We describe and develop a close relationship between two problems that have customarily been regarded as distinct: that of maximizing entropy, and that of minimizing worst-case expected loss. Using a formulation grounded in the equilibrium theory of zero-sum games between Decision Maker and
Nature, these two problems are shown to be dual to each other, the solution to each providing that to the other. Although Topsœdescribed this connection for the Shannon entropy over 20 years ago, it does not appear to be widely known even in that important special case. We here generalize this theory to apply to arbitrary decision problems and loss functions. We indicate how an appropriate generalized definition of entropy can be associated with such a problem, and we show that, subject to certain regularity conditions, the above-mentioned duality continues to apply in this extended context.
This simultaneously provides a possible rationale for maximizing entropy and a tool for finding robust Bayes acts. We also describe the essential identity between the problem of maximizing entropy and that of minimizing a related discrepancy or divergence between distributions. This leads to an extension, to arbitrary discrepancies, of a well-known minimax theorem for the case of Kullback-Leibler divergence (the ``redundancy-capacity theorem'' of information theory). For the important case of families of distributions having certain mean values specified, we develop simple sufficient conditions and methods for identifying the desired solutions.
We introduce the concepts of Bayesian lens, characterizing the bidirectional structure of exact Bayesian inference, and statistical game, formalizing the optimization objectives of approximate inference problems. We prove that Bayesian inversions compose according to the compositional lens pattern, and exemplify statistical games with a number of classic statistical concepts, from maximum likelihood estimation to generalized variational Bayesian methods. This paper is the first in a series laying the foundations for a compositional account of the theory of active inference, and we therefore pa…
CREST（Crest研究中心）；ENSAE（法国国立统计与经济管理学院）；Institut Polytechnique de Paris（巴黎理工学院）；RIKEN Center for AI Project（理化学研究所人工智能项目中心）
In some misspecified settings, the posterior distribution in Bayesian statistics may lead to inconsistent estimates. To fix this issue, it has been suggested to replace the likelihood by a pseudo-likelihood, that is the exponential of a loss function enjoying suitable robustness properties. In this paper, we build a pseudo-likelihood based on the Maximum Mean Discrepancy, defined via an embedding of probability distributions into a reproducing kernel Hilbert space. We show that this MMD-Bayes posterior is consistent and robust to model misspecification. As the posterior obtained in this way mi…
A standard objective in computer experiments is to approximate the behaviour of an unknown function on a compact domain from a few evaluations inside the domain. When little is known about the function, space-filling design is advisable: typically, points of evaluation spread out across the available space are obtained by minimizing a geometrical (for instance, covering radius) or a discrepancy criterion measuring distance to uniformity. The paper investigates connections between design for integration (quadrature design), construction of the (continuous) BLUE for the location model, space-fil…
The aim of this article is to make a contribution to the Bayesian procedure of testing precise hypotheses for parametric models. For this purpose, we define the Bayesian Discrepancy Measure that allows one to evaluate the suitability of a given hypothesis with respect to the available information (prior law and data). To summarise this information, the posterior median is employed, allowing a simple assessment of the discrepancy with a fixed hypothesis. The Bayesian Discrepancy Measure assesses the compatibility of a single hypothesis with the observed data, as opposed to the more common compa…
We study capital process behavior in the fair-coin game and biased-coin games in the framework of the game-theoretic probability of Shafer and Vovk (2001). We show that if Skeptic uses a Bayesian strategy with a beta prior, the capital process is lucidly expressed in terms of the past average of Reality's moves. From this it is proved that the Skeptic's Bayesian strategy weakly forces the strong law of large numbers (SLLN) with the convergence rate of O(\sqrt{\log n/n})$ and if Reality violates SLLN then the exponential growth rate of the capital process is very accurately described in terms o…
Generalised Bayesian inference updates prior beliefs using a loss function, rather than a likelihood, and can therefore be used to confer robustness against possible mis-specification of the likelihood. Here we consider generalised Bayesian inference with a Stein discrepancy as a loss function, motivated by applications in which the likelihood contains an intractable normalisation constant. In this context, the Stein discrepancy circumvents evaluation of the normalisation constant and produces generalised posteriors that are either closed form or accessible using standard Markov chain Monte Ca…
The Mars rover Curiosity carries an instrument called ChemCam to determine the composition of the soil and rocks. ChemCam uses laser-induced breakdown spectroscopy (LIBS) for this purpose. Los Alamos National Laboratory has developed a simulation capability that can predict spectra from ChemCam, but there are major scale differences between the prediction and observation. This presents a challenge when using Bayesian model calibration to determine the unknown physical parameters that describe the LIBS observations. We present an analysis of LIBS data to support ChemCam based on including a str…
