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Bayesian Optimization for Likelihood-Free Inference of Simulator-Based Statistical Models

Michael U. Gutmann, Jukka Corander

arXiv:1501.03291v3stat.MLstat.COstat.ME

TL;DR

Simulator-based models enable realistic generative mechanisms but have intractable likelihoods, making likelihood-free inference computationally expensive. The paper combines probabilistic discrepancy modeling with Bayesian optimization, and reports reductions in required simulations by several orders of magnitude. The approach is most suitable when simulations are costly and modal, low-discrepancy regions matter more than accurately approximating small-probability tails.

  • Problem

    Simulator-based models have intractable likelihoods, while likelihood-free inference requires repeated simulations and lacks knowledge of the parameter–discrepancy relationship.

  • Method

    The paper uses regression to model discrepancies from actively acquired training data, implementing the strategy with Bayesian optimization.

  • Results

    The approach reduces the number of required simulator calls by several orders of magnitude, including about 600 times fewer calls for the Ricker-model posterior estimate after 150 acquired data points.

  • Takeaways & Limitations

    Bayesian optimization can substantially accelerate likelihood-free inference by focusing simulations and modeling discrepancies smoothly.

  • Takeaways & Limitations

    The approach can underrepresent likelihood or posterior tails because it prioritizes low-discrepancy regions, and requires modification when small-probability events are primary.

Abstract

from arXiv · show

Our paper deals with inferring simulator-based statistical models given some observed data. A simulator-based model is a parametrized mechanism which specifies how data are generated. It is thus also referred to as generative model. We assume that only a finite number of parameters are of interest and allow the generative process to be very general; it may be a noisy nonlinear dynamical system with an unrestricted number of hidden variables. This weak assumption is useful for devising realistic models but it renders statistical inference very difficult. The main challenge is the intractability of the likelihood function. Several likelihood-free inference methods have been proposed which share the basic idea of identifying the parameters by finding values for which the discrepancy between simulated and observed data is small. A major obstacle to using these methods is their computational cost. The cost is largely due to the need to repeatedly simulate data sets and the lack of knowledge about how the parameters affect the discrepancy. We propose a strategy which combines probabilistic modeling of the discrepancy with optimization to facilitate likelihood-free inference. The strategy is implemented using Bayesian optimization and is shown to accelerate the inference through a reduction in the number of required simulations by several orders of magnitude.

1. Introduction

Simulator-based models support realistic generative mechanisms but make likelihood-based inference intractable. The paper addresses the resulting computational burden by combining discrepancy modeling with Bayesian optimization.

  • Simulator-based models are parametrized stochastic data-generating mechanisms that can be sampled exactly despite having unknown analytical density forms.They are also called implicit or generative models and can represent realistic scientific hypotheses without analytically tractable probability densities.
  • Likelihood-based inference is unavailable because the density shape and, unlike unnormalized models, its scaling factor are unknown.Consequently, likelihood-free methods designed for unnormalized models do not apply directly.
  • Likelihood-free methods identify parameters by seeking simulated data that resemble the observations, but they face discrepancy-assessment and simulation-cost difficulties.The cost reflects repeated simulation and limited knowledge of how parameters affect discrepancies.
  • The proposed BOLFI strategy combines probabilistic discrepancy modeling with optimization through Bayesian optimization.It targets the computational difficulty of likelihood-free inference.
  • BOLFI reduces the required number of simulations by several orders of magnitude and thereby accelerates inference substantially.

2. Examples of Simulator-Based Statistical Models

The paper presents three simulator-based examples spanning an artificial normal model and realistic ecological and epidemiological systems. These examples illustrate latent variables, nonlinear dynamics, and snapshot observations that complicate inference.

  • Examples: Three examples include a normal distribution, the ecological Ricker model, and bacterial infections in day care centers.The first is artificial; the latter two come from real-data analysis with intractable models.
  • Example 1: Normal distribution: In the normal example, fixing the random quantities makes simulated data a deterministic transformation of the parameter.The same fixed-randomness principle applies to more general simulators.
  • Example 2: Ricker model: The Ricker model combines a latent stochastic nonlinear autoregressive time series with a Poisson observation model.Its parameters govern growth rate, innovation variability, and observation scaling; inference is difficult because the latent states are unobserved.
  • Example 3: Bacterial infections in day care centers: The bacterial-infection model uses a latent continuous-time Markov chain and an observation model to represent transmission dynamics in day care centers.
  • Example 3: Bacterial infections in day care centers: Infection dynamics include clearance, external infection, and infection under co-colonization, with rates combining outside and within-day-care probabilities.
  • Example 3: Bacterial infections in day care centers: Inference from single-time-point attendee snapshots is difficult because the continuous-time process contains infinitely many correlated unobserved variables.

