Maximum entropy (MAXENT) method has a large number of applications in theoretical and applied machine learning, since it provides a convenient non-parametric tool for estimating unknown probabilities. The method is a major contribution of statistical physics to probabilistic inference. However, a systematic approach towards its validity limits is currently missing. Here we study MAXENT in a Bayesian decision theory set-up, i.e. assuming that there exists a well-defined prior Dirichlet density for unknown probabilities, and that the average Kullback-Leibler (KL) distance can be employed for deciding on the quality and applicability of various estimators. These allow to evaluate the relevance of various MAXENT constraints, check its general applicability, and compare MAXENT with estimators having various degrees of dependence on the prior, viz. the regularized maximum likelihood (ML) and the Bayesian estimators. We show that MAXENT applies in sparse data regimes, but needs specific types of prior information. In particular, MAXENT can outperform the optimally regularized ML provided that there are prior rank correlations between the estimated random quantity and its probabilities.
Works, briefly surveyed here, are concerned with two basic methods: Maximum Probability and Bayesian Maximum Probability; as well as with their asymptotic instances: Relative Entropy Maximization and Maximum Non-parametric Likelihood. Parametric and empirical extensions of the latter methods - Empirical Maximum Maximum Entropy and Empirical Likelihood - are also mentioned. The methods are viewed as tools for solving certain ill-posed inverse problems, called Pi-problem, Phi-problem, respectively. Within the two classes of problems, probabilistic justification and interpretation of the respecti…
We study a parametric estimation problem related to moment condition models. As an alternative to the generalized empirical likelihood (GEL) and the generalized method of moments (GMM), a Bayesian approach to the problem can be adopted, extending the MEM procedure to parametric moment conditions. We show in particular that a large number of GEL estimators can be interpreted as a maximum entropy solution. Moreover, we provide a more general field of applications by proving the method to be robust to approximate moment conditions.
We study the parameter estimation problem in mixture models with observational nonidentifiability: the full model (also containing hidden variables) is identifiable, but the marginal (observed) model is not. Hence global maxima of the marginal likelihood are (infinitely) degenerate and predictions of the marginal likelihood are not unique. We show how to generalize the marginal likelihood by introducing an effective temperature, and making it similar to the free energy. This generalization resolves the observational nonidentifiability, since its maximization leads to unique results that are be…
Sony Computer Science Laboratories, Inc.（索尼计算机科学实验室公司）
The issue of discrete probability estimation for samples of small size is addressed in this study. The maximum likelihood method often suffers over-fitting when insufficient data is available. Although the Bayesian approach can avoid over-fitting by using prior distributions, it still has problems with objective analysis. In response to these drawbacks, a new theoretical framework based on thermodynamics, where energy and temperature are introduced, was developed. Entropy and likelihood are placed at the center of this method. The key principle of inference for probability mass functions is th…
This paper addresses the estimation of parameters of a Bayesian network from incomplete data. The task is usually tackled by running the Expectation-Maximization (EM) algorithm several times in order to obtain a high log-likelihood estimate. We argue that choosing the maximum log-likelihood estimate (as well as the maximum penalized log-likelihood and the maximum a posteriori estimate) has severe drawbacks, being affected both by overfitting and model uncertainty. Two ideas are discussed to overcome these issues: a maximum entropy approach and a Bayesian model averaging approach. Both ideas ca…
Many scientifically well-motivated statistical models in natural, engineering, and environmental sciences are specified through a generative process. However, in some cases, it may not be possible to write down the likelihood for these models analytically. Approximate Bayesian computation (ABC) methods allow Bayesian inference in such situations. The procedures are nonetheless typically computationally intensive. Recently, computationally attractive empirical likelihood-based ABC methods have been suggested in the literature. All of these methods rely on the availability of several suitable an…
This paper modifies Jaynes's axioms of plausible reasoning and derives the minimum relative entropy principle, Bayes's rule, as well as maximum likelihood from first principles. The new axioms, which I call the Optimum Information Principle, is applicable whenever the decision maker is given the data and the relevant background information. These axioms provide an answer to the question "why maximize entropy when faced with incomplete information?"
