Conditional and Joint Power Priors

Why the distinction matters

A power prior discounts the contribution of historical data through a power parameter, often denoted \(a_0\). There are two different questions an analysis might ask. One is: what is the prior for the model parameter \(\theta\), given a particular value of \(a_0\)? The other is: what is the prior for both \(\theta\) and \(a_0\), when \(a_0\) itself is uncertain? The first is conditional; the second is joint. A distribution for \(\theta\) at each fixed \(a_0\) does not, by itself, specify a probability distribution for \(a_0\).

This distinction is consequential because the conditional power prior must be normalized separately at each discount value. Once a probability distribution for \(a_0\) is introduced, that normalization determines whether the marginal prior for \(a_0\) is the distribution intended by the analyst. A missing or misplaced normalizing constant can therefore change the model, not just its notation.

Conditional power prior

Let \(D_0\) denote historical data, with likelihood \(L(\theta\mid D_0)\), and let \(\pi_0(\theta)\) be a proper baseline prior. For \(a_0\in[0,1]\), define the conditional normalizing constant

C(a_0)=\int L(\theta\mid D_0)^{a_0}\pi_0(\theta)\,d\theta.

Assume \(C(a_0)\) is finite and positive over the discount values considered. The normalized conditional power prior is

\pi(\theta\mid D_0,a_0) = \frac{L(\theta\mid D_0)^{a_0}\pi_0(\theta)}{C(a_0)}.

For each fixed \(a_0\), this is a probability density over \(\theta\): integrating it over \(\theta\) gives one. At \(a_0=0\), the historical likelihood contributes a factor of one and the prior reduces to \(\pi_0(\theta)\). At \(a_0=1\), the historical likelihood enters at full power and the result is the ordinary posterior based on \(D_0\) and \(\pi_0\). Intermediate values give intermediate powers of the historical likelihood.

The phrase “conditional on \(a_0\)” is essential. This formula describes the distribution of \(\theta\) at a chosen value of \(a_0\); it does not assign probabilities to different values of \(a_0\). If the analysis fixes \(a_0\), this conditional distribution is the relevant power prior. If the analysis treats \(a_0\) as uncertain, a probability distribution for it must also be specified.

Worked Example: A Conditional Beta Power Prior

Suppose a hypothetical historical control study observes 8 responses among 20 patients. Let \(\theta\) be the response probability, and use the baseline prior \(\theta\sim\operatorname{Beta}(1,1)\). The historical likelihood, including its binomial coefficient, is

L(\theta\mid D_0) = {20\choose 8}\theta^8(1-\theta)^{12}.

For a fixed \(a_0=0.5\), the unnormalized conditional density is proportional to \[ \left\{{20\choose 8}\theta^8(1-\theta)^{12}\right\}^{0.5} \theta^{1-1}(1-\theta)^{1-1}. \] The binomial coefficient does not depend on \(\theta\), so it cancels when this density is normalized over \(\theta\). The remaining powers are \(\theta^4(1-\theta)^6\), giving

\theta\mid D_0,a_0=0.5 \sim \operatorname{Beta}(1+8(0.5),\,1+12(0.5)) =\operatorname{Beta}(5,7).

Its mean is \(5/(5+7)=0.4167\). This is the mean of the conditional prior for \(\theta\) at the specified discount, not an estimate that accounts for uncertainty about the discount. The example uses a proper baseline prior, and the beta density is integrable, so the conditional normalizing constant is finite.

From a conditional prior to a joint prior

To model uncertainty in \(a_0\), specify a probability distribution \(\pi(a_0)\) on \([0,1]\). Combining it with the normalized conditional power prior gives the conditional-normalized joint prior

\pi(\theta,a_0\mid D_0) = \pi(\theta\mid D_0,a_0)\pi(a_0) = \frac{L(\theta\mid D_0)^{a_0}\pi_0(\theta)}{C(a_0)} \pi(a_0).

This construction has the intended marginal distribution for the discount before observing current-trial data. Integrating out \(\theta\) gives

\int \pi(\theta,a_0\mid D_0)\,d\theta = \pi(a_0) \frac{\int L(\theta\mid D_0)^{a_0}\pi_0(\theta)\,d\theta}{C(a_0)} = \pi(a_0).

Thus, if \(\pi(a_0)\) is uniform, the marginal prior on \(a_0\) is uniform. This is a statement about the model before current-trial data are observed; after those data are observed, the posterior distribution of \(a_0\) will generally differ.

