Simulation-based Inference : From Approximate Bayesian Computation and Particle Methods to Neural Density Estimation

This doctoral thesis in computational statistics utilizes both Monte Carlo methods(approximate Bayesian computation and sequential Monte Carlo) and machine­-learning methods (deep learning and normalizing flows) to develop novel algorithms for infer­ence in implicit Bayesian models. Implicit models are those for which calculating the likelihood function is very challenging (and often impossible), but model simulation is feasible. The inference methods developed in the thesis are simulation­-based infer­ence methods since they leverage the possibility to simulate data from the implicit models. Several approaches are considered in the thesis: Paper II and IV focus on classical methods (sequential Monte Carlo­-based methods), while paper I and III fo­cus on more recent machine learning methods (deep learning and normalizing flows, respectively).Paper I constructs novel deep learning methods for learning summary statistics for approximate Bayesian computation (ABC). To achieve this paper I introduces the partially exchangeable network (PEN), a deep learning architecture specifically de­signed for Markovian data (i.e., partially exchangeable data).Paper II considers Bayesian inference in stochastic differential equation mixed-effects models (SDEMEM). Bayesian inference for SDEMEMs is challenging due to the intractable likelihood function of SDEMEMs. Paper II addresses this problem by designing a novel a Gibbs­-blocking strategy in combination with correlated pseudo­ marginal methods. The paper also discusses how custom particle filters can be adapted to the inference procedure.Paper III introduces the novel inference method sequential neural posterior and like­lihood approximation (SNPLA). SNPLA is a simulation­-based inference algorithm that utilizes normalizing flows for learning both the posterior distribution and the likelihood function of an implicit model via a sequential scheme. By learning both the likelihood and the posterior, and by leveraging the reverse Kullback Leibler (KL) divergence, SNPLA avoids ad­-hoc correction steps and Markov chain Monte Carlo (MCMC) sampling.Paper IV introduces the accelerated-delayed acceptance (ADA) algorithm. ADA can be viewed as an extension of the delayed­-acceptance (DA) MCMC algorithm that leverages connections between the two likelihood ratios of DA to further accelerate MCMC sampling from the posterior distribution of interest, although our approach introduces an approximation. The main case study of paper IV is a double­-well po­tential stochastic differential equation (DWP­SDE) model for protein-­folding data (reaction coordinate data)

Partially Exchangeable Networks and Architectures for Learning Summary Statistics in Approximate Bayesian Computation

We present a novel family of deep neural architectures, named partially exchangeable networks (PENs) that leverage probabilistic symmetries. By design, PENs are invariant to block-switch transformations, which characterize the partial exchangeability properties of conditionally Markovian processes. Moreover, we show that any block-switch invariant function has a PEN-like representation. The DeepSets architecture is a special case of PEN and we can therefore also target fully exchangeable data. We employ PENs to learn summary statistics in approximate Bayesian computation (ABC). When comparing PENs to previous deep learning methods for learning summary statistics, our results are highly competitive, both considering time series and static models. Indeed, PENs provide more reliable posterior samples even when using less training data.Comment: Forthcoming on the Proceedings of ICML 2019. New comparisons with several different networks. We now use the Wasserstein distance to produce comparisons. Code available on GitHub. 16 pages, 5 figures, 21 table

Accelerating delayed-acceptance Markov chain Monte Carlo algorithms

Delayed-acceptance Markov chain Monte Carlo (DA-MCMC) samples from a probability distribution via a two-stages version of the Metropolis-Hastings algorithm, by combining the target distribution with a "surrogate" (i.e. an approximate and computationally cheaper version) of said distribution. DA-MCMC accelerates MCMC sampling in complex applications, while still targeting the exact distribution. We design a computationally faster, albeit approximate, DA-MCMC algorithm. We consider parameter inference in a Bayesian setting where a surrogate likelihood function is introduced in the delayed-acceptance scheme. When the evaluation of the likelihood function is computationally intensive, our scheme produces a 2-4 times speed-up, compared to standard DA-MCMC. However, the acceleration is highly problem dependent. Inference results for the standard delayed-acceptance algorithm and our approximated version are similar, indicating that our algorithm can return reliable Bayesian inference. As a computationally intensive case study, we introduce a novel stochastic differential equation model for protein folding data.Comment: 40 pages, 21 figures, 10 table

Sequential Neural Posterior and Likelihood Approximation

We introduce the sequential neural posterior and likelihood approximation (SNPLA) algorithm. SNPLA is a normalizing flows-based algorithm for inference in implicit models, and therefore is a simulation-based inference method that only requires simulations from a generative model. SNPLA avoids Markov chain Monte Carlo sampling and correction-steps of the parameter proposal function that are introduced in similar methods, but that can be numerically unstable or restrictive. By utilizing the reverse KL divergence, SNPLA manages to learn both the likelihood and the posterior in a sequential manner. Over four experiments, we show that SNPLA performs competitively when utilizing the same number of model simulations as used in other methods, even though the inference problem for SNPLA is more complex due to the joint learning of posterior and likelihood function. Due to utilizing normalizing flows SNPLA generates posterior draws much faster (4 orders of magnitude) than MCMC-based methods.Comment: 28 pages, 8 tables, 14 figures. The supplementary material is attached to the main pape

Scalable and flexible inference framework for stochastic dynamic single-cell models

