A staple of Bayesian model comparison and hypothesis testing, Bayes factors are often used to quantify the relative predictive performance of two rival hypotheses. The computation of Bayes factors can be challenging, however, and this has contributed to the popularity of convenient approximations such as the BIC. Unfortunately, these approximations can fail in the case of informed prior distributions. Here we address this problem by outlining an approximation to informed Bayes factors for a focal parameter $\theta$. The approximation is computationally simple and requires only the maximum likelihood estimate $\hat\theta$ and its standard error. The approximation uses an estimated likelihood of $\theta$ and assumes that the posterior distribution for $\theta$ is unaffected by the choice of prior distribution for the nuisance parameters. The resulting Bayes factor for the null hypothesis $\mathcal{H}_0: \theta = \theta_0$ versus the alternative hypothesis $\mathcal{H}_1: \theta \sim g(\theta)$ is then easily obtained using the Savage--Dickey density ratio. Three real-data examples highlight the speed and closeness of the approximation compared to bridge sampling and Laplace's method. The proposed approximation facilitates Bayesian reanalyses of standard frequentist results, encourages application of Bayesian tests with informed priors, and alleviates the computational challenges that often frustrate both Bayesian sensitivity analyses and Bayes factor design analyses. The approximation is shown to suffer under small sample sizes and when the posterior distribution of the focal parameter is substantially influenced by the prior distributions on the nuisance parameters. The proposed methodology may also be used to approximate the posterior distribution for $\theta$ under $\mathcal{H}_1$.

翻译：贝叶斯因子作为贝叶斯模型比较与假设检验的核心工具，常用于量化两个竞争假设的相对预测性能。然而，贝叶斯因子的计算极具挑战性，这促使BIC等便捷近似方法被广泛采用。遗憾的是，这些近似在信息先验分布情况下可能失效。本文针对这一问题，提出针对焦点参数θ的信息贝叶斯因子近似方法。该近似方法计算简便，仅需最大似然估计\hatθ及其标准误。方法通过估计θ的似然函数，并假设θ的后验分布不受冗余参数先验分布选择的影响。基于Savage-Dickey密度比，可便捷导出零假设\mathcal{H}_0: θ = θ_0与备择假设\mathcal{H}_1: θ ~ g(θ)的贝叶斯因子。三个实际数据案例表明，相较于桥接抽样与拉普拉斯方法，本近似方法在计算速度与逼近精度方面均具优势。该近似促进了对经典频率学派结果的贝叶斯再分析，鼓励应用信息先验的贝叶斯检验，并缓解了长期困扰贝叶斯敏感性分析与贝叶斯因子设计分析的计算难题。研究表明，当样本量较小或焦点参数的后验分布受冗余参数先验分布的显著影响时，近似效果会恶化。此外，所提方法还可用于近似\mathcal{H}_1条件下θ的后验分布。