelity discretizations of the spatial coordinate, where ∆ x 2 = 5 ×10 − 3 , 1 × 10 − 2 .
The results are in Figure 5, where we plot two of the MF results with γ 0 = 0 .
Discussion and future work
In this work, we introduced a class of multi-ﬁdelity MCMC that uses a low-ﬁdelity unbiasedestimator to reduce the computational cost of sampling while still maintaining the desired limitingtarget distribution of the Markov chain.
In particular, we have demonstrated the use of our frameworkon more advanced MCMC algorithms beyond M-H, such as slice sampling, and to additional settingssuch as optimization.
Our results show a reduction in computation while producing accurate solutionsin comparison with high-ﬁdelity models when it is possible to construct a target estimator that isnot too noisy.
Many future directions remain.
First, applying MF-MCMC to large-scale expensiveapplications has many computational challenges.
Making the method more robust to specializedproblems is important, especially if the estimator is heavy-tailed.
Thus, constructing good proposaldistributions matching properties of the low-ﬁdelity target sequence is crucial, especially for theapplication to high-dimensional problems.
In addition, we have thus far focused on target densitieswhere there is a single computation whose ﬁdelity is varied.
However, in many settings, there may betarget densities with multiple computations that converge at different rates, for example, if the targetdensity includes both an intractable integral and a solution of a linear system.
Our framework canbe extended to that setting by adjusting the proposal distribution, and it is useful to understand howthese rates impact the convergence properties of the sampler. Checklist
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