There is also a distinct construction sometimes encountered in which the joint density is made proportional to \(L(\theta\mid D_0)^{a_0}\pi_0(\theta)\pi(a_0)\), without dividing by \(C(a_0)\) at each \(a_0\). To make that expression a joint probability density, it must be divided by one overall constant, \(\int C(a_0)\pi(a_0)\,da_0\). In this construction the marginal prior for \(a_0\) is proportional to \(C(a_0)\pi(a_0)\), not generally to \(\pi(a_0)\). The two constructions therefore encode different prior distributions on the discount. The formulas, rather than the label used for a construction, identify which model is being used.

How current data update the joint model

Let \(D\) denote the current-trial data, with likelihood \(L(\theta\mid D)\). Under the conditional-normalized joint prior, Bayes’ rule gives a posterior density proportional to the current likelihood times the joint prior:

\pi(\theta,a_0\mid D,D_0) \propto L(\theta\mid D)\, \pi(\theta\mid D_0,a_0)\pi(a_0).

Integrating over \(\theta\) shows that the marginal posterior for \(a_0\) is proportional to its prior density times the current-data predictive probability, conditional on that discount:

\pi(a_0\mid D,D_0) \propto \pi(a_0) \int L(\theta\mid D)\pi(\theta\mid D_0,a_0)\,d\theta.

The integral measures how well the current data are predicted by the conditional prior at that value of \(a_0\). It is sometimes called a marginal likelihood or model evidence for the current data, conditional on \(a_0\). The full posterior is a mixture over discount values: the posterior for \(\theta\) averages the conditional posteriors using the posterior distribution of \(a_0\) as the mixing distribution.

Key distinction: \(\pi(\theta\mid D_0,a_0)\) is a normalized distribution for \(\theta\) at a specified discount. A joint prior \(\pi(\theta,a_0\mid D_0)\) additionally assigns probability to the discount. Under the conditional-normalized construction, its marginal distribution for \(a_0\) is exactly the specified \(\pi(a_0)\).

Worked Example: Updating a Discrete Discount Distribution

Continue with the hypothetical historical data of 8 responses among 20 patients and the \(\operatorname{Beta}(1,1)\) baseline prior. Suppose the prior for \(a_0\) places probability \(0.5\) at each of \(0.25\) and \(0.75\). A hypothetical current study observes one response among one patient. We calculate the current-data predictive probability at each discount.

At \(a_0=0.25\), the conditional prior is \(\operatorname{Beta}(3,4)\). The probability of one response in one trial is its mean: \[ P(D\mid a_0=0.25,D_0)=\frac{3}{3+4}=\frac{3}{7}=0.4286. \] At \(a_0=0.75\), the conditional prior is \(\operatorname{Beta}(7,10)\), giving \[ P(D\mid a_0=0.75,D_0)=\frac{7}{7+10}=\frac{7}{17}=0.4118. \]

The unnormalized posterior probabilities for the two discount values are their prior probabilities multiplied by these predictive probabilities: \[ 0.5(0.4286)=0.2143,\qquad 0.5(0.4118)=0.2059. \] Their sum is \(0.4202\). Normalizing gives \[ P(a_0=0.25\mid D,D_0)=\frac{0.2143}{0.4202}=0.5100,\qquad P(a_0=0.75\mid D,D_0)=\frac{0.2059}{0.4202}=0.4900. \] The observed response makes the lower discount slightly more probable under this model, because its conditional prior assigned slightly higher predictive probability to a response. The difference is small; it does not establish that one historical-data discount is objectively correct. Given the response, the conditional posterior distributions for \(\theta\) are \(\operatorname{Beta}(4,4)\) at \(a_0=0.25\) and \(\operatorname{Beta}(8,10)\) at \(a_0=0.75\). The marginal posterior for \(\theta\) is their mixture with weights \(0.5100\) and \(0.4900\), respectively.

Why the normalizing constant cannot be casually omitted

For a fixed discount, multiplying a likelihood by a positive factor that does not depend on \(\theta\) does not change the normalized conditional density for \(\theta\): that factor appears in both numerator and normalizing constant. But when the discount is random, \(C(a_0)\) varies with \(a_0\). Omitting it from a joint construction can therefore alter the relative prior weights assigned to discount values. This is also why the precise likelihood convention matters in an unnormalized joint construction: a factor constant in \(\theta\) can still depend on \(a_0\) after it is raised to the power \(a_0\).

Worked Example: The Induced Discount Prior Without Conditional Normalization

Use again the hypothetical binomial historical data, \(y_0=8\) responses among \(n_0=20\), with \(\theta\sim\operatorname{Beta}(1,1)\). Consider the unnormalized-in-\(\theta\) construction proportional to \(L(\theta\mid D_0)^{a_0}\pi_0(\theta)\pi(a_0)\), and suppose \(\pi(a_0)\) is uniform on \([0,1]\). Integrating out \(\theta\) shows that its marginal density for \(a_0\) is proportional to \(C(a_0)\).

