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After Bayes, the oldest Bayesian account of enumerative induction is given by Laplace's so-called rule of succession: if all $n$ observed instances of a phenomenon to date exhibit a given character, the probability that the next instance of that phenomenon will also exhibit the character is $\frac{n+1}{n+2}$. Laplace's rule however has the apparently counterintuitive mathematical consequence that the corresponding "universal generalization" (every future observation of this type will also exhibit that character) has zero probability. In 1932, the British scientist J. B. S. Haldane proposed an alternative rule giving a universal generalization the positive probability $\frac{n+1}{n+2} \times \frac{n+3}{n+2}$. A year later Harold Jeffreys proposed essentially the same rule in the case of a finite population. A related variant rule results in a predictive probability of $\frac{n+1}{n+2} \times \frac{n+4}{n+3}$. These arguably elegant adjustments of the original Laplacean form have the advantage that they give predictions better aligned with intuition and common sense. In this paper we discuss J. B. S. Haldane's rule and its variants, placing them in their historical context, and relating them to subsequent philosophical discussions.

Bayesian model-averaged meta-analysis allows quantification of evidence for both treatment effectiveness $\mu$ and across-study heterogeneity $\tau$. We use the Cochrane Database of Systematic Reviews to develop discipline-wide empirical prior distributions for $\mu$ and $\tau$ for meta-analyses of binary and time-to-event clinical trial outcomes. First, we use 50% of the database to estimate parameters of different required parametric families. Second, we use the remaining 50% of the database to select the best-performing parametric families and explore essential assumptions about the presence or absence of the treatment effectiveness and across-study heterogeneity in real data. We find that most meta-analyses of binary outcomes are more consistent with the absence of the meta-analytic effect or heterogeneity while meta-analyses of time-to-event outcomes are more consistent with the presence of the meta-analytic effect or heterogeneity. Finally, we use the complete database - with close to half a million trial outcomes - to propose specific empirical prior distributions, both for the field in general and for specific medical subdisciplines. An example from acute respiratory infections demonstrates how the proposed prior distributions can be used to conduct a Bayesian model-averaged meta-analysis in the open-source software R and JASP.

We outline a Bayesian model-averaged meta-analysis for standardized mean differences in order to quantify evidence for both treatment effectiveness $\delta$ and across-study heterogeneity $\tau$. We construct four competing models by orthogonally combining two present-absent assumptions, one for the treatment effect and one for across-study heterogeneity. To inform the choice of prior distributions for the model parameters, we used 50% of the Cochrane Database of Systematic Reviews to specify rival prior distributions for $\delta$ and $\tau$. The relative predictive performance of the competing models and rival prior distributions was assessed using the remaining 50\% of the Cochrane Database. On average, $\mathcal{H}_1^r$ -- the model that assumes the presence of a treatment effect as well as across-study heterogeneity -- outpredicted the other models, but not by a large margin. Within $\mathcal{H}_1^r$, predictive adequacy was relatively constant across the rival prior distributions. We propose specific empirical prior distributions, both for the field in general and for each of 46 specific medical subdisciplines. An example from oral health demonstrates how the proposed prior distributions can be used to conduct a Bayesian model-averaged meta-analysis in the open-source software R and JASP. The preregistered analysis plan is available at https://osf.io/zs3df/.

Testing the equality of two proportions is a common procedure in science, especially in medicine and public health. In these domains it is crucial to be able to quantify evidence for the absence of a treatment effect. Bayesian hypothesis testing by means of the Bayes factor provides one avenue to do so, requiring the specification of prior distributions for parameters. The most popular analysis approach views the comparison of proportions from a contingency table perspective, assigning prior distributions directly to the two proportions. Another, less popular approach views the problem from a logistic regression perspective, assigning prior distributions to logit-transformed parameters. Reanalyzing 39 null results from the New England Journal of Medicine with both approaches, we find that they can lead to markedly different conclusions, especially when the observed proportions are at the extremes (i.e., very low or very high). We explain these stark differences and provide recommendations for researchers interested in testing the equality of two proportions and users of Bayes factors more generally. The test that assigns prior distributions to logit-transformed parameters creates prior dependence between the two proportions and yields weaker evidence when the observations are at the extremes. When comparing two proportions, we argue that this test should become the new default., Comment: 16 pages, 7 figures

Booming in business and a staple analysis in medical trials, the A/B test assesses the effect of an intervention or treatment by comparing its success rate with that of a control condition. Across many practical applications, it is desirable that (1) evidence can be obtained in favor of the null hypothesis that the treatment is ineffective; (2) evidence can be monitored as the data accumulate; (3) expert prior knowledge can be taken into account. Most existing approaches do not fulfill these desiderata. Here we describe a Bayesian A/B procedure based on Kass and Vaidyanathan (1992) that allows one to monitor the evidence for the hypotheses that the treatment has either a positive effect, a negative effect, or, crucially, no effect. Furthermore, this approach enables one to incorporate expert knowledge about the relative prior plausibility of the rival hypotheses and about the expected size of the effect, given that it is non-zero. To facilitate the wider adoption of this Bayesian procedure we developed the abtest package in R. We illustrate the package options and the associated statistical results with a fictitious business example and a real data medical example.

