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A Study on the Power Parameter in Power Prior Bayesian Analysis

The power prior and its variations have been proven to be a useful class of informative priors in Bayesian inference due to their flexibility in incorporating the historical information by raising the likelihood of the historical data to a fractional power {\delta}. The derivation of the marginal likelihood based on the original power prior,and its variation, the normalized power prior, introduces a scaling factor C({\delta}) in the form of a prior predictive distribution with powered likelihood. In this paper, we show that the scaling factor might be infinite for some positive {\delta} with conventionally used initial priors, which would change the admissible set of the power parameter. This result seems to have been almost completely ignored in the literature. We then illustrate that such a phenomenon may jeopardize the posterior inference under the power priors when the initial prior of the model parameters is improper. The main findings of this paper suggest that special attention should be paid when the suggested level of borrowing is close to 0, while the actual optimum might be below the suggested value. We use a normal linear model as an example for illustrative purposes.


Introduction
The power prior is a class of informative priors constructed from historical data in Bayesian inference. It allows researchers to incorporate historical information via the likelihood function of the historical data raised to a power. The basic principle is to use a power parameter δ (0 ≤ δ ≤ 1) to control the influence of the historical data on the current study.
This information borrowing strategy was introduced by Chen (1998, 2000) with the formulation π(θ | D 0 , δ) ∝ L(θ | D 0 ) δ π 0 (θ), (1.1) where L(θ|D 0 ) is the likelihood based on the historical data D 0 , π 0 (θ) is an initial prior for the parameter of interest θ, and δ may be assumed to be fixed. Despite the fact that π(θ|D 0 , δ) depends on D 0 and δ, it is often treated as an informative prior for the current study and is essentially a powered posterior based on D 0 . It is worth noting that the power prior in (1.1) and the initial prior π 0 (θ) are not necessarily proper, so long as the resulting posterior is proper. In many practical applications, it seems natural to expect most information to come from the historical data rather than the initial prior, and thus, we frequently adopt non-informative priors such as reference priors (Berger et al., 2009) or weakly informative priors as the initial prior in Bayesian analyses. Assuming that the likelihood based on the current data D is of the form L(θ|D), using the power prior defined in (1.1), the posterior of θ when δ is fixed has the form π(θ | D, D 0 , δ) ∝ L(θ | D)π(θ | D 0 , δ), (1.2) where π(θ|D 0 , δ) is considered as a prior based on the historical data.
An important issue in the use of the power prior is to determine the level of borrowing by selecting a sensible power parameter, which can usually be determined either by expert opinions in advance, or using criteria that can reflect the prior-data conflict. Ibrahim et al. (2015) proposed multiple information-based criteria to determine the level of borrowing based on the data compatibility. Among those criteria, the marginal likelihood criterion corresponding to the empirical Bayes solution has recently been extensively studied in the literature (Gravestock and Held, 2017;Nikolakopoulos et al., 2018;Gravestock and Held, 2019;Wiesenfarth and Calderazzo, 2020;Ollier et al., 2020;Pateras et al., 2021). To derive the marginal likelihood, one first integrates out the model parameter θ from the right hand side of (1.2) to obtain where π(θ|D 0 , δ) is the power prior of the form (1.1), with a normalizing constant C(δ) = Θ L(θ|D 0 ) δ π 0 (θ)dθ, and Θ is the parameter space of θ. Since C(δ) is a function of δ, for the purpose of selecting a power parameter, one cannot drop it when comparing different models indexed by δ. To avoid an infinite C(δ), we denote the marginal likelihood as in which the indicator function I A (δ) = 1 if δ ∈ A and 0 otherwise, and the non-empty set is given by We call A the feasible set of the power parameter δ, which is complete when A = [0, 1], and semi-complete if A = (0, 1] and otherwise incomplete. Then the suggested level of borrowing using the empirical Bayes solution could be written as δ EB = arg max δ∈A m(δ|D 0 , D) (Gravestock and Held, 2017), in which the condition of δ ∈ A to ensure a finite C(δ) in (1.4) seems to be ignored. Consequently, this might influence the marginal likelihood in (1.3) for quantifying the data compatibility. Although the formulation guarantees that δ EB ∈ A, the domain of A might be incomplete, and can thus preclude an optimal level of borrowing.
