Stochastic Gradient Bayesian Optimal Experimental Designs
for Simulation-based Inference
Vincent D. Zaballa 1 Elliot E. Hui 1

arXiv:2306.15731v1 [cs.LG] 27 Jun 2023

Abstracttion known as the likelihood p(y|θ, ξ) that is used with theprior of the model parameters p(θ) to estimate the posterior probability of the model parameters given the observed
data, p(θ|y, ξ) (Cranmer et al., 2020). Recent SBI methods
have use deep learning-based models to infer either the intractable likelihood or posterior using density estimators forboth, or classifiers to estimate the likelihood-to-evidencep(y|θ,ξ)p(y,θ|ξ)ratio, p(θ|y,ξ)p(θ|ξ) = p(y|ξ) = p(θ)p(y|ξ) .

Simulation-based inference (SBI) methods tackle
complex scientific models with challenging inverse problems. However, SBI models often facea significant hurdle due to their non-differentiable
nature, which hampers the use of gradient-based
optimization techniques. Bayesian Optimal Experimental Design (BOED) is a powerful approach that aims to make the most efficient use ofexperimental resources for improved inferences.
While stochastic gradient BOED methods haveshown promising results in high-dimensional design problems, they have mostly neglected theintegration of BOED with SBI due to the difficult
non-differentiable property of many SBI simulators. In this work, we establish a crucial connection between ratio-based SBI inference algorithms
and stochastic gradient-based variational inference by leveraging mutual information bounds.
This connection allows us to extend BOED to SBI
applications, enabling the simultaneous optimization of experimental designs and amortized inference functions. We demonstrate our approach ona simple linear model and offer implementation
details for practitioners.

However, inferring the likelihood, posterior, or ratio is
a computationally expensive process that depends on observed data yo , to compute. Recent work questioned thevalidity of this expensive computational process used in SBI
if using the wrong simulator for the true data generating process (Cannon et al.). Naı̈ve conclusions can be made if using
the wrong model of the underlying scientific phenomenon,or the model is not close enough to the real data generatingprocess, which motivates the use of optimal experimental
designs in SBI methods.
Bayesian optimal experimental design (BOED) has shownpromise as a way to optimize experiments given a model, thesimulator, and priors of the parameters of interest. BOED
works by determining the information gain of a proposed
experimental design, ξ, on the parameters of the model ofinterest
IG(y, ξ) = H[p(θ)] − H[p(θ|y, ξ)].

(1)

The information gain can only be evaluated after an experiment but another quantity, the Expected Information Gain
(EIG), I(ξ), can be used as a proxy for the information
gained in an experiment

1. Introduction
Many scientific models are defined by a simulator that defines an output y determined by the inputs, or designs, to asystem, ξ, and parameters that define how the model transforms the inputs to outputs, θ. Inferring a distribution ofparameters given data p(θ|y, ξ) is of central importance inBayesian statistics and can be seen as a form of solving
the inverse problem for a given simulator(Lindley, 1972).
In SBI, a simulator forms an implicit probability distribu-

I(ξ) ≜ Ep(y|ξ) [H[p(θ)] − H[p(θ|y, ξ)]] ,(2)

The intuition behind this process is we must ask ourselves,
which experimental design and outcome would be mostsurprising given what we assume, or know, about the system
when conducting the experiment. This can be rewritten into
the form of calculating the mutual information between theobserved data and unknown parametersp(y|θ, ξ). (3)I(ξ) = MIξ (θ; y) = Ep(θ)p(y|θ,ξ) log
p(y|ξ)

1

Department of Biomedical Engineering, University of California, Irvine, Irvine, CA, USA. Correspondence to: Vincent Zaballa<vzaballa@uci.edu>, Elliot Hui <eehui@uci.edu>.
Published at the Differentiable Almost Everything Workshop of the
40 th International Conference on Machine Learning, Honolulu,
Hawaii, USA. July 2023. Copyright 2023 by the author(s).