At \(a_0=0\), \(L(\theta\mid D_0)^0=1\), so \(C(0)=\int_0^1 1\,d\theta=1\). At \(a_0=1\), the binomial likelihood is the full likelihood, and \[ C(1)=\int_0^1 {20\choose 8}\theta^8(1-\theta)^{12}\,d\theta=\frac{1}{21}. \] The last equality follows because a uniform beta prior gives equal prior-predictive probability \(1/(20+1)\) to each of the 21 possible binomial response counts. Thus, before current data are observed, the unnormalized construction’s marginal density for \(a_0\) at 1 is only \(1/21\) of its density at 0, despite starting with a uniform \(\pi(a_0)\). By contrast, including \(1/C(a_0)\) in the conditional-normalized joint prior makes the marginal density for \(a_0\) exactly uniform. The calculation demonstrates why the two constructions should not be treated as interchangeable.

Assumptions, checks, and common pitfalls

The constructions above require a well-defined likelihood for the historical data, a baseline prior, and finite positive normalizing constants over the discount values assigned positive probability. The conditional-normalized joint prior also requires a proper probability distribution \(\pi(a_0)\). In applications, verify that the parameter space and baseline prior make \(C(a_0)\) finite, and verify that the joint density integrates to one. With continuous \(a_0\), the marginalization check is direct; with a discrete distribution, the corresponding integral is a sum.

A useful diagnostic is to calculate or approximate \(C(a_0)\) at several discount values. If it changes substantially, omitting it is likely to have a substantial effect on the implied discount prior. For the joint posterior, also check that the predictive integral over \(\theta\) is computed under the normalized conditional prior and that the posterior weights over \(a_0\) sum or integrate to one. Numerical integration or Monte Carlo approximations should be checked for stability, particularly where \(C(a_0)\) is small.

Pitfall: treating a conditional density as a joint prior. Knowing \(\pi(\theta\mid D_0,a_0)\) does not specify how probable each \(a_0\) is. Add an explicit \(\pi(a_0)\) to define a joint model.
Pitfall: saying the discount prior remains unchanged after current data. Under the conditional-normalized construction, the specified distribution is the prior marginal. Current data update it through the conditional predictive probability, so the posterior usually differs.
Pitfall: dropping \(C(a_0)\) because it does not involve \(\theta\). It varies with \(a_0\), so it affects the marginal distribution of the discount in a joint model.
Pitfall: interpreting a conditional result as an average over discount uncertainty. A posterior for \(\theta\) at one specified discount is not the marginal posterior under a joint model; the latter averages across discounts using their posterior weights.

The conditional construction is appropriate when the discount is specified and the analysis is explicitly conditional on that value. A joint construction is appropriate when the analysis assigns a distribution to the discount and uses current data to update it. The choice is a modeling decision: each construction implies a different prior structure, and the analysis should state which density is used and what marginal distribution for \(a_0\) it induces.

Check Your Understanding

Answer each question, then compare with the brief solutions below.

What does the conditional power prior \(\pi(\theta\mid D_0,a_0)\) describe, and what additional ingredient is required to define a joint prior?
Under the conditional-normalized joint construction, what is the marginal prior density of \(a_0\) after integrating out \(\theta\)?
For \(y_0=8\) responses among 20 and a \(\operatorname{Beta}(1,1)\) baseline prior, what is the conditional prior for \(\theta\) at \(a_0=0.25\)?
In the discrete-discount example, why did the posterior probability of \(a_0=0.25\) exceed its prior probability?
Why can omitting \(C(a_0)\) change the marginal prior distribution of a random discount?

Solutions. (1) It describes the distribution of \(\theta\) at a specified discount; a probability distribution \(\pi(a_0)\) is additionally required for the conditional-normalized joint prior. (2) It is \(\pi(a_0)\), since the conditional density integrates to one for every \(a_0\). (3) It is \(\operatorname{Beta}(1+8(0.25),1+12(0.25))=\operatorname{Beta}(3,4)\). (4) The current response had predictive probability \(0.4286\) at \(0.25\), slightly higher than \(0.4118\) at \(0.75\), so the data reweighted the prior probabilities in favor of \(0.25\). (5) Because \(C(a_0)\) generally varies with \(a_0\); integrating the unnormalized joint expression over \(\theta\) weights the discount prior by \(C(a_0)\).