Small-variance asymptotics for Bayesian neural networks

Bayesian neural networks (BNNs) are a rich and flexible class of models that have several advantages over standard feedforward networks, but are typically expensive to train on large-scale data. In this thesis, we explore the use of small-variance asymptotics-an approach to yielding fast algorithms from probabilistic models-on various Bayesian neural network models. We first demonstrate how small-variance asymptotics shows precise connections between standard neural networks and BNNs; for example, particular sampling algorithms for BNNs reduce to standard backpropagation in the small-variance limit. We then explore a more complex BNN where the number of hidden units is additionally treated as a random variable in the model. While standard sampling schemes would be too slow to be practical, our asymptotic approach yields a simple method for extending standard backpropagation to the case where the number of hidden units is not fixed. We show on several data sets that the resulting algorithm has benefits over backpropagation on networks with a fixed architecture.2019-01-02T00:00:00

Approximation of epidemic models by diffusion processes and their statistical inference

Multidimensional continuous-time Markov jump processes $(Z(t))$ on $\mathbb{Z}^p$ form a usual set-up for modeling $SIR$-like epidemics. However, when facing incomplete epidemic data, inference based on $(Z(t))$ is not easy to be achieved. Here, we start building a new framework for the estimation of key parameters of epidemic models based on statistics of diffusion processes approximating $(Z(t))$. First, \previous results on the approximation of density-dependent $SIR$-like models by diffusion processes with small diffusion coefficient $\frac{1}{\sqrt{N}}$, where $N$ is the population size, are generalized to non-autonomous systems. Second, our previous inference results on discretely observed diffusion processes with small diffusion coefficient are extended to time-dependent diffusions. Consistent and asymptotically Gaussian estimates are obtained for a fixed number $n$ of observations, which corresponds to the epidemic context, and for $N\rightarrow \infty$. A correction term, which yields better estimates non asymptotically, is also included. Finally, performances and robustness of our estimators with respect to various parameters such as $R_0$ (the basic reproduction number), $N$, $n$ are investigated on simulations. Two models, $SIR$ and $SIRS$, corresponding to single and recurrent outbreaks, respectively, are used to simulate data. The findings indicate that our estimators have good asymptotic properties and behave noticeably well for realistic numbers of observations and population sizes. This study lays the foundations of a generic inference method currently under extension to incompletely observed epidemic data. Indeed, contrary to the majority of current inference techniques for partially observed processes, which necessitates computer intensive simulations, our method being mostly an analytical approach requires only the classical optimization steps.Comment: 30 pages, 10 figure