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Bayesian Experimental Design for Implicit Models by Mutual Information Neural Estimation

Steven Kleinegesse, Michael U. Gutmann

arXiv:2002.08129v3stat.MLcs.LGstat.COstat.ME

TL;DR

For implicit models, difficult posterior estimation and unavailable design gradients make MI-based experimental design challenging. MINEBED optimizes a neural lower bound on MI to determine designs and estimate the posterior; simulations report suitable designs and reasonable posterior distributions at higher design dimensions.

  • Problem

    For implicit models, posteriors are hard to estimate and sample from, and the relationship between design variables and MI is unknown, with gradients generally unavailable.

    Source: §1 · pp. 1–2

  • Method

    MINEBED uses a neural network to maximize a lower bound on MI, jointly determining experimental designs and estimating the posterior.

    Source: §1 · p. 2

  • Results

    Simulation studies found suitable and intuitive experimental designs and reasonable posterior distributions, even at higher design dimensions.

    Source: §4 · p. 9

  • Takeaways & Limitations

    The posterior estimate is available alongside the optimized design, without a separate likelihood-free inference step.

    Source: §1 · p. 2

    The pharmacokinetic example assumes shared parameters across patients and models only the marginal concentration at design time, not each patient's time course.

    Source: §3.2 · p. 6

Abstract

from arXiv · show

Implicit stochastic models, where the data-generation distribution is intractable but sampling is possible, are ubiquitous in the natural sciences. The models typically have free parameters that need to be inferred from data collected in scientific experiments. A fundamental question is how to design the experiments so that the collected data are most useful. The field of Bayesian experimental design advocates that, ideally, we should choose designs that maximise the mutual information (MI) between the data and the parameters. For implicit models, however, this approach is severely hampered by the high computational cost of computing posteriors and maximising MI, in particular when we have more than a handful of design variables to optimise. In this paper, we propose a new approach to Bayesian experimental design for implicit models that leverages recent advances in neural MI estimation to deal with these issues. We show that training a neural network to maximise a lower bound on MI allows us to jointly determine the optimal design and the posterior. Simulation studies illustrate that this gracefully extends Bayesian experimental design for implicit models to higher design dimensions.

1. Introduction

Bayesian experimental design seeks measurements that provide the most information about model parameters, but for implicit models the posterior is difficult to estimate and the design–MI relationship lacks accessible gradients. MINEBED addresses these challenges by maximizing a neural lower bound on MI and also estimating the posterior.

1. Introduction

  • Experimental designs can affect data collection, and mutual information (MI) is used in Bayesian experimental design to value the expected uncertainty reduction in model parameters.MI depends on the full posterior and is sensitive to nonlinear correlations and multimodality that other utility functions may not capture.Source: §1 · p. 1
  • For implicit models, posteriors are hard to estimate and sample from, while the relationship between design variables and MI is unknown and generally lacks available gradients.These obstacles have motivated alternative utilities and gradient-free optimization approaches, which generally scale poorly as design dimensionality increases.Source: §1 · pp. 1–2
  • MINEBED uses neural MI estimation to maximize a lower bound rather than spending resources on accurately estimating MI at non-optimal designs.The authors state that, for a large class of implicit models, the bound can be tightened and maximized by gradient ascent.Source: §1 · p. 2
  • After design optimization, MINEBED also provides an approximate posterior, avoiding a separate likelihood-free inference step.Source: §1 · p. 2

2. The MINEBED Method

MINEBED optimizes a neural lower bound on mutual information to select experimental designs for implicit models and can also estimate the posterior. It uses pathwise gradients when design derivatives are available, with Bayesian optimization as a less scalable fallback when they are not.

