The power prior in Bayesian statistics is used to incorporate historical or external data into the current analysis in a flexible and controlled manner. It was introduced to address situations where prior data are available, but their relevance to the current study might be uncertain or partial. The power prior allows statisticians to adjust the influence of historical data on the current analysis by raising the likelihood function of the historical data to a power between 0 and 1.

Definition

Suppose that

  • current data: $D = (X, y)$
  • historical data: $D_0 = (X_0, y_0)$
  • parameter of intereset: $\theta$
  • $\pi_0(\theta)$: initial prior
  • power parameter: $\alpha_0 \in [0, 1]$

The power prior for $\theta$ is defined as

\[\pi(\theta \mid D_0, \alpha_0) \propto L(\theta \mid D_0)^{\alpha_0}\pi_0(\theta),\]

where $L(\theta \mid D_0)$ is the likelihood function based on the historical data $D_0$.

Resulting Posterior

The posterior distribution of $\theta$ given both the current and historical data is

\[\begin{align*} \pi(\theta \mid D, D_0, \alpha_0) &\propto L(\theta \mid D) \times \pi(\theta \mid D_0, \alpha_0) \\ &= L(\theta \mid D)\times L(\theta \mid D_0)^{\alpha_0}\times \pi_0(\theta). \end{align*}\]

We can see that the power parameter $\alpha_0$ adjusts the influence of the historical data.

  • $\alpha_0 = 0$: No influence from $D_0$, and the prior reduces to $\pi_0(\theta)$.
  • $\alpha_0 = 1$: The historical data $D_0$ is fully incorporated.
  • $\alpha_0 \in (0, 1)$: Partial influence, and discounting $D_0$ accordingly.

By adjusting $\alpha_0$, we can control how much the historical data impacts the prior, allowing for uncertainty about its relevance.

Power Prior in GLMs

Consider the following normal linear regression model.

\[\begin{align*} f(y_i \mid X_i, \theta) = \frac{1}{\sqrt{2\pi\sigma^2}}\exp\left\{-\frac{1}{2\sigma^2}(y_i - X_i^\top\theta)^2\right\}, \\ f(y_{0i} \mid X_{0i}, \theta) = \frac{1}{\sqrt{2\pi\sigma^2}}\exp\left\{-\frac{1}{2\sigma^2}(y_{0i} - X_{0i}^\top\theta)^2\right\}. \end{align*}\]

Assume that we take an improper uniform initial prior for $\theta$ as $\pi_0(\theta) \propto 1$, we have

\[\theta \mid D_0, \alpha_0 \sim \mathcal{N}\left((X_0^\top X_0)^{-1}X_0^\top y_0, \frac{\sigma^2}{\alpha_0}(X_0^\top X)^{-1}\right).\]

Then, the posterior distribution of $\theta$ is given as

\[\theta \mid D, D_0, \alpha_0 \sim \mathcal{N}\left((X^\top X + \alpha_0X_0^\top X_0)^{-1}(X^\top y + \alpha_0 X_0^\top y_0), \sigma^2 (X^\top X + \alpha_0 X_0^\top X_0)^{-1}\right).\]

Remarks

  • The posterior mean deviates according to the power parameter. If the underlying distributions of two data are different, it leads the biased estimator.
  • The posterior variance is reduced according to the power parameter. That is, the power prior leads the variance-reduced estimator.

References

  • Ibrahim, Joseph G., and Ming-Hui Chen. “Power prior distributions for regression models.” Statistical Science (2000): 46-60.
  • Ibrahim, Joseph G., et al. “The power prior: theory and applications.” Statistics in medicine 34.28 (2015): 3724-3749.
  • Neuenschwander, Beat, Michael Branson, and David J. Spiegelhalter. “A note on the power prior.” Statistics in medicine 28.28 (2009): 3562-3566.