In a 1935 paper, and in his book Theory of Probability, Jeffreys developed a methodology for quantifying the evidence in favor of a scientific theory. The centerpiece was a number, now called the Bayes factor, which is the posterior odds of the null hypothesis when the prior probability on the null is one-half. Although there has been much discussion of Bayesian hypothesis testing in the context of criticism of P -values, less attention has been given to the Bayes factor as a practical tool of applied statistics. In this paper we review and discuss the uses of Bayes factors in the context of five scientific applications in genetics, sports, ecology, sociology and psychology.

. The Bayesian approach to probability theory is presented as an alternative to the currently used long-run relative frequency approach, which does not offer clear, compelling criteria for the design of statistical methods. Bayesian probability theory offers unique and demonstrably optimal solutions to well-posed statistical problems, and is historically the original approach to statistics. The reasons for earlier rejection of Bayesian methods are discussed, and it is noted that the work of Cox, Jaynes, and others answers earlier objections, giving Bayesian inference a firm logical and mathematical foundation as the correct mathematical language for quantifying uncertainty. The Bayesian approaches to parameter estimation and model comparison are outlined and illustrated by application to a simple problem based on the gaussian distribution. As further illustrations of the Bayesian paradigm, Bayesian solutions to two interesting astrophysical problems are outlined: the measurement of wea...

This paper shows how particle filtering facilitates likelihood-based inference in dynamic macroeconomic models. The economies can be nonlinear and/or nonnormal. We describe how to use the output from the particle filter to estimate the structural parameters of the model, those characterizing preferences and technology, and to compare different economies. Both tasks can be implemented from either a classical or a Bayesian perspective. We illustrate the technique by estimating a business cycle model with investment-specific technological change, preference shocks, and stochastic volatility.

. The `frequentist' approach to statistics, currently dominating statistical practice in astrophysics, is compared to the historically older Bayesian approach, which is now growing in popularity in other scientific disciplines, and which provides unique, optimal solutions to well-posed problems. The two approaches address the same questions with very different calculations, but in simple cases often give the same final results, confusing the issue of whether one is superior to the other. Here frequentist and Bayesian methods are applied to problems where such a mathematical coincidence does not occur, allowing assessment of their relative merits based on their performance, rather than on philosophical argument. Emphasis is placed on a key distinction between the two approaches: Bayesian methods, based on comparisons among alternative hypotheses using the single observed data set, consider averages over hypotheses; frequentist methods, in contrast, average over hypothetical alternative...

Policy gradient methods are reinforcement learning algorithms that adapt a parameterized policy by following a performance gradient estimate. Conventional policy gradient methods use Monte-Carlo techniques to estimate this gradient. Since Monte Carlo methods tend to have high variance, a large number of samples is required, resulting in slow convergence. In this paper, we propose a Bayesian framework that models the policy gradient as a Gaussian process. This reduces the number of samples needed to obtain accurate gradient estimates. Moreover, estimates of the natural gradient as well as a measure of the uncertainty in the gradient estimates are provided at little extra cost.

This paper presents a method to perform likelihood-based inference in nonlinear dynamic equilibrium economies. This type of models has become a standard tool in quantitative economics. However, existing literature has been forced so far to use moment procedures or linearization techniques to estimate these models. This situation is unsatisfactory: moment procedures suffer from strong small samples biases and linearization depends crucially on the shape of the true policy functions, possibly leading to erroneous answers. We propose the use of Sequential Monte Carlo methods to evaluate the likelihood function implied by the model. Then we can perform likelihood-based inference, either searching for a maximum (Quasi-Maximum Likelihood Estimation) or simulating the posterior using a Markov Chain Monte Carlo algorithm (Bayesian Estimation). We can also compare different models even if they are nonnested and misspecified. To perform classical model selection, we follow Vuong (1989) and use the Kullback-Leibler dis- tance to build Likelihood Ratio Tests. To perform Bayesian model comparison, we build Bayes factors. As an application, we estimate the stochastic neoclassical growth model.

This paper presents a decision-theoretic approach to statistical inference that satisfies the Likelihood Principle (LP) without using prior information. Unlike the Bayesian approach, which also satisfies LP, we do not assume knowledge of the prior distribution of the unknown parameter. With respect to information that can be obtained from an experiment, our solution is more efficient than Wald’s minimax solution. However, with respect to information assumed to be known before the experiment, our solution demands less input than the Bayesian solution. 1

In the problem of parametric statistical inference with a finite parameter space, we study some simple rules for defining posterior upper and lower probabilities directly from the observed likelihood function, without using any prior probabilities. The rules satisfy the likelihood principle and a basic consistency principle ("avoiding sure loss"), they produce vacuous inferences when the likelihood function is constant, and they have other symmetry, monotonicity and continuity properties. The rules can be used to eliminate nuisance parameters, and to interpret the likelihood function and use it in making decisions. To compare the rules, they are applied to the problem of sampling from a finite population. Our results indicate that there are objective statistical methods which can reconcile two general approaches to statistical inference: likelihood inference and coherent inference. Keywords.- Coherence, foundations of statistics, imprecise probabilities, likelihood function, likelihoo...

This paper is devoted primarily to a presentation of some main features of these developments, which seem to have intrinsic as well as historical interest. These developments include an apparently decisive negative outcome. It has seemed to some (including this writer) that any adequate concept of statistical evidence must meet at least certain minimum versions of both of the criteria just indicated. But the difficulties of developing such a concept have become increasingly apparent, and it now seems rather clear that no such adequate concept of statistical evidence can exist.

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