Most real world phenomena such as sunlight distribution under a forest canopy, minerals concentration, stock valuation, exhibit nonstationary dynamics i.e. phenomenon variation changes depending on the locality. Nonstationary dynamics pose both theoretical and practical challenges to statistical machine learning algorithms that aim to accurately capture the complexities governing the evolution of such processes. Typically the nonstationary dynamics are modeled using nonstationary Gaussian Process models (NGPS) that employ local latent dynamics parameterization to correspondingly model the nons…
A key sticking point of Bayesian analysis is the choice of prior distribution, and there is a vast literature on potential defaults including uniform priors, Jeffreys' priors, reference priors, maximum entropy priors, and weakly informative priors. These methods, however, often manifest a key conceptual tension in prior modeling: a model encoding true prior information should be chosen without reference to the model of the measurement process, but almost all common prior modeling techniques are implicitly motivated by a reference likelihood. In this paper we resolve this apparent paradox by pl…
Imperial College London（伦敦帝国学院）；New York University（纽约大学）；University of Liège（列日大学）；Federico Santa María Technical University（费德里科·圣玛丽亚理工大学）
We extend recent work (Brehmer, et. al., 2018) that use neural networks as surrogate models for likelihood-free inference. As in the previous work, we exploit the fact that the joint likelihood ratio and joint score, conditioned on both observed and latent variables, can often be extracted from an implicit generative model or simulator to augment the training data for these surrogate models. We show how this augmented training data can be used to provide a new cross-entropy estimator, which provides improved sample efficiency compared to previous loss functions exploiting this augmented traini…
Understanding the way in which random entities interact is of key interest in numerous scientific fields. This can range from a full characterization of the joint distribution to single scalar summary statistics. In this work we identify a novel relationship between the ubiquitous Shannon's mutual information measure and the central tool for capturing real-valued non-Gaussian distributions, namely the framework of copulas. Specifically, we establish a monotonic relationship between the mutual information and the copula dependence parameter, for a wide range of copula families. In addition to t…
In inverse problems, the parameters of a model are estimated based on observations of the model response. The Bayesian approach is powerful for solving such problems; one formulates a prior distribution for the parameter state that is updated with the observations to compute the posterior parameter distribution. Solving for the posterior distribution can be challenging when, e.g., prior and posterior significantly differ from one another and/or the parameter space is high-dimensional. We use a sequence of importance sampling measures that arise by tempering the likelihood to approach inverse p…
A common problem faced in statistical inference is drawing conclusions from paired comparisons, in which two objects compete and one is declared the victor. A probabilistic approach to such a problem is the Bradley-Terry model, first studied by Zermelo in 1929 and rediscovered by Bradley and Terry in 1952. One obvious area of application for such a model is sporting events, and in particular Major League Baseball. With this in mind, we describe a hierarchical Bayesian version of Bradley-Terry suitable for use in ranking and prediction problems, and compare results from these application domain…
This paper proposes a new Bayesian approach for analysing moment condition models in the situation where the data may be contaminated by outliers. The approach builds upon the foundations developed by Schennach (2005) who proposed the Bayesian exponentially tilted empirical likelihood (BETEL) method, justified by the fact that an empirical likelihood (EL) can be interpreted as the nonparametric limit of a Bayesian procedure when the implied probabilities are obtained from maximizing entropy subject to some given moment constraints. Considering the impact that outliers are thought to have on th…
Transfer entropy, an information-theoretic measure of time-directed information transfer between joint processes, has steadily gained popularity in the analysis of complex stochastic dynamics in diverse fields, including the neurosciences, ecology, climatology and econometrics. We show that for a broad class of predictive models, the log-likelihood ratio test statistic for the null hypothesis of zero transfer entropy is a consistent estimator for the transfer entropy itself. For finite Markov chains, furthermore, no explicit model is required. In the general case, an asymptotic chi-squared dis…
The problem of assigning probability distributions which objectively reflect the prior information available about experiments is one of the major stumbling blocks in the use of Bayesian methods of data analysis. In this paper the method of Maximum (relative) Entropy (ME) is used to translate the information contained in the known form of the likelihood into a prior distribution for Bayesian inference. The argument is inspired and guided by intuition gained from the successful use of ME methods in statistical mechanics. For experiments that cannot be repeated the resulting "entropic prior" is …
