Compartmental models are used for predicting the scale and duration of epidemics, estimating epidemiological parameters such as reproduction numbers, and guiding outbreak control measures [Brauer, 2008, O'Neill, 2010, Kucharski et al., 2020]. They are increasingly important because they allow joint modelling of disease dynamics and multimodal data, such as medical test results, cell phone and transport flow data [Rubrichi et al., 2018, Wu et al., 2020], census and demographic information [Prem et al., 2020]. However, statistical inference in stochastic variants of compartmental models is a major computational challenge [Bretó, 2018]. The likelihood function for model parameters is usually intractable because it involves summation over a prohibitively large number of configurations of latent variables representing counts of subpopulations in disease states which cannot be observed directly. This has lead to the recent development of sophisticated computational methods for approximate inference involving various forms of stochastic simulation [Funk and King, 2020].

We are often interested in identifying the feasible subset of a decision space under multiple constraints to permit effective design exploration. If determining feasibility required computationally expensive simulations, the cost of exploration would be prohibitive. Bayesian search is data-efficient for such problems: starting from a small dataset, the central concept is to use Bayesian models of constraints with an acquisition function to locate promising solutions that may improve predictions of feasibility when the dataset is augmented. At the end of this sequential active learning approach with a limited number of expensive evaluations, the models can accurately predict the feasibility of any solution obviating the need for full simulations. In this paper, we propose a novel acquisition function that combines the probability that a solution lies at the boundary between feasible and infeasible spaces (representing exploitation) and the entropy in predictions (representing exploration). Experiments confirmed the efficacy of the proposed function.

Model-based reinforcement learning (RL) algorithms allow us to combine model-generated data with those collected from interaction with the real system in order to alleviate the data efficiency problem in RL. However, designing such algorithms is often challenging because the bias in simulated data may overshadow the ease of data generation. A potential solution to this challenge is to jointly learn and improve model and policy using a universal objective function. In this paper, we leverage the connection between RL and probabilistic inference, and formulate such an objective function as a variational lower-bound of a log-likelihood. This allows us to use expectation maximization (EM) and iteratively fix a baseline policy and learn a variational distribution, consisting of a model and a policy (E-step), followed by improving the baseline policy given the learned variational distribution (M-step). We propose model-based and model-free policy iteration (actor-critic) style algorithms for the E-step and show how the variational distribution learned by them can be used to optimize the M-step in a fully model-based fashion. Our experiments on a number of continuous control tasks show that despite being more complex, our model-based (E-step) algorithm, called {\em variational model-based policy optimization} (VMBPO), is more sample-efficient and robust to hyper-parameter tuning than its model-free (E-step) counterpart. Using the same control tasks, we also compare VMBPO with several state-of-the-art model-based and model-free RL algorithms and show its sample efficiency and performance.

For testing conditional independence (CI) of a response $Y$ and a predictor $X$ given covariates $Z$, the recently introduced model-X (MX) framework has been the subject of active methodological research, especially in the context of MX knockoffs and their successful application to genome-wide association studies. In this paper, we build a theoretical foundation for the MX CI problem, yielding quantitative explanations for empirically observed phenomena and novel insights to guide the design of MX methodology. We focus our analysis on the conditional randomization test (CRT), whose validity conditional on $Y,Z$ allows us to view it as a test of a point null hypothesis involving the conditional distribution of $X$. We use the Neyman-Pearson lemma to derive the most powerful CRT statistic against a point alternative as well as an analogous result for MX knockoffs. We define CRT-style analogs of $t$- and $F$-tests with explicit critical values, and show that they have uniform asymptotic Type-I error control under the assumption that only the first two moments of $X$ given $Z$ are known, a significant relaxation of MX. We derive expressions for the power of these tests against local semiparametric alternatives using Le Cam's local asymptotic normality theory, explicitly capturing the prediction error of the underlying learning algorithm. Finally, we pave the way for estimation in the MX setting by drawing connections to semiparametric statistics and causal inference. Thus, this work forms explicit bridges from MX to both classical statistics (testing) and modern causal inference (estimation).

Graphs are ubiquitous in modelling relational structures. Recent endeavours in machine learning for graph-structured data have led to many architectures and learning algorithms. However, the graph used by these algorithms is often constructed based on inaccurate modelling assumptions and/or noisy data. As a result, it fails to represent the true relationships between nodes. A Bayesian framework which targets posterior inference of the graph by considering it as a random quantity can be beneficial. In this paper, we propose a novel non-parametric graph model for constructing the posterior distribution of graph adjacency matrices. The proposed model is flexible in the sense that it can effectively take into account the output of graph-based learning algorithms that target specific tasks. In addition, model inference scales well to large graphs. We demonstrate the advantages of this model in three different problem settings: node classification, link prediction and recommendation.