Early BOED work focused on estimating the mutual information then using that estimate as the surrogate function in1

Stochastic Gradient Bayesian Optimal Experimental Designs for Simulation-based Inference

drawn from a prior θ ∼ p(θ), we can obtain an amortized likelihood pϕ (y|θ) or posterior pϕ (θ|y) by training
a neural density estimator, such as a normalizing flow, withparameters ϕ, or estimate the likelihood-to-evidence ratio
exp fϕ (θ, y) ≈ p(y|θ)p(y) , by training a classifier to distinguishparameters used to simulate an observed values, y. Different SBI methods can be used in inference for downstream
applications depending on the desiderata of the inferencetask. For example, one might use an amortized posterior
approximation if there are many different data samples toevaluate, whereas an ensemble of ratios (Hermans et al.) has
been shown to perform more robustly on Simulation-BasedCalibration (SBC) tests (Talts et al.) at the cost of increased
computational complexity.

an outer optimizer, such as Bayesian optimization (Foster
et al., 2019b; Kleinegesse & Gutmann, 2019). This double
loop of optimization was inefficient and lead to developmentof methods to simultaneously optimize the design and mutual information in a single optimization process. However,this unified optimization depended on an unnormalized likelihood and posterior approximation (Foster et al., 2019a) oran implicit likelihood with a simulator that has a differentiable functional form (Kleinegesse & Gutmann, 2021).
We present a method to simultaneously optimize designs
and the mutual information for the remaining set of models, implicit likelihoods without a differentiable simulator,
which are typically used in the SBI literature. We additionally make a link to how we can use a generative model inContrastive Precitive Coding. We show:

There are many SBI methods proposed for approximating
the likelihood, posterior, or ratio. We review the relevant
ones to our method here. See (Lueckmann et al., 2021) fora more thorough review and benchmark of SBI methods.

• A differentiable objective for simultaneously optimizing the mutual information and likelihood for SBIbased models.

Neural Likelihood Estimation We can use data fromthe joint distribution to train a conditional neural densitybased likelihood function. If we take a dataset of samples{yn , θn }1:N obtained from a simulator as previously described, we can train a conditional density estimator pϕ (y|θ)
to modelP
the likelihood by maximizing the total log likelihood of n log pϕ (yn |θn ), which is approximately equivalent to minimizing the loss

• A connection between Likelihood-Free based methods for BOED and contrastive ratio estimation (CRE)
methods for SBI models.
• Experimental validation of the unified objective on asimple linear model.

2. BackgroundL(ϕ) = Ep(θ) (DKL (p(y|θ)∥pϕ (y|θ)) + const,

Previous work in SBI methods have focused on improving
methods based on given, observed, data yo , (Papamakarios & Murray, 2016; Papamakarios et al., 2018; Durkan
et al., 2020; Greenberg et al., 2019) whereas BOED has
focused on determining an optimal design ξ ∗ , based on various bounds of MI between y and θ. While these aims seemto be unrelated, we will show how they can be performed
simultaneously for SBI methods that rely on potentiallystochastic simulators that act as black-box functions.

(4)

where the Kullback-Leibler divergence is minimized whenpϕ (y|θ) approaches p(y|θ). SBI methods would then condition this likelihood on observed data, yo , and refine thelikelihood estimate by resetting the prior to become the newposterior samples via Markov Chain Monte Carlo (MCMC)
sampling of the approximate likelihood p(θ) := p(θ|yo ) ∝pϕ (yo |θ)p(θ) and training a new neural density estimator of
the likelihood (Papamakarios et al., 2018; Lueckmann et al.,2018). This is Sequential Neural Likelihood (SNL) which
we forego as we focus on the preliminary step of optimizing
an experimental design without yo .

2.1. Simulation-Based Inference
In many scientific disciplines, it is desirable to infer a distribution of parameters θ, of a potentially stochastic model, or
simulator, given observations, yo . The closed-box simulator
may depend on random numbers z, such as in stochastic
differential equations, and previous experimental designs ξ,such that the simulator takes the form y = g(θ, ξ, z). Whena likelihood is not available, Approximate Bayesian Computation (ABC) methods can be used, (Sisson et al., 2018)
which aim to infer the likelihood of parameters of the simulator that are within an ϵ ball, Bϵ (y), of the observed data
y := yo , resulting in the likelihood p(∥y −yo ∥ < ϵ|θ). However, recent SBI methods have outperform ABC methods in
inference tasks (Lueckmann et al., 2021). By using a simulator to simulate the joint data distribution (θ, y) ∼ p(y|θ),2.2. Bayesian Optimal Experimental Design
Following from equation 3, (Foster et al., 2019a) proposed
the prior contrastive estimation (PCE) lower bound of theMI"#p(y|θ0 , ξ)
IP CE (ξ, L) ≜ E log 1 PL
, (5)ℓ=0 p(y|θℓ , ξ)
L+1

