Optimal Priors for the Discounting Parameter of the Normalized Power Prior

TL;DR

This paper investigates the behavior of the discounting parameter in normalized power priors for generalized linear models, proving convergence properties and proposing methods to construct optimal priors that balance borrowing and conflict.

Contribution

It provides theoretical insights into the marginal posterior of the discounting parameter and introduces procedures for eliciting optimal beta priors based on divergence and error minimization.

Findings

Posterior for the discounting parameter converges to zero with data discrepancy.

Optimal priors differ significantly from uniform priors.

Proposed priors effectively balance borrowing and conflict.

Abstract

The power prior is a popular class of informative priors for incorporating information from historical data. It involves raising the likelihood for the historical data to a power, which acts as discounting parameter. When the discounting parameter is modelled as random, the normalized power prior is recommended. In this work, we prove that the marginal posterior for the discounting parameter for generalized linear models converges to a point mass at zero if there is any discrepancy between the historical and current data, and that it does not converge to a point mass at one when they are fully compatible. In addition, we explore the construction of optimal priors for the discounting parameter in a normalized power prior. In particular, we are interested in achieving the dual objectives of encouraging borrowing when the historical and current data are compatible and limiting borrowing when they are in conflict. We propose intuitive procedures for eliciting the shape parameters of a beta prior for the discounting parameter based on two minimization criteria, the Kullback-Leibler divergence and the mean squared error. Based on the proposed criteria, the optimal priors derived are often quite different from commonly used priors such as the uniform prior.

Tables

Table 1. Table 1: Posterior mean (variance) of μ 𝜇 \mu for the normal i.i.d. case

	$d_{\text{obs}}=0$	$d_{\text{obs}}$=0.5	$d_{\text{obs}}$=1	$d_{\text{obs}}$=1.5
$d_{\text{MTD}}$=0.5	1.5 (0.022)	1.85 (0.026)	2.32 (0.036)	2.87 (0.036)
$d_{\text{MTD}}$=1	1.5 (0.019)	1.82 (0.026)	2.36 (0.042)	2.93 (0.035)
$d_{\text{MTD}}$=1.5	1.5 (0.018)	1.78 (0.020)	2.19 (0.040)	2.84 (0.039)

Table 2. Table 2: MSE for different prior choices and percent reduction of MSE of the optimal prior compared to the uniform prior

	Optimal Prior	Beta$(1,1)$	Beta$(2,2)$	Percent Reduction of MSE,
				Optimal Prior vs. $\mathrm{beta}(1,1)$
$d_{\text{MTD}}=0.5$	0.054	0.057	0.057	5%
$d_{\text{MTD}}=1$	0.063	0.069	0.079	9%
$d_{\text{MTD}}=1.5$	0.052	0.059	0.067	12%

Table 3. Table 3: Bias and variance decomposition for different prior choices

	Optimal Prior	Beta$(1,1)$	Beta$(2,2)$
Bias2
$d_{\text{MTD}}=0.5$	0.011	0.015	0.018
$d_{\text{MTD}}=1$	0.005	0.012	0.025
$d_{\text{MTD}}=1.5$	0.003	0.006	0.015
Variance
$d_{\text{MTD}}=0.5$	0.043	0.042	0.039
$d_{\text{MTD}}=1$	0.058	0.057	0.054
$d_{\text{MTD}}=1.5$	0.049	0.053	0.052

Table 4. Table 4: pediatric lupus trial: posterior mean, standard deviation, and 95% credible interval for β 𝛽 \beta

	Mean	SD	95% Credible Interval
$\mathrm{beta}{(5.5,5.5)}^1$	0.47	0.16	(0.15, 0.79)
$\mathrm{beta}{(2,5)}^2$	0.47	0.21	(0.04, 0.89)
$\mathrm{beta}(1,1)$	0.47	0.18	(0.12, 0.83)
$\mathrm{beta}(2,2)$	0.47	0.17	(0.13, 0.79)
$\mathrm{beta}(0.5,0.5)$	0.47	0.17	(0.12, 0.81)

Table 5. Table 5: Melanoma trial: posterior mean, standard deviation and 95% credible interval for β 𝛽 \beta

	Mean	SD	95% Credible Interval
$\mathrm{beta}{(0.7,1.5)}^1$	0.05	0.19	(-0.33, 0.43)
$\mathrm{beta}{(5.5,3)}^2$	-0.01	0.17	(-0.35, 0.33)
$\mathrm{beta}(1,1)$	0.04	0.19	(-0.32, 0.42)
$\mathrm{beta}(2,2)$	0.03	0.18	(-0.34, 0.40)
$\mathrm{beta}(0.5,0.5)$	0.05	0.19	(-0.33, 0.41)