On the other hand, as a natural Bayesian response to the uncertainty of δ, we can assign a hyperprior π 0 (δ), which results in a hierarchical power prior. When a weakly-informative prior π 0 (δ) is used, hypothetically, the posterior of δ would reflect the data compatibility in a semi-automatic way. Therefore Ibrahim and Chen (2000) proposed the joint power prior by specifying a prior for (θ, δ) jointly with the form π(θ, δ | D 0 ) ∝ L(θ | D 0 ) δ π 0 (θ)π 0 (δ). (1.5) It is noted that with the prior in (1.5), the likelihood principle (Birnbaum, 1962) is violated, since using various forms of the historical likelihood differ by a multiplicative positive constant c 0 would result in different posteriors (Their posteriors would be differed by c δ 0 ; see discussions in Neuenschwander et al., 2009 andYe et al., 2022). We provide a simple example in the Appendix to illustrate its consequence. The prior in (1.5) simply specifies a joint prior of (θ, δ) directly (Ibrahim et al., 2015). If one first specifies a prior π 0 (δ), then specifies a conditional prior of θ given δ using the power prior in (1.1), the normalizing factor C(δ) in π(θ|D 0 , δ) should not be dropped as well. This is due to the fact that the power δ is treated as a parameter. Therefore Duan et al. (2006) proposed the following modified power prior, named as the normalized power prior, given by which obviously obeys the likelihood principle since a multiplicative constant before L(θ|D 0 ) will be canceled out (Neuenschwander et al., 2009). This prior exists under a non-restrictive assumption of a non-empty set for A, since 1 ∈ A as long as the initial prior leads to a proper posterior in a conventional Bayesian analysis. Of particular note is that in the current literature, almost all normalized power prior formulae omit I A (δ) and assume either δ ∈ [0, 1] with a proper prior π 0 (θ) or δ ∈ (0, 1] with an improper prior π 0 (θ). In this study, we will show that the latter may not be the case even for the commonly used improper priors that can yield proper posteriors, such as some reference priors. This finding justifies the importance of studying the range of the admissible δ and/or the feasible set A in the marginal likelihood (1.3) and the normalized power prior (1.6). However, such importance has not been given sufficient consideration in the past.
The major contribution of this study is to mathematically examine an important but almost completely ignored key point that, with a commonly used improper initial prior that is believed to be objective, the feasible parameter space A of δ might be restricted to only a subregion of (0, 1], i.e., A may be incomplete. With the exception of Duan et al. (2006), almost all the research articles by default assume δ ∈ (0, 1]. As a result, either the marginal likelihood in (1.3) or the normalized power prior (1.6) might not be able to accurately quantify the data compatibility. This is because A could possibly exclude the optimal range of δ. We further prove that, under certain conditions, the feasible set A is a convex set with a lower limit δ * ≥ 0, which suggests that the phenomenon described above might happen when the heterogeneity between D 0 and D is strong. For instance, when the optimal level of borrowing under strong heterogeneity is close to 0, the set A can exclude it by definition. Albeit its impact on the analysis could be mild in many scenarios, when using the empirical Bayes or the normalized power prior, researchers should be vigilant against the use of certain predominantly used improper priors as the initial objective prior, such as the reference prior in a normal linear model. The root cause is that, despite the improper prior with a full likelihood can yield a proper posterior, the same prior with a fractional likelihood may lead to an improper posterior for some δ.
The rest of the paper is organized as follows. In Section 2, we establish theoretical results regarding the propriety of the power priors in general. Explicit results under the normal linear model are provided in Section 3 with several commonly used initial priors. In Section 4, we conduct a numerical study to illustrate the undesirable behavior when using the power prior with empirical Bayes or the normalized power prior in a normal linear model. We further discuss the implications of our results in Section 5.

Propriety of the Power Priors
In this section, we discuss some fundamental issues regarding the propriety of the power priors in general, which shed light on the form of a feasible set A defined in (1.4). Proofs of the theorems are provided in the Appendix. For all of the following results, we consider the power prior model with the form π(θ|D 0 , δ) ∝ L(θ|D 0 ) δ π 0 (θ) as described in (1.1).
Theorem 1. When the initial prior π 0 (θ) is proper, the power prior is always proper with a non-negative δ. Therefore the range of feasible δ defined in (1.3) is complete, i.e., It is well known that with a proper prior and a non-degenerate (full) likelihood, the set of observations in which the posterior is improper is a Lebesgue null set. This guarantees that the propriety of the posterior holds almost surely with a proper prior. Theorem 1 demonstrates that using a proper (initial) prior π 0 (θ) with a fractional likelihood would be a safe choice since it can also almost surely guarantee posterior propriety. However, we show in the following theorem that, this may not be the case when an improper (initial) prior of θ is used.