3. Inference Methods for Simulator-Based Statistical Models

Simulator-based inference replaces the unavailable likelihood with approximations built from discrepancies between observed and simulated summary statistics. These approximations support posterior inference but face computational costs, finite-simulation variability, nonsmoothness, and bandwidth or threshold trade-offs.

  • General properties: Inference methods compare observed and simulated summary statistics through a discrepancy and use it to construct an approximate likelihood.Simulated summary statistics are generated by simulating data and reducing them to selected features.
  • General properties: Summary statistics reduce dimensionality and filter information, but the resulting likelihood targets the summaries rather than the full data unless they are sufficient.The summary-based likelihood is valid for inference given the summaries, not necessarily for inference given the original observations.
  • General properties: The analytical form of the summary-statistic likelihood remains unknown, so practical approximations replace intractable expectations with finite simulation averages.The paper distinguishes finite-resource approximations ˆL from limiting approximations ˜L obtained with infinitely many computational resources.
  • Computational issues: For finite N, reducing simulation cost can decrease estimate accuracy and make approximate likelihoods nonsmooth, complicating gradient-based optimization.The paper notes that small-N stochastic gradients are unreliable and require small stepsizes, making optimization costly.
  • Bayesian optimization: Bayesian optimization models the discrepancy probabilistically and selects new evaluations by balancing exploration against exploitation.The approach updates its belief after each new evaluation and uses the model to guide efficient minimization.
  • Nonparametric approximation: Small thresholds can make nonparametric likelihood estimation computationally expensive because accepted discrepancies become vanishingly rare.Although the limiting approximation becomes proportional to the likelihood for small nh, estimating it by sample averages requires many simulations.
  • Nonparametric approximation: Average discrepancies and kernel-based approximate likelihoods can have optima at approximately the same parameter values, while the discrepancy objective avoids choosing a kernel bandwidth.The paper presents this alignment as advantageous for the bandwidth-independent objective ˆJ_N.

4. Computational Difficulties in the Standard Inference Approach

The standard approach faces computational and statistical difficulties because approximate likelihoods are stochastic, may be nonsmooth, and require costly simulations across parameter values.

  • Approximate likelihoods behave like stochastic black-box functions because simulations approximate intractable expectations, and each evaluation requires simulating N data sets.
  • Reducing N lowers computational cost but can decrease estimate accuracy, creating a trade-off between computational and statistical efficiency.
  • For finite N, approximate likelihoods may lack smoothness, complicating their optimization.
  • Uniformly simulating N data sets across parameter space wastes resources, especially when individual simulations are expensive.

5. Framework to Increase the Computationally Efficiency

The proposed framework replaces uniform sample averaging with regression and active data acquisition, using Bayesian optimization to focus simulations where discrepancies are small.

  • The framework combines optimization with probabilistic modeling to increase the efficiency of likelihood-free inference.
  • From Sample Average to Regression Based Approximations: Regression approximates intractable expectations underlying computable likelihood approximations, rather than relying only on sample averages.
  • From Sample Average to Regression Based Approximations: The discrepancy Δθ is modeled as the response variable, with parameters θ serving as covariates in the regression function J(θ).
  • Bayesian Optimization: Training data are actively concentrated near the minimizer of J(θ), combining regression and optimization to reduce unnecessary simulations.
  • Bayesian Optimization: Bayesian optimization acquires new evidence while fitting a nonlinear regression model for the black-box discrepancy function.
  • Benefits and Limitations: The approach can share observed discrepancies across parameter values under a smoothness assumption about J, reducing the simulations needed for a target accuracy.
  • Benefits and Limitations: It targets low-discrepancy regions directly, which is especially useful when simulating data sets is time consuming.
  • Benefits and Limitations: The training-set size becomes the computational–statistical trade-off, allowing accuracy to vary across parameter-space regions.

6. Implementing the Framework with Bayesian Optimization

The framework models the discrepancy as a noisy Gaussian process and uses Bayesian optimization to select informative parameter evaluations. Its acquisition rule balances exploiting low predicted discrepancy against exploring uncertain regions.

  • Bayesian optimization: Bayesian optimization models an evaluable black-box objective probabilistically, then updates that model after each selected evaluation.The paper applies this paradigm to discrepancy functions in likelihood-free inference.
  • Gaussian-process model: The discrepancy is modeled with a Gaussian process having a prior mean, covariance function, and additive Gaussian observation noise.The paper uses squared exponential covariance functions and either constant or quadratic prior means.
  • Implementation choices: The Gaussian-process implementation learns hyperparameters by maximizing leave-one-out log predictive probability, without a systematic comparison to alternative learning methods.The authors report satisfactory results but note that Bayesian hyperparameter learning is an alternative.
  • Gaussian-process model: The posterior mean provides a surrogate for the objective, while posterior variance quantifies uncertainty at each parameter value.Both quantities are available analytically from the Gaussian-process posterior.
  • Acquisition rule: The lower confidence bound acquisition function combines posterior mean and variance to trade off exploitation against exploration.Minimizing the posterior mean favors currently promising points, whereas posterior variance favors uncertain points.
  • Acquisition rule: A stochastic acquisition rule samples around the acquisition minimizer to avoid repeatedly selecting one point and to improve spatial exploration.Initialization uses a small set of points, with Sobol sequences offered as a more even-spreading option than uniform pseudorandom numbers.