Understanding the inherited nature of how biological processes dynamically change over time and exhibit intra- and inter-individual variability, due to the different responses to environmental stimuli and when interacting with other processes, has been a major focus of systems biology. The rise of single-cell fluorescent microscopy has enabled the study of those phenomena. The analysis of single-cell data with mechanistic models offers an invaluable tool to describe dynamic cellular processes and to rationalise cell-to-cell variability within the population. However, extracting mechanistic information from single-cell data has proven difficult. This requires statistical methods to infer unknown model parameters from dynamic, multi-individual data accounting for heterogeneity caused by both intrinsic (e.g. variations in chemical reactions) and extrinsic (e.g. variability in protein concentrations) noise. Although several inference methods exist, the availability of efficient, general and accessible methods that facilitate modelling of single-cell data, remains lacking. Here we present a scalable and flexible framework for Bayesian inference in state-space mixed-effects single-cell models with stochastic dynamic. Our approach infers model parameters when intrinsic noise is modelled by either exact or approximate stochastic simulators, and when extrinsic noise is modelled by either time-varying, or time-constant parameters that vary between cells. We demonstrate the relevance of our approach by studying how cell-to-cell variation in carbon source utilisation affects heterogeneity in the budding yeast Saccharomyces cerevisiae SNF1 nutrient sensing pathway. We identify hexokinase activity as a source of extrinsic noise and deduce that sugar availability dictates cell-to-cell variability

Which method is best for the induction of labour?: A systematic review, network meta-analysis and cost-effectiveness analysis

Background: More than 150,000 pregnant women in England and Wales have their labour induced each year. Multiple pharmacological, mechanical and complementary methods are available to induce labour. Objective: To assess the relative effectiveness, safety and cost-effectiveness of labour induction methods and, data permitting, effects in different clinical subgroups. Methods: We carried out a systematic review using Cochrane methods. The Cochrane Pregnancy and Childbirth Group’s Trials Register was searched (March 2014). This contains over 22,000 reports of controlled trials (published from 1923 onwards) retrieved from weekly searches of OVID MEDLINE (1966 to current); Cochrane Central Register of Controlled Trials (The Cochrane Library); EMBASE (1982 to current); Cumulative Index to Nursing and Allied Health Literature (1984 to current); ClinicalTrials.gov; the World Health Organization International Clinical Trials Registry Portal; and hand-searching of relevant conference proceedings and journals. We included randomised controlled trials examining interventions to induce labour compared with placebo, no treatment or other interventions in women eligible for third-trimester induction. We included outcomes relating to efficacy, safety and acceptability to women. In addition, for the economic analysis we searched the Database of Abstracts of Reviews of Effects, and Economic Evaluations Databases, NHS Economic Evaluation Database and the Health Technology Assessment database. We carried out a network meta-analysis (NMA) using all of the available evidence, both direct and indirect, to produce estimates of the relative effects of each treatment compared with others in a network. We developed a de novo decision tree model to estimate the cost-effectiveness of various methods. The costs included were the intervention and other hospital costs incurred (price year 2012–13). We reviewed the literature to identify preference-based utilities for the health-related outcomes in the model. We calculated incremental cost-effectiveness ratios, expected costs, utilities and net benefit. We represent uncertainty in the optimal intervention using cost-effectiveness acceptability curves. Results: We identified 1190 studies; 611 were eligible for inclusion. The interventions most likely to achieve vaginal delivery (VD) within 24 hours were intravenous oxytocin with amniotomy [posterior rank 2; 95% credible intervals (CrIs) 1 to 9] and higher-dose (≥ 50 μg) vaginal misoprostol (rank 3; 95% CrI 1 to 6). Compared with placebo, several treatments reduced the odds of caesarean section, but we observed considerable uncertainty in treatment rankings. For uterine hyperstimulation, double-balloon catheter had the highest probability of being among the best three treatments, whereas vaginal misoprostol (≥ 50 μg) was most likely to increase the odds of excessive uterine activity. For other safety outcomes there were insufficient data or there was too much uncertainty to identify which treatments performed ‘best’. Few studies collected information on women’s views. Owing to incomplete reporting of the VD within 24 hours outcome, the cost-effectiveness analysis could compare only 20 interventions. The analysis suggested that most interventions have similar utility and differ mainly in cost. With a caveat of considerable uncertainty, titrated (low-dose) misoprostol solution and buccal/sublingual misoprostol had the highest likelihood of being cost-effective. Limitations: There was considerable uncertainty in findings and there were insufficient data for some planned subgroup analyses. Conclusions: Overall, misoprostol and oxytocin with amniotomy (for women with favourable cervix) is more successful than other agents in achieving VD within 24 hours. The ranking according to safety of different methods was less clear. The cost-effectiveness analysis suggested that titrated (low-dose) oral misoprostol solution resulted in the highest utility, whereas buccal/sublingual misoprostol had the lowest cost. There was a high degree of uncertainty as to the most cost-effective intervention

Sequential Neural Posterior and Likelihood Approximation

We introduce the sequential neural posterior and likelihood approximation (SNPLA) algorithm. SNPLA is a normalizing flows-based algorithm for inference in implicit models, and therefore is a simulation-based inference method that only requires simulations from a generative model. SNPLA avoids Markov chain Monte Carlo sampling and correction-steps of the parameter proposal function that are introduced in similar methods, but that can be numerically unstable or restrictive. By utilizing the reverse KL divergence, SNPLA manages to learn both the likelihood and the posterior in a sequential manner. Over four experiments, we show that SNPLA performs competitively when utilizing the same number of model simulations as used in other methods, even though the inference problem for SNPLA is more complex due to the joint learning of posterior and likelihood function. Due to utilizing normalizing flows SNPLA generates posterior draws much faster (4 orders of magnitude) than MCMC-based methods