Remark. When the initial prior π 0 (θ) is improper, the power prior may or may not be proper, even if the posterior with the corresponding full likelihood is proper. In other words, Note that unlike the proof of Theorem 1, here Jensen's inequality can no longer guarantee a finite upper bound for an improper density π 0 (θ). More details can be found in the Appendix. A primary example is also provided in Section 3 for the normal linear model. This suggests that when using the marginal likelihood criterion (1.3) or the normalized power prior (1.6) with an improper initial prior on θ, the feasible set A might be restricted to a subregion of (0, 1], which is incomplete. To better understand their properties, we have the following theorem.
Theorem 2. If the power prior with a positive power parameter δ * is proper, then for all Based on the Theorem 2 and all the related results above, we come up with the following corollary regarding the feasible parameter space A of δ in the formulation of the marginal likelihood (1.3) and the normalized power prior (1.6).
Corollary 2.1. The feasible parameter space A of δ in the marginal likelihood (1.3) and the normalized power prior (1.6) is a convex set with upper limit 1 and lower limit δ * , where

Investigation on the Feasible Set for the Normal Linear Model
In this section we provide some results under the commonly used normal linear model, with detailed derivations in the Appendix. These results will serve as examples to illustrate some of the findings obtained in Section 2, and will be further used for a numerical study in Section 4. We derive the results for normal linear regression model with a common unknown variance σ 2 , while the result for a known variance is given by Ibrahim et al. (2015). The model is specified as where I n is the n × n identity matrix, N n (0 n , σ 2 I n ) denotes the n-dimensional multivariate normal distribution with mean 0 n , the n-dimensional column vector of zeroes, and covariance matrix σ 2 I n . The dimension of the vector Y is also n and that of β is p. Similarly, for historical data we assume Y 0 = X 0 β + ǫ 0 , with ǫ 0 ∼ N n 0 (0 n 0 , σ 2 I n 0 ), and both X ′ 0 X 0 and X ′ X are positive definite. We adopt similar notations to Ye et al. (2022) but consider the prior with a more general form where R is a known p × p real-valued positive-definite matrix, µ 0 is a known p × 1 real vector, t and b are non-negative real numbers, k = 0 or 1. This class of prior includes several commonly used ones in the linear model as special cases. For example, when t > 1 + p/2, b > 0, k = 1 and R is positive definite, the prior (3.1) is the proper conjugate multivariate normal-inverse-gamma prior denoted as N p -Γ −1 (µ 0 , R, t − p/2 − 1, b). If t = 1 + p/2, b = 0, k = 1, and matrix R = g −1 (X ′ X) with g is a positive constant, the prior has the form which is the well-known Zellner's g-prior (Zellner, 1986). When t = 1, k = b = 0, the general form reduces to a reference prior π 0 (β, σ 2 ) ∝ 1/σ 2 (Berger et al., 2009). (b) If the initial prior (3.1) satisfies t ≥ 1 + p/2, then the power prior is proper for all δ ∈ (0, 1]. This includes Zellner's g-prior and the proper conjugate normal-inversegamma prior. Next, we provide some closed-form results to guide the choice of δ when we use the power prior with a deterministic information-based criterion. In addition to the empirical Bayes approach described above based on the marginal likelihood criterion in (1.3), we include another information criterion, the deviance information criterion (DIC) (Spiegelhalter et al., 2002), which is extensively used for model selection in Bayesian statistics. For notational simplicity, we definê Then we have the following result.

(a) The marginal likelihood in (1.3) is of the form
The DIC for a model with a certain value of δ, denoted as DIC(δ|D 0 , D), up to a constant, can be expressed as In both parts, ν 0 = n 0 δ−p , and It is worth noting that in the case when π 0 (β, σ 2 ) is the reference prior, the power prior (with fixed δ) for δ < p/n 0 is still appropriately defined, since after incorporating the full likelihood of the current data D, the resulting posterior is still proper. However, the normalized power prior is only defined with A = (p/n 0 , 1]. Likewise, when using the empirical Bayes method to choose δ EB , it must be within the range of A, while using other criterion, for example the DIC, can yield different yet more suitable results. This will be illustrated via numerical examples below.

Numerical Examples
We conduct two numerical studies to explore how data heterogeneity could impact the choice of δ using different criteria and initial priors, and assess its impact on making posterior inference. We primarily illustrate our findings with the widely used marginal likelihood criterion, which is also equivalent to the use of the posterior mode of δ with the normalized power prior under a uniform initial prior on δ.