2. The MINEBED Method

  • MINEBED maximizes a neural MI lower bound to find experimental designs and estimate the posterior for implicit models.The method uses the MINE-f lower bound; the paper does not claim this bound is superior to other MI lower bounds.Source: §2 · p. 2

2.1. Experimental Design with Gradients

  • Pathwise differentiation through the sampling path supplies design gradients for joint ascent on the network and designs when the model Jacobian or its automatic-differentiation product is available.The paper identifies differentiable observation processes and models implemented in differential programming frameworks as applicable cases.Source: §2.1 · p. 3

2.2. Experimental Design without Gradients

  • When design gradients cannot be computed or reasonably approximated, Bayesian optimization is a fallback, at the expense of reduced scalability.The described procedure uses a Gaussian-process surrogate with a Matérn-5/2 kernel and expected improvement, retraining the neural network at each design evaluation.Source: §2.2 · p. 4

2.3. Estimating the Posterior Distribution

  • Once the MI lower bound is tight, the learned network relates to the posterior and prior, allowing posterior estimation after design optimization without additional work.Prior samples can be reweighted for amortized posterior sampling, or the posterior density can be evaluated for use with MCMC.Source: §2.3 · p. 4

3. Experiments

The experiments apply the approach to noisy linear, pharmacokinetic, and gas-leak models. They examine optimal designs and resulting parameter posteriors across different numbers of measurements.

3. Experiments

  • The experiments use gradient ascent for the noisy linear and PK models, then a gas-leak simulator without sampling-path gradients to demonstrate design optimization in that setting.The gas-leak model also tests a ten-dimensional design with five measurements.Source: §3 · p. 4, §3.3 · p. 8

3.1. Noisy Linear Model

  • For D = 1 in the noisy linear model, the optimal design is d* = −10, and the MI lower bound converges near 2.5, close to its numerical reference.The design is at the boundary, where the linear model has a larger signal-to-noise ratio.Source: §3.1 · p. 5, Figure 1 · p. 5
  • At D = 100, the noisy linear model converges to an MI lower bound with only small bias from the reference, but its optimal designs avoid the boundaries.The D = 10 design strategy does not transfer optimally to D = 100; measurements should instead focus around zero.Source: Figure 1 · p. 5, §3.1 · p. 6

3.2. Pharmacokinetic Model

  • For one PK measurement, the MI lower bound converges near 1 with negligible bias from the reference, but the posterior estimates ke relatively poorly.At the selected low sampling time, the data determine ka/V relatively accurately, while the posterior mean is closer to truth for V than for ka.Source: §3.2 · p. 7, Figure 3 · p. 7, Figure 4 · p. 7
  • With ten PK measurements, the designs form three clusters, and all parameters are estimated more precisely and accurately than with one measurement.Early, middle, and late clusters help identify ka/V, the third parameter, and reduce the effect of ka, respectively.Source: §3.2 · p. 7, Figure 4 · p. 7

3.3. Locating a Gas Leak

  • In the gas-leak experiment, optimal measurements lie upstream near grid edges, with an additional interior site to help localize the leak.The simulator lacks sampling-path gradients, so the design is optimized with Bayesian optimization.Source: §3.3 · pp. 8–9, Figure 5 · p. 8
  • With five gas measurements, the posterior is more concentrated than with one, and one of its two modes covers the true leak location.The bimodality arises from uncertainty in wind direction; with wind direction known, the posterior is unimodal.Source: Figure 6 · p. 8, §3.3 · p. 9

4. Conclusions

The paper presents MINEBED, which maximizes a neural-network lower bound on mutual information to find experimental designs and their corresponding posteriors for implicit models. Simulation studies find suitable designs and reasonable posteriors at higher design dimensions.

4. Conclusions

  • MINEBED uses one neural network to maximize a mutual-information lower bound and determine both optimal designs and corresponding posterior distributions.The approach applies when the data-generating distribution is intractable but sampling is possible.Source: §4 · p. 9
  • When design gradients are available, gradient ascent updates network parameters and designs together, enhancing scalability.Without design gradients, the network parameters can still be updated by gradient ascent while designs are optimized with methods such as Bayesian optimization.Source: §4 · p. 9
  • Simulation studies found suitable, intuitive experimental designs and reasonable posterior distributions, including at higher design dimensions.Source: §4 · p. 9
  • The authors identify sequential experimental design, model discrimination, and future-observation prediction as directions for further investigation.Source: §4 · p. 9

Supplementary Material for Bayesian Experimental Design for Implicit Models

The supplement describes a reference mutual-information calculation for the noisy linear model. It approximates the likelihood and uses a sample average to estimate mutual information for independent measurements.