where the expectation is over p(θ0 )p(y|θ0 , ξ)p(θ1:L ) andξ is the proposed design, θ0 is the original parameter thatgenerated data y, and L is the number of contrastive samples. The PCE bound is appropriate in BOED when the2

Stochastic Gradient Bayesian Optimal Experimental Designs for Simulation-based Inference

prior and posterior are similar enough that p(θ) is a suitableproposal distribution for p(y|ξ). This bound has low variance but is upper-bounded by log L, potentially leading tolarge bias but still demonstrated adequate performance onvarious benchmarks. Unfortunately, this bound requires a
tractable likelihood function, which is not available in SBI
applications.

optimization"#pϕ (y|θ0 , ξ)E log 1 PL
+ λ · log pϕ (y|θ0 , ξ) , (7)ℓ=0 pϕ (y|θl , ξ)
1+L

where the expectation is over p(θ0 )p(y|θ0 , ξ)p(θ1:L ).
3.2. Connection to Generative MI Estimation

3. SBI-based BOED

The mutual information bound proposed by (Foster et al.,2019a) for PCE is similar to Contrastive Predictive Coding(CPC) (Poole et al., 2019; Oord et al., 2018), but wherea generative model replaces a discriminative one and therandom variable X corresponds to observed data and randomvariable Y to the prior distribution. In our formulation thebound of the MI depends on both the amount of training
tr → ∞ and number of contrastive samples L → ∞ toapproach the true MI. The generative approach to CPC canbe simplified as

3.1. Likelihood Free PCE
We take inspiration from previous SBI and BOED methods
to allow optimization of designs with respect to closed-box
simulators that are modeled using normalizing flows. We
start by noting how the loss function of contrastive ratio
estimation (CRE) (Durkan et al., 2020) lower bounds PCE
exp(fϕ (θ, y))exp(fϕ (θ, y))≤ log 1 PL
log PL
exp(f
(θ
,
y))
ϕ ℓ
ℓ=0 exp(fϕ (θℓ , y))
ℓ=0
1+L
= log

IP CE (ϕ) := EP [log pϕ (x|y) − log pϕ (x)],pϕ (y|θ0 , ξ),PL1ℓ=0 pϕ (y|θl , ξ)
1+L

where P is a random variable representing the joint distribution we obtain from our simulators (x, y) ∼ p(x|y)p(y)and pϕ (x) implicitly depends on the number of contrastive
samples L to approximate the marginal likelihood.

where L is the number of contrastive samples and fϕ isa discriminative classifier, which holds for a single batchof data and constant experimental design, i.e. when ξ isconstant. We exchange an explicit likelihood in PCE witha neural density estimator to create Likelihood-Free PCE(LF-PCE). We now have a MI lower bound#pϕ (y|θ0 , ξ),I(ξ, ϕ, L) ≥ E log 1 PLℓ=0 pϕ (y|θl , ξ)
1+L
"

(8)

4. Experimental Evaluation
4.1. Noisy Linear Model
We follow (Kleinegesse & Gutmann, 2020) and evaluate
optimal designs on a classic noisy linear model where a response variable y has a linear relationship with experimental
designs ξ, which is determined by values of the model parameters θ = [θ0 , θ1 ], which model the offset and gradient.
We would like to optimize the value of D measurements toestimate the posterior of θ, and so create a design vector
ξ = [ξ1 , . . . , ξD ]T . Each design, ξi returns a measurement
yi , which results in the data vector y = [y1 , . . . , yD ]T . We
assume non-Gaussian noise sources, otherwise evaluating
the posterior and MI would be trivial. We use a Gaussian
noise source N (ϵ; 0, 1) and Gamma noise source Γ(ν; 2, 2).
The model is then(6)

