Bayesian Sample Size Calculations for SMART Studies

In the management of most chronic conditions characterized by the lack of universally effective treatments, adaptive treatment strategies (ATSs) have been growing in popularity as they offer a more individualized approach, and sequential multiple assignment randomized trials (SMARTs) have gained attention as the most suitable clinical trial design to formalize the study of these strategies. While the number of SMARTs has increased in recent years, their design has remained limited to the frequentist setting, which may not fully or appropriately account for uncertainty in design parameters and hence not yield appropriate sample size recommendations. Specifically, standard frequentist formulae rely on several assumptions that can be easily misspecified. The Bayesian framework offers a straightforward path to alleviate some of these concerns. In this paper, we provide calculations in a Bayesian setting to allow more realistic and robust estimates that account for uncertainty in inputs through the `two priors approach. Additionally, compared to the standard formulae, this methodology allows us to rely on fewer assumptions, integrate pre-trial knowledge, and switch the focus from the standardized effect size to the minimal detectable difference. The proposed methodology is evaluated in a thorough simulation study and is implemented to estimate the sample size for a full-scale SMART of an Internet-Based Adaptive Stress Management intervention based on a pilot SMART conducted on cardiovascular disease patients from two Canadian provinces.

Bayesian inference for continuous-time hidden Markov models with an unknown number of states

We consider the modeling of data generated by a latent continuous-time Markov jump process with a state space of finite but unknown dimensions. Typically in such models, the number of states has to be pre-specified, and Bayesian inference for a fixed number of states has not been studied until recently. In addition, although approaches to address the problem for discrete-time models have been developed, no method has been successfully implemented for the continuous-time case. We focus on reversible jump Markov chain Monte Carlo which allows the trans-dimensional move among different numbers of states in order to perform Bayesian inference for the unknown number of states. Specifically, we propose an efficient split-combine move which can facilitate the exploration of the parameter space, and demonstrate that it can be implemented effectively at scale. Subsequently, we extend this algorithm to the context of model-based clustering, allowing numbers of states and clusters both determined during the analysis. The model formulation, inference methodology, and associated algorithm are illustrated by simulation studies. Finally, We apply this method to real data from a Canadian healthcare system in Quebec.

Estimation of Optimal Dynamic Treatment Regimes via Gaussian Process Emulation: A Technical Report

Causal inference of treatment effects is a challenging undertaking in it of itself; inference for sequential treatments leads to even more hurdles. In precision medicine, one additional ambitious goal may be to infer about effects of dynamic treatment regimes (DTRs) and to identify optimal DTRs. Conventional methods for inferring about DTRs involve powerful semi-parametric estimators. However, these are not without their strong assumptions. Dynamic Marginal Structural Models (MSMs) are one semi-parametric approach used to infer about optimal DTRs in a family of regimes. To achieve this, investigators are forced to model the expected outcome under adherence to a DTR in the family; relatively straightforward models may lead to bias in the optimum. One way to obviate this difficulty is to perform a grid search for the optimal DTR. Unfortunately, this approach becomes prohibitive as the complexity of regimes considered increases. In recently developed Bayesian methods for dynamic MSMs, computational challenges may be compounded by the fact that at each grid point, a posterior mean must be calculated. We propose a manner by which to alleviate modelling difficulties for DTRs by using Gaussian process optimization. More precisely, we show how to pair this optimization approach with robust estimators for the causal effect of adherence to a DTR to identify optimal DTRs. We examine how to find the optimum in complex, multi-modal settings which are not generally addressed in the DTR literature. We further evaluate the sensitivity of the approach to a variety of modeling assumptions in the Gaussian process.