In an intercept-only model (normal population without covariates), the criteria (3.2) and (3.3) can be simplified to a function of the sample statisticsȳ 0 −ȳ when the sample variances of D 0 and D are the same. We setȳ 0 −ȳ at various levels which creates different degrees of heterogeneity, and plot its relationship with the selected δ in Figure 1. In this experiment, the sample sizes of D 0 and D are n 0 = n = 10, with standard deviations s = s 0 = 0.5 respectively, and the sample mean of the current data isȳ = 0. The solid curve (labeled as EB1) is the δ EB using reference (initial) prior π 0 (µ, σ 2 ) ∝ 1/σ 2 , and each point on the dotted curve is the δ associated with the minimum DIC with the same reference prior. The dashed curve (labeled as EB2) is the δ EB with π 0 (µ, σ 2 ) = π 0 (σ 2 )π 0 (µ|σ 2 ), where π 0 (σ 2 ) ∝ 1/σ 2 and π 0 (µ|σ 2 ) ∼ N(0, 10 4 σ 2 ). Note that the initial prior used in EB1 is equivalent to the use of the prior of the form (3.1) when p = 1, with t = 1 and k = b = 0.
The general trend in Figure 1 shows that the optimal δ will decrease with the increase ofȳ 0 −ȳ, which reflects the prior-data conflict in an expected way. However, the desired δ should be very close to 0 when the discrepancy betweenȳ 0 andȳ is large. This is not the case when using an empirical Bayes with reference prior (EB1), since the range of feasible δ is (0.1, 1], which precludes values below 0.1. In other words, enforcing δ ∈ (0.1, 1] will possibly result in borrowing more information than the optimal choice of δ under strong heterogeneity. This suggests that both the empirical Bayes and the normalized power prior should be cautiously used with improper initial priors. To assess its impact on the inferential results for model parameters, in the following experiment, we consider a linear regression model with an intercept and three covariates so the regression parameter is β = (β 1 , β 2 , β 3 , β 4 ) ′ , and the variance parameter is σ 2 . To generate different levels of heterogeneity between the historical and the current data, we simulate current data D with β = (β 1 , β 2 , β 3 , β 4 ) ′ = (1, 1, 1, 1) ′ and simulate historical data D 0 with β = (1, 1, 1, β 04 ) ′ , where β 04 takes a grid of the values between 1 and 3. We generate 10 4 data sets for each scenario, with sample sizes n = n 0 = 20 and the covariates are generated from the uniform distribution on (0, 1). For each dataset, we first choose the optimal δ with criteria similar to those in Figure 1, and display the results of the average δ over the 10 4 samples in Figure 2 (left). For each data set, we also calculate the posterior mean of β 4 with the power priors using the corresponding δ, and report the logarithm of the mean squared error (logMSE) in Figure 2 (right). Likewise, the empirical Bayes with the reference prior of the form π 0 (β, σ 2 ) ∝ 1/σ 2 is denoted as EB1, and the empirical Bayes with the prior of the form π 0 (β, σ 2 ) ∝ π 0 (σ 2 )π 0 (β|σ 2 ), where π 0 (σ 2 ) ∝ 1/σ 2 and π 0 (β|σ 2 ) ∼ N 4 (0 4 , 10 4 σ 2 I 4 ), is denoted as EB2. The DIC is obtained by using the reference prior described above as the initial prior, but selecting the optimal δ via the DIC. Setting δ as random and using the normalized power prior (with π 0 (δ) ∼ unif(0, 1)) with (initial) reference prior provides similar results to the EB1, so it is not shown in the plots. Similar to Figure 1, we observe from Figure 2 (left) that on average, the borrowing strength will decrease when the level of heterogeneity between D 0 and D increases, but the lower limit is p/n 0 = 0.2 when a reference prior is used with the empirical Bayes (this is also the lower bound of δ if using the normalized power prior). Therefore, when estimating the parameter β 4 , at least 20% of the information will be borrowed on average from historical data regardless of the strength of heterogeneity. As a result, the logMSE increases monotonically with β 0,4 − β 4 . On the other hand, the lower limit of δ in the other two approaches is 0, so eventually their logMSE can be maintained to the same level as no borrowing.
Overall we can conclude that, when using either the empirical Bayes with a fixed δ, or the normalized power prior in a normal linear model, an improper initial prior may prevent the selection of an optimal δ. The primary example is the reference prior, which is commonly used in Bayesian analysis with the full likelihood. On the other hand, under these scenarios, the DIC and other information criteria could be better alternatives.

Concluding Remarks
In this study, we have discussed a critical yet largely ignored key point for the use of the power priors or its modified form, the normalized power prior, to borrow information from historical data in conducting a new study. By establishing general results regarding the propriety of a powered posterior (i.e., using a fractional likelihood with power δ) under various initial priors, we showed that the lower limit for A, the feasible set of δ, is not necessarily 0 with an improper initial prior, even if an initial prior can yield a proper posterior in conventional Bayesian inferences (i.e., δ = 1). We thus advocated the use of a more rigorous formulation for the normalized power prior, as well as for the formula of the marginal likelihood. These formulations are provided in (1.3) and (1.6), which account for the aforementioned restriction on the power parameter.