where the expectation is over p(θ0 )p(y|θ0 , ξ)p(θ1:L ). We
now can simultaneously optimizes designs and parametersof a neural density estimator. If we are to use a normalizing
flow as exp fϕ (y, θ, ξ) = pϕ (y|θ, ξ), then the PCE lower
bound of the MI holds since the distribution is normalized
as normalizing flows are bounded functions (Papamakarios
et al., 2019). We note that this can be an unstable objective
as the data distribution of the flow will change as experimental designs change. However, the result is that it returns
an amortized likelihood that can be evaluated on observed
experimental data to return a posterior density or used indownstream inference algorithms, such as SNL. Finally,using a normalizing flow allows us to take gradients withrespect to designs ξ, which we derive in Appendix A.

y = θ0 1 + θ1 ∗ ξ + ϵ + ν,

(9)where ϵ = [ϵ1 , . . . , ϵD ]T and ν = [ν1 , . . . , νD ]T are i.i.d.
samples. We evaluate LF-PCE on this model and examine
how changing the λ regularization parameter in (7) influences the resulting mutual information bound and design
quality for both models.

Practical implementation of LF-PCE loss For LF-PCE
training, stability of the density estimator is a challenge
when optimizing the MI lower bound. To address this, we
added a regularization term, λ, to both loss functions to help
stabilize the training of the density estimator during design

For each design dimension, D, we randomly initialize designs ξ ∈ [−10, 10]. For LF-PCE, we chose N = 10, thenumber of non-contrastive samples y ∼ p(y|ξ, θ0 ), and3

Stochastic Gradient Bayesian Optimal Experimental Designs for Simulation-based Inference

Figure 1. Comparison of the EIG across design dimensions, type of BOED, and λ regularization for the noisy linear model examining themoving average over N=10 samples. For the single design dimension, LF-PCE with no λ regularization outperforms in estimating a lower
bound of the MI, which can translate to more informative experimental designs. In the higher-dimension design cases, LF-PCE increases
its EIG with more designs, which is expected, but sees diminishing returns when expanding from 10D to 100D design evaluations. In
the 100-dimensional design case, we see the benefit of using λ regularization to stabilize the training of a neural density estimator inhigh-dimensional input space at the cost of slightly lower EIG.

M = 50 contrastive samples for all experiments. For theneural spline flow, we chose 5 bijector layers, each with
4 bins, and 4 resnet multilayer perceptrons, each with 128
dimensions, for the neural network-based conditional networks. For both the neural density estimator’s parameters ϕ,and the designs ξ, we use the Adam optimizer (Kingma &Ba, 2014) with β1 = 0.9 and β2 = 0.99, with learning rate
α = 1e−3 for the neural density estimator and α = 1e−2
for design optimization.

Design Dimension
D=1
D=10
D=100

θ0

θ1

1.29 ± 2.98
0.07 ± 1.40
1.35 ± 0.52

5.20 ± 0.41
4.87 ± 0.16
4.81 ± 0.20

Table 1. Posterior estimates mean and 68% interquartile range after observing ξ ∗ values for each design dimension only using theamortized likelihood approximation provided by the neural density
estimator used in the LF-PCE training. The held-out parameter values that were used to generate yo were θ true = [2, 5]. More design
dimensions approach the true held-out parameter with increasing
precision.

Examining the graph of the mutual information in Figure 1,
we see that LF-PCE lower bound steadily increases for all
values of lambda; however, the stability of the optimizationof the generative model’s parameters diverges in higher
design dimensions whenever λ = 0. We see a general trend
between exploration and exploitation in changing valuesof λ, where higher λ values lead to lower MI lower-boundestimates and potentially more homogenous designs.

5. Discussion
We demonstrated a novel information bounds, ILF −P CE ,to perform gradient-based BOED using black-box simulators present in many SBI applications and obtained lower
bounds of the EIG on a toy model across a range of experimental design dimensions to showcase its scalability.
Optimizing designs in SBI applications provides a valuable
preconditioning step to typical sequential SBI methods suchas SNL that are based on observed experimental designs.
Sidestepping Bayesian optimization can also help to accelerate model testing and feedback from real-world data. Future
work will examine the tradeoff between design diversity for
improved entropy reduction and neural density estimator robustness, similar to the exploration and exploitation tradeoff
present in Bayesian optimization.