6.2 Inferring the Regression Function of the Discrepancies

Bayesian optimization learns the discrepancy regression function by modeling observed discrepancy values and actively selecting new parameter locations. Log-Gaussian modeling and stochastic acquisition improve exploration and approximation stability in the illustrated examples.

  • Discrepancy modeling: For nonnegative discrepancies, the paper considers modeling either ∆θ directly or its logarithm, with both approaches working well in practice.The direct model estimates J with the posterior mean, while the log model uses the corresponding log-normal mean.
  • Gaussian example: In the Gaussian-mean example, the discrepancy is the squared difference between empirical and observed means, modeled with a Gaussian process.The example uses a constant prior mean and a deterministic acquisition rule.
  • Ricker model: For the Ricker model, individual synthetic-likelihood realizations are noisy at N = 50, whereas their averages are smoother and optimized in the appropriate region.Bayesian optimization approximates the limiting negative log synthetic likelihood from noisy evaluations.
  • Ricker model: The Ricker illustration uses up to T = 150 training tuples, including nine initialization points, to construct the approximation.The synthetic likelihood was computed using code that returned averaged log-likelihood estimates rather than the individual discrepancies.
  • Procedure: Bayesian optimization estimates the discrepancy regression function by acquiring new evaluations according to the current probabilistic belief.The posterior is updated after each acquisition using Bayes’ theorem.
  • Ricker model: Log-Gaussian modeling improves exploration and approximation, while stochastic acquisition spreads points more evenly and produces more stable, accurate approximations.Deterministic acquisition can produce little spatial exploration or clustered acquisitions.

6.3 Model-Based Nonparametric Likelihood Approximation

The learned discrepancy model yields a computable likelihood approximation that can support several inference procedures without further simulator runs. In the day care example, this approach reduces simulation cost substantially, while scalability and modeling assumptions remain important boundaries.

  • Likelihood approximation: The probabilistic discrepancy model produces a computable likelihood approximation whose evaluation requires no additional simulator runs.Its tractable posterior mean and variance also permit derivative computation.
  • Likelihood approximation: The approximate likelihood can support maximization, curvature-based error bars, hybrid Monte Carlo, or importance-sampling inference.The paper uses iterative importance sampling with a Gaussian-mixture auxiliary density.
  • Computational savings: 50 simulations were required for an entire likelihood curve in the model-based approach, versus 300 simulations for one sample-based likelihood evaluation.Across a 50-point grid, the model-based result therefore required 300 times fewer simulations.
  • Computational savings: In the day care example, each model-based likelihood curve used 50 simulations, compared with 15,000 for the standard sample-average approach.This yields a factor of 300 in computational savings.
  • Limitations and extensions: The straightforward Gaussian-process implementation has O(t^3) matrix-inversion complexity, limiting acquisitions to a few thousand.Sparse Gaussian processes and basis-function regression can reduce the complexity to O(tm^2).
  • Limitations and extensions: Higher-dimensional likelihood-free inference requires more training data and makes both regression and acquisition-function optimization more difficult.High-dimensional Bayesian optimization commonly relies on structural assumptions such as low-dimensional or additive objectives.

7. Applications

Applications to the Ricker and day care infection models show that BOLFI can recover informative posterior results while substantially reducing simulator use. The examples also illustrate how acquired data reveal parameter correlations and how model-based posteriors compare with reference methods.

  • Overview: BOLFI reduces the required number of simulations by several orders of magnitude in applications to both complete models.The applications are the Ricker model and the bacterial infection model for day care centers.
  • Ricker model: As Ricker training data increase, the learned regression function indicates weaker confinement along σ and negative correlation between log r and φ conditional on σ.The estimated optimum location changes less than the regression-function shape between t = 50 and t = 500.
  • Ricker model: In the Ricker model, 150 acquired data points already produce a posterior solution matching the MCMC reference.The model-based posterior is obtained through iterative importance sampling without further simulator runs.
  • Ricker model: The largest model-based versus MCMC posterior differences occur for σ, whose model-based posterior mean is closer to the data-generating value σo = 0.3.For the other parameters, model-based posterior means are also closer to ground truth than the MCMC means.
  • Day care model: In the day care model, the estimated regression function moves the optimal co-infection parameter away from the boundary as training data accumulate.Its isocontours become oblique, indicating negative conditional correlations among all three parameters.
  • Day care model: The day care model-based posterior means quickly match reference results, but its credibility intervals are broader because inference focuses on the modal region.The broader posteriors may serve as auxiliary densities for iterative ABC methods.