Equations

$$\pi (\theta |D_0,a_0)\propto L(\theta |D_0{)}^{a_0}{\pi }_0(\theta ),$$

$$\pi (\theta ,a_0|D_0)=\pi (\theta |D_0,a_0)\pi (a_0)=\frac{L(\theta |D_0{)}^{a_0}{\pi }_0(\theta )}{c(a_0)}\pi (a_0),$$

$$p(y|\theta )=q(y)\exp \left(y\theta -b(\theta )\right),$$

$$L(\theta |D)=Q(D)\exp \left(\sum _{i=1}^n{y}_i\theta -nb(\theta )\right),$$

$$[L(\theta |D_0){]}^{a_0}=Q(D_0{)}^{a_0}\exp \left(a_0\left[\sum _{i=1}^{n_0}{y}_{0i}\theta -n_{0}b(\theta )\right]\right),$$

$$\pi (\theta ,a_0|D,D_0)$$

$$\pi (a_0|D,D_0)=\int \pi (\theta ,a_0|D,D_0)d\theta \propto \int L(\theta |D)\frac{L(\theta |D_0{)}^{a_0}{\pi }_0(\theta )}{c(a_0)}\pi (a_0)d\theta .$$

$$\tilde{\pi }(a_0|D,D_0)=\frac{\sqrt{\frac{r{a}_0}{r{a}_0+1}}\pi (a_0)}{{\int }_0^1\sqrt{\frac{r{a}_0}{r{a}_0+1}}\pi (a_0)d{a}_0}$$

$$p(y_i|x_i,\beta ,\varphi )=q(y_i,\varphi )\exp \{{\varphi }^{-1}[y_i{x}_i^{\prime }\beta -b(x_i^{\prime }\beta )]\}.$$

$$L(\beta |D)=Q(Y)\exp \left(\sum _{i=1}^n{y}_i{x}_i^{\prime }\beta -\sum _{i=1}^{n}b(x_i^{\prime }\beta )\right),$$

$$[L(\beta |D_0){]}^{a_0}=Q(Y_0{)}^{a_0}\exp \left(a_0\left[\sum _{i=1}^{n_0}{y}_{0i}{x}_{0i}^{\prime }\beta -\sum _{i=1}^{n}b(x_{0i}^{\prime }\beta )\right]\right),$$

$$\pi (\beta ,a_0|D,D_0)$$

$$\pi (a_0|D,D_0)$$

$$\tilde{\pi }(a_0|X,Y,X_0,Y_0)=\frac{{\left(\frac{a_0}{a_0+1}\right)}^{\frac{p}{2}}\pi (a_0)}{{\int }_0^1{\left(\frac{a_0}{a_0+1}\right)}^{\frac{p}{2}}\pi (a_0)d{a}_0},$$

$$\pi (a_0)\cdot \frac{|{\hat{\Sigma }}_g{|}^{1/2}}{|{\tilde{\Sigma }}_k{|}^{1/2}}\exp \left\{-n[g_n(\hat{\beta })-k_n(\tilde{\beta })]\right\},$$

$$K({\alpha }_0,{\beta }_0)=wKL({\pi }^{\ast }(a_0),{\pi }_1(a_0))+(1-w)KL({\pi }_{\text{MTD}}(a_0),{\pi }_2(a_0)).$$

$$KL(p,q)$$

$$\text{MSE}({\mu }^{\ast })=$$

$$w\text{MSE}({\mu }^{\ast }={\bar{y}}_0)+(1-w)\text{MSE}({\mu }^{\ast }={\bar{y}}_0+d_{\text{MTD}}),$$

$$I_p-A(\epsilon )\le L^{\prime \prime }(\theta ){\left[L^{\prime \prime }(\theta )\right]}^{-1}\le I_p+A(\epsilon ),$$

$${\int }_{\Theta \setminus B_r({\hat{\theta }}_n)}{p}_n(t\mid x)dt\to 0,a.s.,$$

$$p_n(\theta |x)\to N_p\left({\hat{\theta }}_n,{\left[-L^{\prime \prime }({\hat{\theta }}_n)\right]}^{-1}\right),$$

$$L(\theta |D)$$

$$\frac{1}{c(a_0)}L(\theta |D_0{)}^{a_0}{\pi }_0(\theta )$$

$${\pi }^{\ast }(a_0|D_0,D,{\alpha }_0,{\beta }_0)\equiv$$

$$\to$$

$$\propto$$

$$=$$

$$\pi (a_0|D_0,D,{\alpha }_0,{\beta }_0)=$$

$$\to$$

$$=$$

$$A={\int }_0^1{a}_0^{{\alpha }_0-1}(1-a_0{)}^{{\beta }_0-1}f(a_0)\exp \left\{-nh(a_0)\right\}d{a}_0.$$

Full text

Keywords— Bayesian analysis; Clinical trial; Normalized power prior; Power prior.