Using LF-PCE we obtain an amortized neural density estimator of the likelihood that is able to perform inference on
observed data evaluated at the optimal design. For example,p(θ|yo , ξ ∗ ) ∝ pϕ (yo |θ, ξ ∗ )p(θ) by MCMC sampling. We
evaluate the posterior densities after optimizing on the LFPCE lower bound in Appedix B and can see the mean andinterquartile range in Table 1. We note that we were able
to arrive at accurate and precise posterior estimates using
the neural density estimator that simultaneously optimized
an optimal design ξ ∗ , without any post-processing such as
using SNL or Sampling Importance Resampling.

Acknowledgements
This research was funded by the National Institute of General Medical Sciences (NIGMS) of the National Institutes
4

Stochastic Gradient Bayesian Optimal Experimental Designs for Simulation-based Inferenceof Health (NIH) under award number 1F31GM145188-01.
We would like to thank Adam Foster and members of the
Elowitz Lab for helpful discussions.

Kleinegesse, S. and Gutmann, M. U. Bayesian experimental
design for implicit models by mutual information neural estimation, February 2020. URL http://arxiv.
org/abs/2002.08129. arXiv: 2002.08129 Publication Title: arXiv.

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Stochastic Gradient Bayesian Optimal Experimental Designs for Simulation-based Inference

A. Design Gradients of LF-PCE
For LF-PCE, we need unbiased gradient estimators of the information bounds. A normalizing flow can be seen as areparameterized distribution, which allows for calculating the gradient with respect to designs ∇ξ f −1 (u; θ, ξ). In practice,since we are evaluating the log probability of a data point, we would actually evaluate the inverse direction of a flow∇ξ f (y; θ, ξ) at the base distribution pu (u), which is usually a Gaussian distribution and evaluated by maximum likelihood.
More formally, following equation 4, the gradient with respect to ξ is∇ξ L(ξ) ≈ −X1∇ξ
log pu (f −1 (yn ; ϕ, θ, ξ) + log| det J (f −1 )(yn ; ϕ, θ, ξ)|),
N
n

(10)

which is tractable as long as we can compute f −1 , its Jacobian determinant, and evaluate the base density, pu (u), which is
tractable for a base Gaussian distribution. Given this gradient, we can plug this into the gradient of LF-PCE to estimate thegradient of the information bound:∂ILF −P CE∂g∂= Ep(θ0 )p(y|θ,ξ)q(θ1:L |y)+g·log pϕ(y|θ0 , ξ) ,
(11)
∂ξ
∂ξ
∂ξ
where
g(y, θ0:L , ϕ, ξ) = log

1
L+1

pϕ (y|θ0 , ξ).
PLℓ=0 pϕ (y|θℓ , ξ)

(12)

B. Evaluation of Linear Model Designs and Posteriors

We evaluated the efficacy of the neural density estimator trained using the LF-PCE loss function to infer a held out true
parameter value in Figure 2 by MCMC. We provide a quantitative evaluation of the posteriors in Table 1. The posteriors canbe improved by computationally efficient methods such as Sampling Importance Resampling, or used in SBI algorithms thatuse sequential methods to refine the neural density estimator.

Figure 2. Comparison of the prior density the posterior achieved by the different design dimensional normalizing flows evaluated at an
optimal design p(θ|yo , ξ ∗ ) ∝ pφ (yo |θ, ξ ∗ )p(θ). The red cross denotes the true model parameters.

C. Evaluation of Posterior Predictive Distribution
As a reference, we plot the prior and posterior predictive plots for the 1-dimensional optimal design in Figure 3. An insight
into the optimal experimental design problem is that the designs closer to where the prior distribution has more noise willlead to more clarification in a performed experiment, which is why the most optimal designs will be at the boundaries forthe noisy linear model.

7

Stochastic Gradient Bayesian Optimal Experimental Designs for Simulation-based Inference

Figure 3. Prior predictive (blue) and posterior predictive (orange) distributions with the ground truth liner model (dotted red) for the single
design case where D=1.

8