8. Conclusions

The paper frames BOLFI as a computational strategy for likelihood-free inference and situates it among Gaussian-process approaches to modeling likelihoods, discrepancies, and simulator outputs.

  • BOLFI models discrepancies with a Gaussian process and actively acquires training data where discrepancies tend to be small.The posterior Gaussian-process model is used to approximate the intractable likelihood.
  • Modeling discrepancies differs from modeling the likelihood because it can delay kernel and bandwidth selection and use all simulated data.
  • BOLFI differs from Meeds and Welling’s approach in both the modeled quantities and the use of Bayesian optimization to design Gaussian-process training data.
  • Earlier computer-code research modeled deterministic simulator outputs with Gaussian processes to construct emulators for computationally heavy calculations.
  • The basic Bayesian optimization algorithm is not a limitation of likelihood-free inference; alternative and more advanced algorithms fit the proposed framework.
  • High-dimensional likelihood-free inference and acquisition rules tailored to discrepancy modeling remain future research directions.The paper focused on approximating modal areas of the intractable likelihood.

Appendix A. Proof of Proposition 1

The appendix derives Proposition 1 by decomposing the objective, rewriting one term through empirical moments, and identifying the limiting regression relationship.

  • The objective Ĵ_N is split into two terms for the proof.
  • Term T2 is rewritten using the empirical mean μ̂θ and covariance matrix Σ̂θ.
  • For Cθ = Σ̂θ, the proof uses E_N[Φθ] = μ̂θ to establish the claimed result.
  • Replacing the empirical average E_N with an expectation yields an analogous relation for the limiting quantities ℓ̃s and J_g(θ).
  • By definition of J_g, ℓ̃s can be viewed as a regression function with θ as covariates and Δ_g as the response variable.

Appendix B. Using the Prior Distribution of the Parameters in Bayesian

Using the prior density in Bayesian optimization shifts acquisitions toward posterior modal regions, improving posterior approximation when the prior and likelihood modes are separated.

  • When the prior is far from the likelihood mode, focusing acquisitions on the likelihood mode produces less accurate posterior modal areas.
  • Including the prior density shifts acquisitions toward the posterior’s modal area and improves posterior approximation.This comes at the cost of a less accurate approximation of the likelihood mode.
  • The demonstrated prior-shifting setup uses 20 deterministic acquisitions and a Gaussian-process model with constant mean function.

Appendix C. Bayesian Optimization with a Deterministic versus a Stochastic

The appendix compares deterministic and stochastic acquisition rules for log-Gaussian-process modeling in the Ricker model, focusing on approximation quality near the negative log synthetic-likelihood minimum.

  • The stochastic comparison uses the corresponding stochastic acquisition rule in the same log-Gaussian-process Ricker-model setup.
  • Both acquisition rules satisfactorily approximate the negative log synthetic likelihood around its minimum.
  • Deterministic acquisition often selects nearly identical parameters, producing vertically clustered acquired data points.
  • Figure 13 contrasts Bayesian optimization without the prior density against prior-shifted acquisitions using the prior pdf pθ.
  • Figures 14 and 15 show estimated regression functions Ĵ(t) for deterministic and stochastic acquisition rules, respectively.
  • The deterministic comparison uses minimization of the acquisition function, with the plot range restricted to (3.2, 4.2).

Appendix D. Ricker Model Inferred with a Markov Chain Monte Carlo Algorithm

The Ricker-model appendix examines random-walk MCMC inference with a log synthetic likelihood, focusing on transition-kernel settings and simulation-count or random-seed effects.

  • Experimental setup: The Ricker model was inferred using a log synthetic likelihood and a random-walk MCMC algorithm based on publicly available Wood (2010) code.The chain used 100,000 iterations, started at θo = (3.8, 0.3, 10), and discarded the first 25,000 samples.
  • Transition kernels: With Wood's proposal standard deviation for σ set to 0.1, the chain became stuck near σ = 0 even when N = 5,000.Figure 16 reports this behavior for the original transition-kernel choice.
  • Transition kernels: Reducing the proposal standard deviation by a factor of 10 produced more reasonable inference results.The passage presents proposal-scale adjustment as the key change to the transition kernel in this experiment.
  • Simulation count and visualization: N = 5,000 was ten times larger than the simulation count used in the original Wood (2010) work.The appendix visualizes rescaled histograms with dashed marker curves and kernel density estimates with solid curves.
  • Simulation count and random seed: The appendix also examines how the number of simulated data sets N and the random-number-generator seed affect the Ricker example under Wood's method.These effects are presented in Figure 17 using the same visualization format as Figure 16.
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