Consistent and fast inference in compartmental models of epidemics using Poisson Approximate Likelihoods

Addressing the challenge of scaling-up epidemiological inference to complex and heterogeneous models, we introduce Poisson Approximate Likelihood (PAL) methods. In contrast to the popular ODE approach to compartmental modelling, in which a large population limit is used to motivate a deterministic model, PALs are derived from approximate filtering equations for finite-population, stochastic compartmental models, and the large population limit drives consistency of maximum PAL estimators. Our theoretical results appear to be the first likelihood-based parameter estimation consistency results which apply to a broad class of partially observed stochastic compartmental models and address the large population limit. PALs are simple to implement, involving only elementary arithmetic operations and no tuning parameters, and fast to evaluate, requiring no simulation from the model and having computational cost independent of population size. Through examples we demonstrate how PALs can be used to: fit an age-structured model of influenza, taking advantage of automatic differentiation in Stan; compare over-dispersion mechanisms in a model of rotavirus by embedding PALs within sequential Monte Carlo; and evaluate the role of unit-specific parameters in a meta-population model of measles

A State-Space Perspective on Modelling and Inference for Online Skill Rating

This paper offers a comprehensive review of the main methodologies used for skill rating in competitive sports. We advocate for a state-space model perspective, wherein players' skills are represented as time-varying, and match results serve as the sole observed quantities. The state-space model perspective facilitates the decoupling of modeling and inference, enabling a more focused approach highlighting model assumptions, while also fostering the development of general-purpose inference tools. We explore the essential steps involved in constructing a state-space model for skill rating before turning to a discussion on the three stages of inference: filtering, smoothing and parameter estimation. Throughout, we examine the computational challenges of scaling up to high-dimensional scenarios involving numerous players and matches, highlighting approximations and reductions used to address these challenges effectively. We provide concise summaries of popular methods documented in the literature, along with their inferential paradigms and introduce new approaches to skill rating inference based on sequential Monte Carlo and finite state-spaces. We close with numerical experiments demonstrating a practical workflow on real data across different sports

Dynamic Bayesian Neural Networks

We define an evolving in time Bayesian neural network called a Hidden Markov neural network. The weights of a feed-forward neural network are modelled with the hidden states of a Hidden Markov model, whose observed process is given by the available data. A filtering algorithm is used to learn a variational approximation to the evolving in time posterior over the weights. Training is pursued through a sequential version of Bayes by Backprop Blundell et al. 2015, which is enriched with a stronger regularization technique called variational DropConnect. The experiments test variational DropConnect on MNIST and display the performance of Hidden Markov neural networks on time series

Inference in Stochastic Epidemic Models via Multinomial Approximations

We introduce a new method for inference in stochastic epidemic models which uses recursive multinomial approximations to integrate over unobserved variables and thus circumvent likelihood intractability. The method is applicable to a class of discrete-time, finite-population compartmental models with partial, randomly under-reported or missing count observations. In contrast to state-of-the-art alternatives such as Approximate Bayesian Computation techniques, no forward simulation of the model is required and there are no tuning parameters. Evaluating the approximate marginal likelihood of model parameters is achieved through a computationally simple filtering recursion. The accuracy of the approximation is demonstrated through analysis of real and simulated data using a model of the 1995 Ebola outbreak in the Democratic Republic of Congo. We show how the method can be embedded within a Sequential Monte Carlo approach to estimating the time-varying reproduction number of COVID-19 in Wuhan, China, recently published by Kucharski et al. 2020

Approximating optimal SMC proposal distributions in individual-based epidemic models

Many epidemic models are naturally defined as individual-based models: where we track the state of each individual within a susceptible population. Inference for individual-based models is challenging due to the high-dimensional state-space of such models, which increases exponentially with population size. We consider sequential Monte Carlo algorithms for inference for individual-based epidemic models where we make direct observations of the state of a sample of individuals. Standard implementations, such as the bootstrap filter or the auxiliary particle filter are inefficient due to mismatch between the proposal distribution of the state and future observations. We develop new efficient proposal distributions that take account of future observations, leveraging the properties that (i) we can analytically calculate the optimal proposal distribution for a single individual given future observations and the future infection rate of that individual; and (ii) the dynamics of individuals are independent if we condition on their infection rates. Thus we construct estimates of the future infection rate for each individual, and then use an independent proposal for the state of each individual given this estimate. Empirical results show order of magnitude improvement in efficiency of the sequential Monte Carlo sampler for both SIS and SEIR models