Stochastic Reweighted Gradient Descent

Despite the strong theoretical guarantees that variance-reduced finite-sum optimization algorithms enjoy, their applicability remains limited to cases where the memory overhead they introduce (SAG/SAGA), or the periodic full gradient computation they require (SVRG/SARAH) are manageable. A promising approach to achieving variance reduction while avoiding these drawbacks is the use of importance sampling instead of control variates. While many such methods have been proposed in the literature, directly proving that they improve the convergence of the resulting optimization algorithm has remained elusive. In this work, we propose an importance-sampling-based algorithm we call SRG (stochastic reweighted gradient). We analyze the convergence of SRG in the strongly-convex case and show that, while it does not recover the linear rate of control variates methods, it provably outperforms SGD. We pay particular attention to the time and memory overhead of our proposed method, and design a specialized red-black tree allowing its efficient implementation. Finally, we present empirical results to support our findings.

Adaptive Importance Sampling for Finite-Sum Optimization and Sampling with Decreasing Step-Sizes

Reducing the variance of the gradient estimator is known to improve the convergence rate of stochastic gradient-based optimization and sampling algorithms. One way of achieving variance reduction is to design importance sampling strategies. Recently, the problem of designing such schemes was formulated as an online learning problem with bandit feedback, and algorithms with sub-linear static regret were designed. In this work, we build on this framework and propose Avare, a simple and efficient algorithm for adaptive importance sampling for finite-sum optimization and sampling with decreasing step-sizes. Under standard technical conditions, we show that Avare achieves $mathcal{O}(T^{2/3})$ and $mathcal{O}(T^{5/6})$ dynamic regret for SGD and SGLD respectively when run with $mathcal{O}(1/t)$ step sizes. We achieve this dynamic regret bound by leveraging our knowledge of the dynamics defined by the algorithm, and combining ideas from online learning and variance-reduced stochastic optimization. We validate empirically the performance of our algorithm and identify settings in which it leads to significant improvements.

Bayesian Doubly Robust Causal Inference via Loss Functions

Frequentist inference has a well-established supporting theory for doubly robust causal inference based on the potential outcomes framework, which is realized via outcome regression (OR) and propensity score (PS) models. The Bayesian counterpart, however, is not obvious as the PS model loses its balancing property in joint modeling. In this paper, we propose a natural and formal Bayesian solution by bridging loss-type Bayesian inference with a utility function derived from the notion of a pseudo-population via the change of measure. Consistency of the posterior distribution is shown with correctly specified and misspecified OR models. Simulation studies suggest that our proposed method can estimate the true causal effect more efficiently and achieve the frequentist coverage if either the OR model is correctly specified or fit with a flexible function of the confounders, compared to the previous Bayesian approach via the Bayesian bootstrap. Finally, we apply this novel Bayesian method to assess the impact of speed cameras on the reduction of car collisions in England.

The role of exchangeability in causal inference

The notion of exchangeability has been recognized in the causal inference literature in various guises, but only rarely in the original Bayesian meaning as a symmetry property between individual units in statistical inference. Since the latter is a standard ingredient in Bayesian inference, we argue that in Bayesian causal inference it is natural to link the causal model, including the notion of confounding and definition of causal contrasts of interest, to the concept of exchangeability. Here we relate the Bayesian notion of exchangeability to alternative conditions for unconfounded inferences, commonly stated using potential outcomes, and define causal contrasts in the presence of exchangeability in terms of limits of posterior predictive expectations for further exchangeable units. While our main focus is in a point treatment setting, we also investigate how this reasoning carries over to longitudinal settings.

Bayesian Clustering for Continuous-Time Hidden Markov Models

We develop clustering procedures for longitudinal trajectories based on a continuous-time hidden Markov model (CTHMM) and a generalized linear observation model. Specifically in this paper, we carry out finite and infinite mixture model-based clustering for a CTHMM and achieve inference using Markov chain Monte Carlo (MCMC). For a finite mixture model with prior on the number of components, we implement reversible-jump MCMC to facilitate the trans-dimensional move between different number of clusters. For a Dirichlet process mixture model, we utilize restricted Gibbs sampling split-merge proposals to expedite the MCMC algorithm. We employ proposed algorithms to the simulated data as well as a real data example, and the results demonstrate the desired performance of the new sampler.