Timely and accurate forecasts of seasonal influenza would assist public health decision-makers in planning intervention strategies, efficiently allocating resources, and possibly saving lives. For these reasons, influenza forecasts are consequential. Producing timely and accurate influenza forecasts, however, have proven challenging due to noisy and limited data, an incomplete understanding of the disease transmission process, and the mismatch between the disease transmission process and the data-generating process. In this paper, we introduce a dynamic Bayesian (DB) flu forecasting model that…
Bayesian model updating based on Gaussian Process (GP) models has received attention in recent years, which incorporates kernel-based GPs to provide enhanced fidelity response predictions. Although most kernel functions provide high fitting accuracy in the training data set, their out-of-sample predictions can be highly inaccurate. This paper investigates this problem by reformulating the problem on a consistent probabilistic foundation, reviewing common choices of kernel covariance functions, and proposing a new Bayesian model selection for kernel function selection, aiming to create a balanc…
Maddox, et al. [9, 10] establish Bayesian methods for estimating home-team in-game win probability for college and NBA basketball. This paper introduces a Bayesian approach for estimating in-game home-team win probability for Division-I FBS college (American) football that uses expected number of remaining possessions and expected score as two predictors. Models for estimating these are presented and compared. These, along with other predictors are introduced into two Bayesian approaches for the final estimate of in-game home-team win probability. To illustrate utility, methods are applied to …
Two new Bayesian methods for estimating and predicting in-game home team win probabilities are proposed. The first method has a prior that adjusts as a function of lead differential and time elapsed. The second is an adjusted version of the first, where the adjustment is a linear combination of the Bayesian estimator with a time-weighted pre-game win probability. The proposed methods are compared to existing methods, showing the new methods perform better for both estimation and prediction. The utility is illustrated via an application to the 2016 NCAA Division 1 Championship game.
Maddox, et al. (2022) establish a new win probability estimation for college basketball and compared the results with previous methods of Stern (1994), Desphande and Jensen (2016) and Benz (2019). This paper proposes modifications to the approach of Maddox, et al. (2022) for the NBA game and investigates the performance of the model. Enhancements to the model are developed, and the resulting adjusted model is compared with existing methods and to the ESPN counterpart. To illustrate utility, all methods are applied to the November 23, 2019 game between the Chicago Bulls and Charlotte Hornets.
This paper presents an in-depth statistical analysis of an experiment designed to measure the extent to which players in a simple game behave according to a popular behavioral economic model. The p-beauty contest is a multi-player number guessing game that has been widely used to study strategic behavior. This paper describes beauty contest experiments for an audience of data analysts, with a special focus on a class of models for game play called k-step thinking models, which allow each player in the game to employ an idiosyncratic strategy. We fit a Bayesian statistical model to estimate the…
A Semi-Bayesian Nonparametric Estimator of the Maximum Mean Discrepancy Measure: Applications in Goodness-of-Fit Testing and Generative Adversarial Networks
University of Hong Kong（香港大学）；University of Maryland（马里兰大学）
A Bayesian Non-parametric Approach to Generative Models: Integrating Variational Autoencoder and Generative Adversarial Networks using Wasserstein and Maximum Mean Discrepancy
Department of Statistics and Actuarial Science University of Hong Kong（统计与精算科学系香港大学）
In-game win probability models, which provide a sports team's likelihood of winning at each point in a game based on historical observations, are becoming increasingly popular. In baseball, basketball and American football, they have become important tools to enhance fan experience, to evaluate in-game decision-making, and to inform coaching decisions. While equally relevant in soccer, the adoption of these models is held back by technical challenges arising from the low-scoring nature of the sport. In this paper, we introduce an in-game win probability model for soccer that addresses the shor…
Measuring contributions is a classical problem in cooperative game theory where the Shapley value is the most well-known solution concept. In this paper, we establish the convergence property of the Shapley value in parametric Bayesian learning games where players perform a Bayesian inference using their combined data, and the posterior-prior KL divergence is used as the characteristic function. We show that for any two players, under some regularity conditions, their difference in Shapley value converges in probability to the difference in Shapley value of a limiting game whose characteristic…
Bryn Mawr College（布林莫尔学院）；University of Pennsylvania（宾夕法尼亚大学）
The complexity of game play in online multiplayer games has generated strong interest in modeling the different play styles or strategies used by players for success. We develop a hierarchical Bayesian regression approach for the online multiplayer game Battlefield 3 where performance is modeled as a function of the roles, game type, and map taken on by that player in each of their matches. We use a Dirichlet process prior that enables the clustering of players that have similar player-specific coefficients in our regression model, which allows us to discover common play styles amongst our sam…
Florida International University（佛罗里达国际大学）；Ganzfried Research（Ganzfried研究院）；Stanford University（斯坦福大学）
