Bayes factors, in many cases, have been proven to bridge the classic -value based significance testing and bayesian analysis of posterior odds. This paper discusses this phenomena within the binomial A B testing setup (applicable for example to conversion testing). It is shown that the bayes factor is controlled by the of success ratios in two tested groups, which can be further bounded by the Welch statistic. As a result, bayesian sample bounds almost match frequentionist's sample bounds. The link between Jensen-Shannon divergence and Welch's test as well as the derivation are an elegant application of tools from information geometry.
Bounds eq:1 and eq:2 are close to each other by a constant factor (a different small factor is necessary to make the bound small in both the bayesian credibility and p-value sense). The difference (under the normalized constant) is illustrated on fig:example , for the case when one wants to test a relative uplift of @math @math e ^ - ^2 2 $, this is true for p-values much lower than the standard threshold of 0.05. In some sense, the bayesian approach is more conservative and less reluctant to reject than frequentionist tests; this conclusion is shared with other works @cite_1 .
@cite_1
1599813212 2562674776 2610835111 2110097068
We address the problem of ranking question answerers according to their credibility, characterized here by the probability that a given question answerer (user) will be awarded a best answer on a question given the answerer’s question-answering history. This probability (represented by θ ) is considered to be a hidden variable that can only be estimated statistically from specific observations associated with the user, namely the number b of best answers awarded, associated with the number n of questions answered. The more specific problem addressed is the potentially high degree of uncertainty associated with such credibility estimates when they are based on small numbers of answers. We address this problem by a kind of Bayesian smoothing. The credibility estimate will consist of a mixture of the overall population statistics and those of the specific user. The greater the number of questions asked, the greater will be the contribution of the specific user statistics relative to those of the overall population. We use the Predictive Stochastic Complexity (PSC) as an accuracy measure to evaluate several methods that can be used for the estimation. We compare our technique (Bayesian Smoothing (BS)) with maximum a priori (MAP) estimation, maximum likelihood (ML) estimation and Laplace smoothing. We consider in this paper the problem of sampling a high-dimensional probability distribution @math having a density with respect to the Lebesgue measure on @math , known up to a normalization constant @math . Such problem naturally occurs for example in Bayesian inference and machine learning. Under the assumption that @math is continuously differentiable, @math is globally Lipschitz and @math is strongly convex, we obtain non-asymptotic bounds for the convergence to stationarity in Wasserstein distance of order @math and total variation distance of the sampling method based on the Euler discretization of the Langevin stochastic differential equation, for both constant and decreasing step sizes. The dependence on the dimension of the state space of these bounds is explicit. The convergence of an appropriately weighted empirical measure is also investigated and bounds for the mean square error and exponential deviation inequality are reported for functions which are measurable and bounded. An illustration to Bayesian inference for binary regression is presented to support our claims. Software packages usually report the results of statistical tests using p-values. Users often interpret these by comparing them to standard thresholds, e.g. 0.1 , 1 and 5 , which is sometimes reinforced by a star rating (***, **, *). In this article, we consider an arbitrary statistical test whose p-value p is not available explicitly, but can be approximated by Monte Carlo samples, e.g. by bootstrap or permutation tests. The standard implementation of such tests usually draws a fixed number of samples to approximate p. However, the probability that the exact and the approximated p-value lie on different sides of a threshold (the resampling risk) can be high, particularly for p-values close to a threshold. We present a method to overcome this. We consider a finite set of user-specified intervals which cover [0,1] and which can be overlapping. We call these p-value buckets. We present algorithms that, with arbitrarily high probability, return a p-value bucket containing p. We prove that for both a bounded resampling risk and a finite runtime, overlapping buckets need to be employed, and that our methods both bound the resampling risk and guarantee a finite runtime for such overlapping buckets. To interpret decisions with overlapping buckets, we propose an extension of the star rating system. We demonstrate that our methods are suitable for use in standard software, including for low p-values occurring in multiple testing settings, and that they can be computationally more efficient than standard implementations. Given samples from distributions p and q, a two-sample test determines whether to reject the null hypothesis that p = q, based on the value of a test statistic measuring the distance between the samples. One choice of test statistic is the maximum mean discrepancy (MMD), which is a distance between embeddings of the probability distributions in a reproducing kernel Hilbert space. The kernel used in obtaining these embeddings is critical in ensuring the test has high power, and correctly distinguishes unlike distributions with high probability. A means of parameter selection for the two-sample test based on the MMD is proposed. For a given test level (an upper bound on the probability of making a Type I error), the kernel is chosen so as to maximize the test power, and minimize the probability of making a Type II error. The test statistic, test threshold, and optimization over the kernel parameters are obtained with cost linear in the sample size. These properties make the kernel selection and test procedures suited to data streams, where the observations cannot all be stored in memory. In experiments, the new kernel selection approach yields a more powerful test than earlier kernel selection heuristics.