预测对数损失下幂先验的最优折扣参数
The Optimal Discounting Parameter of the Power Prior under Predictive Log-Loss
AI总结：
本文在预测对数损失下推导出幂先验最优折扣参数的闭式解，为自适应数据借用提供基准，解释归一化幂先验的退化现象，表明正确折扣的常数幂值优于子集划分或衰减幂值。
AI中文摘要：
Ibrahim和Chen提出的幂先验通过将历史似然提升至幂值a₀∈[0,1]，将历史数据融入贝叶斯分析，该幂值的选择一直是个开放问题。本文在预测对数损失下给出了闭式解：对于含d个参数的模型、大小为N₀的历史样本，以及历史与当前数据生成分布间的平均Kullback-Leibler散度$\bar{D}_0$，最优幂值为$a_0^{*}=d/(2N_0\bar{D}_0+d)$。等价地，最优借用的有效样本量服从调和法则$1/E^{*}=1/N_0+2\bar{D}_0/d$：兼容数据被完全合并，且任何差异都会将借用信息限制在$d/(2\bar{D}_0)$个观测值。该结果对多项数据是精确的，可扩展至光滑参数族，该法则为自适应借用提供了基准，解释了归一化幂先验已报道的退化现象，并表明既不对数据进行子集划分也不衰减幂值，不会优于正确折扣的常数。
英文摘要：
The power prior of Ibrahim and Chen incorporates historical data into a Bayesian analysis by raising the historical likelihood to a power $a_0 \in [0, 1]$. The choice of the exponent has remained an open question. This paper gives a closed-form answer under the predictive log-loss. For a model with $d$ parameters, a historical sample of size $N_0$, and average Kullback--Leibler divergence $\bar{D}_0$ between the historical and current data-generating distributions, the optimal exponent is $a_0^{*} = d/(2 N_0 \bar{D}_0 + d)$. Equivalently, the optimally borrowed effective sample size obeys the harmonic law $1/E^{*} = 1/N_0 + 2\bar{D}_0/d$: compatible data are pooled in full, and any difference caps the borrowed information at $d/(2\bar{D}_0)$ observations. The result is exact for multinomial data and extends to smooth parametric families. The law benchmarks adaptive borrowing, explains the reported degeneracy of the normalized power prior, and shows that neither subsetting the data nor decaying the exponent improves on the correctly discounted constant.