We propose a novel framework that unifies and extends existing methods of transfer learning (TL) for regression. To bridge a pretrained source model to the model on a target task, we introduce a density-ratio reweighting function, which is estimated through the Bayesian framework with a specific prior distribution. By changing two intrinsic hyperparameters and the choice of the density-ratio model, the proposed method can integrate three popular methods of TL: TL based on cross-domain similarity regularization, a probabilistic TL using the density-ratio estimation, and fine-tuning of pretrained neural networks. Moreover, the proposed method can benefit from its simple implementation without any additional cost; the model can be fully trained using off-the-shelf libraries for supervised learning in which the original output variable is simply transformed to a new output. We demonstrate its simplicity, generality, and applicability using various real data applications.

Sparse stochastic variational inference allows Gaussian process models to be applied to large datasets. The per iteration computational cost of inference with this method is $\mathcal{O}(\tilde{N}M^2+M^3),$ where $\tilde{N}$ is the number of points in a minibatch and $M$ is the number of `inducing features', which determine the expressiveness of the variational family. Several recent works have shown that for certain priors, features can be defined that remove the $\mathcal{O}(M^3)$ cost of computing a minibatch estimate of an evidence lower bound (ELBO). This represents a significant computational savings when $M\gg \tilde{N}$. We present a construction of features for any stationary prior kernel that allow for computation of an unbiased estimator to the ELBO using $T$ Monte Carlo samples in $\mathcal{O}(\tilde{N}T+M^2T)$ and in $\mathcal{O}(\tilde{N}T+MT)$ with an additional approximation. We analyze the impact of this additional approximation on inference quality.

This article reviews how organisms learn and recognize the world through the dynamics of neural networks from the perspective of Bayesian inference, and introduces a view on how such dynamics is described by the laws for the entropy of neural activity, a paradigm that we call thermodynamics of the Bayesian brain. The Bayesian brain hypothesis sees the stimulus-evoked activity of neurons as an act of constructing the Bayesian posterior distribution based on the generative model of the external world that an organism possesses. A closer look at the stimulus-evoked activity at early sensory cortices reveals that feedforward connections initially mediate the stimulus-response, which is later modulated by input from recurrent connections. Importantly, not the initial response, but the delayed modulation expresses animals' cognitive states such as awareness and attention regarding the stimulus. Using a simple generative model made of a spiking neural population, we reproduce the stimulus-evoked dynamics with the delayed feedback modulation as the process of the Bayesian inference that integrates the stimulus evidence and a prior knowledge with time-delay. We then introduce a thermodynamic view on this process based on the laws for the entropy of neural activity. This view elucidates that the process of the Bayesian inference works as the recently-proposed information-theoretic engine (neural engine, an analogue of a heat engine in thermodynamics), which allows us to quantify the perceptual capacity expressed in the delayed modulation in terms of entropy.

Real-world data with underlying structure, such as pictures of faces, are hypothesized to lie on a low-dimensional manifold. This manifold hypothesis has motivated state-of-the-art generative algorithms that learn low-dimensional data representations. Unfortunately, a popular generative model, normalizing flows, cannot take advantage of this. Normalizing flows are based on successive variable transformations that are, by design, incapable of learning lower-dimensional representations. In this paper we introduce noisy injective flows (NIF), a generalization of normalizing flows that can go across dimensions. NIF explicitly map the latent space to a learnable manifold in a high-dimensional data space using injective transformations. We further employ an additive noise model to account for deviations from the manifold and identify a stochastic inverse of the generative process. Empirically, we demonstrate that a simple application of our method to existing flow architectures can significantly improve sample quality and yield separable data embeddings.

When a missing process depends on the missing values themselves, it needs to be explicitly modelled and taken into account while doing likelihood-based inference. We present an approach for building and fitting deep latent variable models (DLVMs) in cases where the missing process is dependent on the missing data. Specifically, a deep neural network enables us to flexibly model the conditional distribution of the missingness pattern given the data. This allows for incorporating prior information about the type of missingness (e.g. self-censoring) into the model. Our inference technique, based on importance-weighted variational inference, involves maximising a lower bound of the joint likelihood. Stochastic gradients of the bound are obtained by using the reparameterisation trick both in latent space and data space. We show on various kinds of data sets and missingness patterns that explicitly modelling the missing process can be invaluable.