The method of Maximum (relative) Entropy (ME) is used to translate the information contained in the known form of the likelihood into a prior distribution for Bayesian inference. The argument is guided by intuition gained from the successful use of ME methods in statistical mechanics. For experiments that cannot be repeated the resulting "entropic prior" is formally identical with the Einstein fluctuation formula. For repeatable experiments, however, the expected value of the entropy of the likelihood turns out to be relevant information that must be included in the analysis. As an example the…
The conditional maximum-entropy method (abbreviated here as C-MaxEnt) is formulated for selecting prior probability distributions in Bayesian statistics for parameter estimation. This method is inspired by a statistical-mechanical approach to systems governed by dynamics with largely-separated time scales and is based on three key concepts: conjugate pairs of variables, dimensionless integration measures with coarse-graining factors and partial maximization of the joint entropy. The method enables one to calculate a prior purely from a likelihood in a simple way. It is shown in particular how …
Given a sequence composed of a limit number of characters, we try to "read" it as a "text". This involves to segment the sequence into "words". The difficulty is to distinguish good segmentation from enormous number of random ones.Aiming at revealing the nonrandomness of the sequence as strongly as possible, by applying maximum likelihood method, we find a quantity called Segmentation Entropy that can be used to fulfill the duty. Contrary to commonplace where maximum entropy principle was applied to obtain good solution, we choose to {\em minimize} the segmentation entropy to obtain good segme…
Maximum likelihood estimation is a valuable tool often applied to inverse problems in quantum theory. Estimation from small data sets can, however, have non unique solutions. We discuss this problem and propose to use Jaynes maximum entropy principle to single out the most unbiased maximum-likelihood guess.
We compare an entropy estimator $H_z$ recently discussed in [10] with two estimators $H_1$ and $H_2$ introduced in [6][7]. We prove the identity $H_z \equiv H_1$, which has not been taken into account in [10]. Then, we prove that the statistical bias of $H_1$ is less than the bias of the ordinary likelihood estimator of entropy. Finally, by numerical simulation we verify that for the most interesting regime of small sample estimation and large event spaces, the estimator $H_2$ has a significant smaller statistical error than $H_z$.
We introduce an algorithm for estimating the entropy of pairwise, probabilistic graph models by leveraging bridges between social communities and an accurate entropy estimator on sparse samples. We propose using a measure of investment from the sociological literature, Burt's structural constraint, as a heuristic for identifying bridges that partition a graph into conditionally independent components. We combine this heuristic with the Nemenman-Shafee-Bialek entropy estimator to obtain a faster and more accurate estimator. We demonstrate it on the pairwise maximum entropy, or Ising, models of …
During the MaxEnt 2002 workshop in Moscow, Idaho, Tony Vignaux asked again a few simple questions about using Maximum Entropy or Bayesian approaches for the famous Dice problems which have been analyzed many times through this workshop and also in other places. Here, there is another analysis of these problems. I hope that, this paper will answer a few questions of Tony and other participants of the workshop on the situations where we can use Maximum Entropy or Bayesian approaches or even the cases where we can actually use both of them. Keywords: Dice problems and probability theory, Maximum …
The main object of this paper is to show how we can use classical probabilistic methods such as Maximum Entropy (ME), maximum likelihood (ML) and/or Bayesian (BAYES) approaches to do microscopic and macroscopic data fusion. Actually ME can be used to assign a probability law to an unknown quantity when we have macroscopic data (expectations) on it. ML can be used to estimate the parameters of a probability law when we have microscopic data (direct observation). BAYES can be used to update a prior probability law when we have microscopic data through the likelihood. When we have both microscopi…
Using the DAMA/LIBRA anomaly as an example, we formalise the notion of halo-independence in the context of Bayesian statistics and quantified maximum entropy. We consider an infinite set of possible profiles, weighted by an entropic prior and constrained by a likelihood describing noisy measurements of modulated moments by DAMA/LIBRA. Assuming an isotropic dark matter (DM) profile in the galactic rest frame, we find the most plausible DM profiles and predictions for unmodulated signal rates at DAMA/LIBRA. The entropic prior contains an a priori unknown regularisation factor, $β$, that describe…
