We outline an approximation to informed Bayes factors for a focal parameter $\theta$ that 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 fail 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 methodology may also be used to approximate the posterior distribution for $\theta$ under $\mathcal{H}_1$.

翻译：我们为一个核心参数 $\theta ${theta$} 概要列出一个知情的 Bayes 系数的近似值,该参数只需要最大可能性估算 $\hat\theta$ $ 及其标准错误。 近似值使用一个估计值 $\theta$ 美元, 并假定美元后方分配值不会因先前分配参数的选择而受到影响。 由此得出的无效假设 $\mathcal{H} 0:\theta =\theta_ 0$ 相对于替代假设 $\mathcal{H}1:\theta\ sim g(theta)$ 然后使用Savage- Dickey 密度比率很容易获得。 三个真实数据示例显示, 与桥梁取样和 Laplace 方法相比, 近似值分布速度和接近度不受影响。 拟议的近似点有助于 Bayesian 标准频率结果的重新分析, 鼓励应用事先知情的 Bayes 测试, 减轻计算挑战, 常常阻碍 Bayes 敏感度分析和 Bayes 系数设计分析。 在小样本 $1 参数下显示, 近似值在先前分配时, 基值下, 比例值的分布也会影响前的分布方法。