On Similarities between Inference in Game Theory and Machine Learning

In this paper, we elucidate the equivalence between inference in game theory and machine learning. Our aim in so doing is to establish an equivalent vocabulary between the two domains so as to facilitate developments at the intersection of both fields, and as proof of the usefulness of this approach, we use recent developments in each field to make useful improvements to the other. More specifically, we consider the analogies between smooth best responses in fictitious play and Bayesian inference methods. Initially, we use these insights to develop and demonstrate an improved algorithm for learning in games based on probabilistic moderation. That is, by integrating over the distribution of opponent strategies (a Bayesian approach within machine learning) rather than taking a simple empirical average (the approach used in standard fictitious play) we derive a novel moderated fictitious play algorithm and show that it is more likely than standard fictitious play to converge to a payoff-dominant but risk-dominated Nash equilibrium in a simple coordination game. Furthermore we consider the converse case, and show how insights from game theory can be used to derive two improved mean field variational learning algorithms. We first show that the standard update rule of mean field variational learning is analogous to a Cournot adjustment within game theory. By analogy with fictitious play, we then suggest an improved update rule, and show that this results in fictitious variational play, an improved mean field variational learning algorithm that exhibits better convergence in highly or strongly connected graphical models. Second, we use a recent advance in fictitious play, namely dynamic fictitious play, to derive a derivative action variational learning algorithm, that exhibits superior convergence properties on a canonical machine learning problem (clustering a mixture distribution)

Approximate Models and Robust Decisions

Decisions based partly or solely on predictions from probabilistic models may be sensitive to model misspecification. Statisticians are taught from an early stage that "all models are wrong", but little formal guidance exists on how to assess the impact of model approximation on decision making, or how to proceed when optimal actions appear sensitive to model fidelity. This article presents an overview of recent developments across different disciplines to address this. We review diagnostic techniques, including graphical approaches and summary statistics, to help highlight decisions made through minimised expected loss that are sensitive to model misspecification. We then consider formal methods for decision making under model misspecification by quantifying stability of optimal actions to perturbations to the model within a neighbourhood of model space. This neighbourhood is defined in either one of two ways. Firstly, in a strong sense via an information (Kullback-Leibler) divergence around the approximating model. Or using a nonparametric model extension, again centred at the approximating model, in order to `average out' over possible misspecifications. This is presented in the context of recent work in the robust control, macroeconomics and financial mathematics literature. We adopt a Bayesian approach throughout although the methods are agnostic to this position

Automatic Differentiation Variational Inference

Probabilistic modeling is iterative. A scientist posits a simple model, fits it to her data, refines it according to her analysis, and repeats. However, fitting complex models to large data is a bottleneck in this process. Deriving algorithms for new models can be both mathematically and computationally challenging, which makes it difficult to efficiently cycle through the steps. To this end, we develop automatic differentiation variational inference (ADVI). Using our method, the scientist only provides a probabilistic model and a dataset, nothing else. ADVI automatically derives an efficient variational inference algorithm, freeing the scientist to refine and explore many models. ADVI supports a broad class of models-no conjugacy assumptions are required. We study ADVI across ten different models and apply it to a dataset with millions of observations. ADVI is integrated into Stan, a probabilistic programming system; it is available for immediate use

Multivariate Bayesian Predictive Synthesis in Macroeconomic Forecasting

We develop the methodology and a detailed case study in use of a class of Bayesian predictive synthesis (BPS) models for multivariate time series forecasting. This extends the recently introduced foundational framework of BPS to the multivariate setting, with detailed application in the topical and challenging context of multi-step macroeconomic forecasting in a monetary policy setting. BPS evaluates-- sequentially and adaptively over time-- varying forecast biases and facets of miscalibration of individual forecast densities, and-- critically-- of time-varying inter-dependencies among them over multiple series. We develop new BPS methodology for a specific subclass of the dynamic multivariate latent factor models implied by BPS theory. Structured dynamic latent factor BPS is here motivated by the application context-- sequential forecasting of multiple US macroeconomic time series with forecasts generated from several traditional econometric time series models. The case study highlights the potential of BPS to improve of forecasts of multiple series at multiple forecast horizons, and its use in learning dynamic relationships among forecasting models or agents