A.1. Reference MI Calculation

  • The supplement's reference calculation approximates mutual information using nested Monte Carlo and an approximation to the likelihood at the reference design.Source: §A.1 · p. 12
  • For D independent measurements, the likelihood factorizes across measurements, and a sample average of the marginal likelihood is used in the mutual-information approximation.Source: §A.1 · p. 12
  • For the noisy linear model, a kernel density estimate from 50,000 noise samples approximates the noise density used to estimate mutual information.The calculation uses N = 5,000 and M = 500.Source: §A.1 · p. 12

A.2. Hyper-Parameter Optimisation

The hyperparameter study selects a one-hidden-layer network size by comparing validation estimates of the mutual-information lower bound. The reported best size for the 10-dimensional linear model is 150 hidden units.

A.2. Hyper-Parameter Optimisation

  • The study trained one-hidden-layer networks with 50, 100, 150, 200, or 250 hidden units for 50,000 epochs.It used learning rates lψ = 10^-4 and ld = 10^-2, then evaluated the lower bound on a validation set.Source: §A.2 · p. 12
  • For the 10-dimensional linear model, validation scores are summarized as means and standard deviations across one-layer networks with different hidden-unit counts.Source: Figure 7 · p. 12
  • In the separate convergence plot, the 10-dimensional linear-model design dimensions end in three clusters, while the one-dimensional design follows its own trajectory.Source: Figure 8 · p. 12
  • H = 150 is the most appropriate tested network size for maximizing the mutual-information lower bound in the 10-dimensional linear model.Source: §A.2 · p. 12, Figure 7 · p. 12

A.3. Convergence of Designs

The convergence studies examine designs and posterior densities for linear models across design dimensions. In the 100-dimensional case, the reported design outperforms several reference strategies, while the posterior comparison shows narrower modes closer to the true parameters as design dimensions increase.

A.3. Convergence of Designs

  • The one-dimensional design converges quickly, while the 10-dimensional dimensions converge more slowly into three clusters: boundary designs for estimating slope and near-zero designs for estimating offset.Source: §A.3 · p. 13
  • In the 100-dimensional model, the mutual-information lower bound converges smoothly above the 1- and 10-dimensional values and close to the reference MI.The run used a five-layer network with 50 hidden units per layer and 10,000 samples.Source: §A.4 · p. 13, Figure 9 · p. 13
  • As design dimensions increase from 1 to 100, posterior densities become narrower, with modes closer to the true parameters.Source: Figure 10 · p. 13, §A.4 · p. 13
  • The 100-dimensional design dimensions converge within [-8, 8], with more centered near zero, unlike the 1- and 10-dimensional designs that mostly lie at -10 and 10.Source: §A.4 · p. 13
  • MI = 5.78 exceeded boundary-only (4.61), boundary-plus-zero (4.83), equally spaced (4.91), and random designs (mean 4.92, SD 0.08; maximum 5.11), supporting a locally optimal 100-dimensional design.Source: §A.4 · p. 13

B.1. Reference MI Calculation

The PK example uses a nested Monte Carlo approximation to obtain a reference MI at the optimal design. Although MI is intractable, the conditional data-generating distribution is available because both noise sources are Gaussian.

B.1. Reference MI Calculation

  • A nested Monte Carlo approximation of MI provides a reference value at the optimal PK design d* = [t*1, ..., t*D].The calculation uses N = 5,000 and M = 500.Source: §B.1 · p. 14
  • The PK model's conditional data-generating distribution p(yj | tj, θ) can be written explicitly because both noise sources are Gaussian.Here θ = [ka, ke, V].Source: §B.1 · p. 14
  • The numerical MI approximation uses the conditional distribution and a sample average of p(y | d*).The reported sample sizes are N = 5,000 and M = 500.Source: §B.1 · p. 14

B.2. Hyper-Parameter Optimisation

For the PK model, validation scores select a neural-network hyperparameter combination, and the optimized designs converge at different rates for one and ten dimensions. The ten-dimensional design spans early, middle, and late measurement times.

