On Bayes factor functions

We describe Bayes factors functions based on the sampling distributions of \emph{z}, \emph{t}, $χ^2$, and \emph{F} statistics, using a class of inverse-moment prior distributions to define alternative hypotheses. These non-local alternative prior distributions are centered on standardized effects, which serve as indices for the Bayes factor function. We compare the conclusions drawn from resulting Bayes factor functions to those drawn from Bayes factors defined using local alternative prior specifications and examine their frequentist operating characteristics. Finally, an application of Bayes factor functions to replicated experimental designs in psychology is provided.

💡 Research Summary

The paper introduces the concept of Bayes factor functions (BFFs) as a way to display how Bayes factors change as a function of hyper‑parameters that define the alternative prior. Rather than reporting a single Bayes factor for a hypothesis test, the authors propose to plot the Bayes factor against a scientifically meaningful index—typically a standardized effect size—thereby visualising the whole family of Bayes factors that would result from different prior specifications.

The authors focus on the common test statistics z, t, χ² and F, whose null distributions are well known and whose non‑centrality parameters are scalar. For the alternative hypothesis they adopt non‑local priors that vanish at the null value: an inverse‑moment prior I(λ|τ,ν) for the non‑centrality λ of z and t tests, and an inverse‑gamma prior IG(λ|τ,ν/2) for the squared non‑centrality of χ² and F tests. These priors have modes at ±√(τ/(ν+1)) (or at √(τ) for the squared case) and are indexed by τ and ν. By choosing τ = n(ν+1)ω₁²/2, where n is the sample size and ω₁ is a hypothesised standardized effect, the prior’s mode is placed exactly at the effect size of interest. Consequently the BFF can be expressed as a mapping ω₁ → BF₁₀(z|ω₁,ν), which the paper calls a Bayes factor function.

Theoretical results are presented in four theorems (3.1–3.4). For each of the four test statistics, when the alternative prior is inverse‑moment or inverse‑gamma and τ grows linearly with n, the Bayes factor in favour of the null under a true null hypothesis converges to zero at an exponential rate Oₚ(exp(−c√n)). This is dramatically faster than the polynomial rate Oₚ(n^{−r−½}) obtained with traditional local priors (normal, gamma, etc.). Conversely, when the alternative hypothesis is true, the Bayes factor in favour of the null decays exponentially Oₚ(exp(−cn)) for any continuous prior that is positive at the true non‑null parameter. Thus the non‑local priors provide faster evidence accumulation for both sides of the test.

To assess finite‑sample performance, the authors employ the equality of expected posterior probabilities (DEP) derived by Wagensmakers and Grasman (2025). By assigning equal prior model probabilities, DEP guarantees that the expected posterior probability of each model when it is true is the same. The authors compute DEP for pairs of prior families (inverse‑moment vs. normal, inverse‑gamma vs. gamma, Cauchy, etc.) and show that the non‑local priors yield higher expected posterior probabilities for both the null and the alternative across a range of simulated scenarios.

The paper also demonstrates how BFFs can be used in replicated experimental designs common in psychology. Because τ is proportional to n, each replication’s prior automatically scales with its sample size, leading to a hierarchical model for the standardized effect ω across studies. This hierarchical BFF framework allows meta‑analytic synthesis by estimating a common effect distribution rather than simply averaging individual Bayes factors. An applied example with replicated psychology experiments illustrates that the non‑local BFFs produce more consistent conclusions and reduce the apparent replication failure rate compared with traditional Bayes factors based on local priors.

Practical guidance is offered for choosing ν (the shape parameter controlling the prior’s tail thickness) and for indexing the BFF by ω. The authors recommend ν in the range 1–5 for most applications, and they stress that once a target standardized effect ω₁ is specified, τ follows directly from the formula above, eliminating the need for arbitrary scale choices.

In summary, the paper makes three major contributions: (1) formal definition of Bayes factor functions as a tool for visualising prior sensitivity; (2) theoretical and empirical evidence that inverse‑moment and inverse‑gamma non‑local priors provide superior asymptotic and finite‑sample operating characteristics compared with conventional local priors; and (3) a concrete methodology for applying BFFs to replicated studies and meta‑analysis, thereby enhancing reproducibility in the social and behavioural sciences. The work bridges Bayesian hypothesis testing with practical concerns about prior specification, offering a transparent and robust alternative to the single‑value Bayes factor paradigm.