Neural Parameter Estimation with Incomplete Data

Advancements in artificial intelligence (AI) and deep learning have led to neural networks being used to generate lightning-speed answers to complex questions, to paint like Monet, or to write like Proust. Leveraging their computational speed and flexibility, neural networks are also being used to facilitate fast, likelihood-free statistical inference. However, it is not straightforward to use neural networks with data that for various reasons are incomplete, which precludes their use in many applications. A recently proposed approach to remedy this issue inputs an appropriately padded data vector and a vector that encodes the missingness pattern to a neural network. While computationally efficient, this "masking" approach can result in statistically inefficient inferences. Here, we propose an alternative approach that is based on the Monte Carlo expectation-maximization (EM) algorithm. Our EM approach is likelihood-free, substantially faster than the conventional EM algorithm as it does not require numerical optimization at each iteration, and more statistically efficient than the masking approach. This research represents a prototype problem that indicates how improvements could be made in AI by introducing Bayesian statistical thinking. We compare the two approaches to missingness using simulated incomplete data from two models: a spatial Gaussian process model, and a spatial Potts model. The utility of the methodology is shown on Arctic sea-ice data and cryptocurrency data.

Spatial Bayesian Neural Networks

Statistical models for spatial processes play a central role in statistical analyses of spatial data. Yet, it is the simple, interpretable, and well understood models that are routinely employed even though, as is revealed through prior and posterior predictive checks, these can poorly characterise the spatial heterogeneity in the underlying process of interest. Here, we propose a new, flexible class of spatial-process models, which we refer to as spatial Bayesian neural networks (SBNNs). An SBNN leverages the representational capacity of a Bayesian neural network; it is tailored to a spatial setting by incorporating a spatial "embedding layer" into the network and, possibly, spatially-varying network parameters. An SBNN is calibrated by matching its finite-dimensional distribution at locations on a fine gridding of space to that of a target process of interest. That process could be easy to simulate from or we have many realisations from it. We propose several variants of SBNNs, most of which are able to match the finite-dimensional distribution of the target process at the selected grid better than conventional BNNs of similar complexity. We also show that a single SBNN can be used to represent a variety of spatial processes often used in practice, such as Gaussian processes and lognormal processes. We briefly discuss the tools that could be used to make inference with SBNNs, and we conclude with a discussion of their advantages and limitations.