Statistical procedures such as Bayes factor model selection and Bayesian model averaging require the computation of normalizing constants (e.g., marginal likelihoods). These normalizing constants are notoriously difficult to obtain, as they usually involve high-dimensional integrals that cannot be solved analytically. Here we introduce an R package that uses bridge sampling (Meng & Wong, 1996; Meng & Schilling, 2002) to estimate normalizing constants in a generic and easy-to-use fashion. For models implemented in Stan, the estimation procedure is automatic. We illustrate the functionality of the package with three examples.

Across the empirical sciences, few statistical procedures rival the popularity of the frequentist t-test. In contrast, the Bayesian versions of the t-test have languished in obscurity. In recent years, however, the theoretical and practical advantages of the Bayesian t-test have become increasingly apparent and various Bayesian t-tests have been proposed, both objective ones (based on general desiderata) and subjective ones (based on expert knowledge). Here we propose a flexible t-prior for standardized effect size that allows computation of the Bayes factor by evaluating a single numerical integral. This specification contains previous objective and subjective t-test Bayes factors as special cases. Furthermore, we propose two measures for informed prior distributions that quantify the departure from the objective Bayes factor desiderata of predictive matching and information consistency. We illustrate the use of informed prior distributions based on an expert prior elicitation effort.

Bayesian analysis often concerns an evaluation of models with different dimensionality as is necessary in, for example, model selection or mixture models. To facilitate this evaluation, transdimensional Markov chain Monte Carlo (MCMC) relies on sampling a discrete indexing variable to estimate the posterior model probabilities. However, little attention has been paid to the precision of these estimates. If only few switches occur between the models in the transdimensional MCMC output, precision may be low and assessment based on the assumption of independent samples misleading. Here, we propose a new method to estimate the precision based on the observed transition matrix of the model-indexing variable. Assuming a first order Markov model, the method samples from the posterior of the stationary distribution. This allows assessment of the uncertainty in the estimated posterior model probabilities, model ranks, and Bayes factors. Moreover, the method provides an estimate for the effective sample size of the MCMC output. In two model-selection examples, we show that the proposed approach provides a good assessment of the uncertainty associated with the estimated posterior model probabilities.

The marginal likelihood plays an important role in many areas of Bayesian statistics such as parameter estimation, model comparison, and model averaging. In most applications, however, the marginal likelihood is not analytically tractable and must be approximated using numerical methods. Here we provide a tutorial on bridge sampling (Bennett, 1976; Meng & Wong, 1996), a reliable and relatively straightforward sampling method that allows researchers to obtain the marginal likelihood for models of varying complexity. First, we introduce bridge sampling and three related sampling methods using the beta-binomial model as a running example. We then apply bridge sampling to estimate the marginal likelihood for the Expectancy Valence (EV) model---a popular model for reinforcement learning. Our results indicate that bridge sampling provides accurate estimates for both a single participant and a hierarchical version of the EV model. We conclude that bridge sampling is an attractive method for mathematical psychologists who typically aim to approximate the marginal likelihood for a limited set of possibly high-dimensional models.

Many questions in experimental mathematics are fundamentally inductive in nature. Here we demonstrate how Bayesian inference --the logic of partial beliefs-- can be used to quantify the evidence that finite data provide in favor of a general law. As a concrete example we focus on the general law which posits that certain fundamental constants (i.e., the irrational numbers $\pi$, $e$, $\sqrt2$, and $\ln{2}$) are normal; specifically, we consider the more restricted hypothesis that each digit in the constant's decimal expansion occurs equally often. Our analysis indicates that for each of the four constants, the evidence in favor of the general law is overwhelming. We argue that the Bayesian paradigm is particularly apt for applications in experimental mathematics, a field in which the plausibility of a general law is in need of constant revision in light of data sets whose size is increasing continually and indefinitely., Comment: 17 pages (including references), 7 figures, revised version