What is the influence of this result on Bayesian inference? We showed in a normal linear model, with the reference prior as the initial prior, the lower limit of feasible δ is p/n 0 when using the normalized power prior or the marginal likelihood criterion. Such impact could have a strong bearing on parameter estimates especially when the sample size of the historical data is small or moderate with some covariates, whereas the heterogeneity between the historical and the current data is strong. Therefore, an ideal level of borrowing may fall below the lower limit p/n 0 . This might be encountered especially in the case when one splits the whole historical dataset into multiple small ones and borrows each of them individually (Banbeta et al., 2019). In these cases, we should avoid using either the marginal likelihood criterion or the normalized power prior and choose other criteria instead. In reality, it is also suggested to use multiple information criteria when possible, while another option is to use a (vague) proper initial prior.
More generally, sampling from the posterior based on a fractional likelihood is not only used for informative prior elicitation, but also a technique widely used in Bayesian computation for more generic problems. For instance, Friel and Pettitt (2008) calculated the normalizing constant based on ideas of thermodynamic integration or path sampling (Gelman and Meng, 1998) with the identity where the feasible set A is similarly defined as in (1.3). We use the same notation z as in Gelman and Meng (1998) to denote the normalizing constant in a general Bayesian model, and L(θ|D) is the likelihood based on the data D. Note that under this general setting we do not consider the historical data D 0 , so essentially D takes the role of D 0 in (1.1), and now (1.1) becomes a powered posterior. Then π(θ|D, δ * ) ∝ π(θ)L(θ|D) δ * , where π(θ) denotes an arbitrary prior for θ such that z = Θ π(θ)L(θ|D)dθ. The integrand is approximated by sampling from a sequence of the posterior densities based on different powered likelihoods where 0 ≤ t 1 < . . . < t s ≤ 1, and the integral is approximated by the trapezoidal rule. Often π(θ) is assumed to be proper such that the sequence of the corresponding powered posteriors is believed to be proper, while Theorem 1 provided the evidence towards this belief. When an improper prior is used, our result indicated that the starting point t 1 is not necessarily very close to 0. If an improper prior that can yield a proper posterior based on the powered likelihood with t 1 > 0, Theorem 2 demonstrated that sampling from all the subsequent powered posteriors with t is valid.

Appendix: Additional Examples and Proofs of Theorems
An example to illustrate the differences between (1.5) and (1.6) For independent Bernoulli trials with y 0 successes out of the n 0 trials in D 0 , suppose the probability of success is θ. Then the historical likelihood based on the product of independent Bernoulli densities is L(θ|D 0 ) = θ y 0 (1 − θ) n 0 −y 0 . Assuming that π 0 (θ) ∼ Beta(a 1 , a 2 ), with a 1 > 0, a 2 > 0, and π 0 (δ) is a proper prior for δ, the joint power prior using (1.5) is of the form where π J (θ, δ|D 0 ) stands for a joint power prior. If we use the normalized power prior (1.6), the prior is of the form These two priors differ by a multiplicative factor B(δy 0 + a 1 , δ(n 0 − y 0 ) + a 2 ) which is not a constant. Furthermore, if we use the likelihood based on the sufficient statistics y 0 , which follows a binomial distribution, the joint power prior denoted as π * J (θ, δ|D 0 ) is of the form This is clearly different from π J (θ, δ|D 0 ), which indicates a violation of the likelihood principle (Birnbaum, 1962;Duan et al., 2006;Neuenschwander et al., 2009). The normalized power prior remains unchanged since the extra term cancels in the numerator and denominator.
Some explanations on the Remark in Section 2 We first show that the inequality in (A.1) in the proof of Theorem 1 is no longer valid if the density π 0 (θ) is not a valid (normalized) probability density function. Suppose π 0 (θ) = cπ * (θ), where Θ π * (θ)dθ = 1, and c is a positive number. The Jensen's inequality indicates that This does not satisfy the inequality in (A.1). Moreover, when c is not finite, Jensen's inequality fails to provide a finite upper bound. Therefore to utilize the Jensen's inequality, one has to consider the normalizing constant, and therefore the result in (A.1) cannot be generalized to an improper prior π 0 (θ). A primary example to illustrate that the powered posterior based on D 0 can be either proper or improper with an improper initial prior is given in the last paragraph of Section 3 in a normal linear model.
This completed the derivation of Result 2.