Power Prior

Definition

A power prior is an informative prior distribution constructed from historical data by raising the historical-data likelihood to a power $a_0 \in [0,1]$, which discounts how much the past study influences the current analysis. Introduced by Ibrahim and Chen (and co-authors), it is a principled way to incorporate data from a previous, similar study into a Bayesian analysis.

Key Ideas

• Given historical data $D_0$ with likelihood $L(\theta \mid D_0)$ and an initial prior $\pi_0(\theta)$, the power prior is $$\pi(\theta \mid D_0, a_0) \propto L(\theta \mid D_0)^{a_0}\,\pi_0(\theta).$$
• The scalar $a_0$ interpolates between no borrowing ($a_0 = 0$, prior reduces to $\pi_0$) and full pooling ($a_0 = 1$, historical data treated like current data).
• $a_0$ can be fixed (sensitivity analysis over a grid) or given its own hyperprior (a joint / normalized power prior), letting the data determine the degree of borrowing.

Why It Matters

Power priors give a transparent, tunable mechanism for borrowing strength from prior studies — valuable when current data are limited (small trials, rare events) or when a modeling assumption (such as the existence of a surviving fraction) is itself grounded in earlier evidence. They connect elicitation to actual data rather than subjective guesses.

Related

• Cure-Rate Model
• Variable Selection
• Chen-Ibrahim-Sinha (1999)
• Chen-Ibrahim-Yiannoutsos (1999)
• Chen-Dey-Sinha (2000)
• Ming-Hui Chen
• Joseph G. Ibrahim