B.2. Hyper-Parameter Optimisation

  • H = 300 and λ = 0.8 achieve the highest mean validation score among the tested PK-model network sizes and learning-rate multipliers.The comparison evaluates one-layer ReLU networks over the specified H and λ ranges.Source: §B.2 · p. 14

B.3. Convergence of Designs

  • The one-dimensional design converges after about 2,000 epochs, while the ten-dimensional design takes roughly 30,000 and clusters at early, middle, and late measurement times.The one-dimensional design settles at a low measurement time; the clusters match those found by Ryan et al. (2014).Source: §B.3 · p. 14, Figure 12 · p. 14
  • For one measurement, the low-time design effectively removes the elimination-rate effect and estimates the ratio ka/V.For ten measurements, late times reduce the effect of ka, while middle times help identify the remaining parameter.Source: §B.3 · p. 14

B.4. Full Posterior Plots

The PK posterior plots show tighter marginal distributions and joint-posterior modes closer to the true parameters for ten measurements than for one. In the gas leak model with five measurements, the highest validation score occurs at H = 200.

B.4. Full Posterior Plots

  • For the PK model, D = 10 yields narrower marginals and joint-posterior modes closer to the true parameters than D = 1; ka and V remain correlated, so their ratio is estimated relatively well.The true parameter vector is θtrue = [1.5, 0.15, 15].Source: §B.4 · pp. 14–15, Figure 13 · p. 15, Figure 14 · p. 15

C.1. Hyperparameter Optimisation

  • H = 200 achieves the highest validation score among the tested one-layer network sizes for the gas leak model with D = 5 measurements.The mean MI lower-bound estimates and standard deviations are shown in Figure 15.Source: §C.1 · p. 15, Figure 15 · p. 15

C.2. Additional Plots

The additional gas leak plots show Bayesian-optimization evaluations converging to a high-MI region and compare source-location posteriors under a known wind direction. With five measurements and wind fixed at 45°, the posterior is unimodal.

C.2. Additional Plots

  • For one gas leak measurement, Bayesian-optimization evaluations quickly converge to the MI lower-bound optimum in the grid's northeast corner.Figure 16 displays the GP posterior mean and standard deviation, with the evaluations marked on the surface.Source: §C.2 · p. 15, Figure 16 · p. 16
  • With wind direction fixed at 45°, the five-measurement source-location posterior is unimodal, unlike the case where wind direction is marginalized out.Figure 17 compares posterior densities for one and five measurements and marks the true source location.Source: §C.2 · p. 15, Figure 17 · p. 16

D. Comparison with Bayesian D-Optimality

The section compares mutual information and Bayesian D-Optimality for choosing measurement times to estimate frequency in an oscillatory model. Their optimal times differ, reflecting how each utility treats posterior multimodality.

D. Comparison with Bayesian D-Optimality

  • The comparison uses noisy measurements of sin(ωt) to estimate frequency ω, with measurement time t as the design variable.The setup sets σ_noise=0.1, ω_true=0.5, and a uniform prior over [0, π].Source: D. Comparison with Bayesian D-Optimality · p. 16
  • MINEBED converged to t*=2.19 versus t*=1.66 for BD-Opt, and took significantly less time to converge.BD-Opt favours smaller measurement times.Source: D. Comparison with Bayesian D-Optimality · p. 16
  • Mutual information accounts for multiple explanations of observations, whereas BD-Opt uses the covariance matrix and penalizes multimodality that increases posterior variance.The two utility functions therefore favour subtly different measurement times.Source: D. Comparison with Bayesian D-Optimality · p. 16
  • BD-Opt posterior modes are closer together than MINEBED modes, consistent with its preference for posteriors with smaller variances.Figure 18 compares the posterior densities for MINEBED data (top) and BD-Opt data (bottom) against reference computations and the true parameter value.Source: D. Comparison with Bayesian D-Optimality · p. 17, Figure 18 · p. 17

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