Two fundamental problems in computational game theory are computing a Nash equilibrium and learning to exploit opponents given observations of their play (opponent exploitation). The latter is perhaps even more important than the former: Nash equilibrium does not have a compelling theoretical justification in game classes other than two-player zero-sum, and for all games one can potentially do better by exploiting perceived weaknesses of the opponent than by following a static equilibrium strategy throughout the match. The natural setting for opponent exploitation is the Bayesian setting where…
The current standard Bayesian approach to model calibration, which assigns a Gaussian process prior to the discrepancy term, often suffers from issues of unidentifiability and computational complexity and instability. When the goal is to quantify uncertainty in physical parameters for extrapolative prediction, then there is no need to perform inference on the discrepancy term. With this in mind, we introduce Gibbs posteriors as an alternative Bayesian method for model calibration, which updates the prior with a loss function connecting the data to the parameter. The target of inference is the …
National University of Singapore（新加坡国立大学）；Massachusetts Institute of Technology（麻省理工学院）
This paper presents a recursive reasoning formalism of Bayesian optimization (BO) to model the reasoning process in the interactions between boundedly rational, self-interested agents with unknown, complex, and costly-to-evaluate payoff functions in repeated games, which we call Recursive Reasoning-Based BO (R2-B2). Our R2-B2 algorithm is general in that it does not constrain the relationship among the payoff functions of different agents and can thus be applied to various types of games such as constant-sum, general-sum, and common-payoff games. We prove that by reasoning at level 2 or more a…
Some statistical models are specified via a data generating process for which the likelihood function cannot be computed in closed form. Standard likelihood-based inference is then not feasible but the model parameters can be inferred by finding the values which yield simulated data that resemble the observed data. This approach faces at least two major difficulties: The first difficulty is the choice of the discrepancy measure which is used to judge whether the simulated data resemble the observed data. The second difficulty is the computationally efficient identification of regions in the pa…
When computer codes are used for modeling complex physical systems, their unknown parameters are tuned by calibration techniques. A discrepancy function may be added to the computer code in order to capture its discrepancy with the real physical process. By considering the validation question of a computer code as a Bayesian selection model problem, Damblin et al. (2016) have highlighted a possible confounding effect in certain configurations between the code discrepancy and a linear computer code by using a Bayesian testing procedure based on the intrinsic Bayes factor. In this paper, we inve…
In games, as well as many user-facing systems, adapting content to users' preferences and experience is an important challenge. This paper explores a novel method to realize this goal in the context of dynamic difficulty adjustment (DDA). Here the aim is to constantly adapt the content of a game to the skill level of the player, keeping them engaged by avoiding states that are either too difficult or too easy. Current systems for DDA rely on expensive data mining, or on hand-crafted rules designed for particular domains, and usually adapts to keep players in the flow, leaving no room for the d…
Uncertainties from model parameters and model discrepancy from small-scale models impact the accuracy and reliability of predictions of large-scale systems. Inadequate representation of these uncertainties may result in inaccurate and overconfident predictions during scale-up to larger models. Hence multiscale modeling efforts must quantify the effect of the propagation of uncertainties during upscaling. Using a Bayesian approach, we calibrate a small-scale solid sorbent model to Thermogravimetric (TGA) data on a functional profile using chemistry-based priors. Crucial to this effort is the re…
In nuclear reactor system design and safety analysis, the Best Estimate plus Uncertainty (BEPU) methodology requires that computer model output uncertainties must be quantified in order to prove that the investigated design stays within acceptance criteria. "Expert opinion" and "user self-evaluation" have been widely used to specify computer model input uncertainties in previous uncertainty, sensitivity and validation studies. Inverse Uncertainty Quantification (UQ) is the process to inversely quantify input uncertainties based on experimental data in order to more precisely quantify such ad-h…
Game theory finds nowadays a broad range of applications in engineering and machine learning. However, in a derivative-free, expensive black-box context, very few algorithmic solutions are available to find game equilibria. Here, we propose a novel Gaussian-process based approach for solving games in this context. We follow a classical Bayesian optimization framework, with sequential sampling decisions based on acquisition functions. Two strategies are proposed, based either on the probability of achieving equilibrium or on the Stepwise Uncertainty Reduction paradigm. Practical and numerical a…
In this article, we present a recently released R package for Bayesian calibration. Many industrial fields are facing unfeasible or costly field experiments. These experiments are replaced with numerical/computer experiments which are realized by running a numerical code. Bayesian calibration intends to estimate, through a posterior distribution, input parameters of the code in order to make the code outputs close to the available experimental data. The code can be time consuming while the Bayesian calibration implies a lot of code calls which makes studies too burdensome. A discrepancy might …