1 Introduction

The power prior (Ibrahim and Chen, 2000) is a popular class of informative priors that allow the incorporation of historical data through a tempering of the likelihood. It is constructed by raising the historical data likelihood to a power $a_{0}$, where $0\leq a_{0}\leq 1$. The discounting parameter $a_{0}$ can be fixed or modelled as random. When it is modelled as random and estimated jointly with other parameters of interest, the normalized power prior (Duan et al., 2006) is recommended as it appropriately accounts for the normalizing constant necessary for forming the correct joint prior distribution (Neuenschwander et al., 2009). Many extensions of the power prior and the normalized power prior have been developed. Banbeta et al. (2019) develop the dependent and robust dependent normalized power priors which allow dependent discounting parameters for multiple historical datasets. When the historical data model contains only a subset of covariates currently of interest and the historical information may not be equally informative for all parameters in the current analysis, Boonstra and Barbaro (2020) propose an extension of the power prior that adaptively combines a prior based upon the historical information with a variance-reducing prior that shrinks parameter values toward zero.

The power prior and the normalized power prior have been shown to have several desirable properties. Ibrahim et al. (2003) show that the power prior defines an optimal class of priors in the sense that it minimizes a convex combination of Kullback-Leibler (KL) divergences between a distribution based on no incorporation of historical data and a distribution based on completely pooling the historical and current data. Ye et al. (2022) prove that the normalized power prior minimizes the expected weighted KL divergence similar to the one in Ibrahim et al. (2003) with respect to the marginal distribution of the discounting parameter. They also prove that if the prior on $a_{0}$ is non-decreasing and if the difference between the sufficient statistics of the historical and current data is negligible from a practical standpoint, the marginal posterior mode of $a_{0}$ is close to one. Carvalho and Ibrahim (2021) show that the normalized power prior is always well-defined when the initial prior is proper, and that, viewed as a function of the discounting parameter, the normalizing constant is a smooth and strictly convex function. Neelon and O’Malley (2010) show through simulations that for large datasets, the normalized power prior may result in more downweighting of the historical data than desired. Han et al. (2022) point out that the normalizing constant might be infinite for $a_{0}$ values close to zero with conventionally used improper priors on $a_{0}$, in which case the optimal $a_{0}$ value might be lower than the suggested value. Despite the aforementioned research, there has not been any theoretical investigation of the asymptotic properties of the normalized power prior when the historical and current datasets are discrepant.

Many empirical Bayes-type approaches have been developed to adaptively determine the discounting parameter. For example, Gravestock and Held (Gravestock et al., 2017; Gravestock and Held, 2019) propose to set $a_{0}$ to the value that maximizes the marginal likelihood. Liu (2018) proposes choosing $a_{0}$ based on the p-value for testing the compatibility of the current and historical data. Bennett et al. (2021) propose using an equivalence probability weight and a weight based on tail area probabilities to assess the degree of agreement between the historical and current control data for cases with binary outcomes. Pan et al. (2017) propose the calibrated power prior, where $a_{0}$ is defined as a function of a congruence measure between the historical and current data. The function which links $a_{0}$ and the congruence measure is prespecified and calibrated through simulation. While these empirical Bayes approaches shed light on the choice of $a_{0}$, there has not been any fully Bayesian approach based on an optimal prior on $a_{0}$.

In this work, we first explore the asymptotic properties of the normalized power prior when the historical and current data are fully compatible (i.e., the sufficient statistics of the two datasets are equal) or incompatible (i.e., the sufficient statistics of the two datasets have some non-zero difference). We prove that for generalized linear models (GLMs) utilizing a normalized power prior, the marginal posterior distribution of $a_{0}$ converges to a point mass at zero if there is any discrepancy between the historical and current data. When the historical and current data are fully compatible, the asymptotic distribution of the marginal posterior of $a_{0}$ is derived for GLMs; we note that it does not concentrate around one. Secondly, we propose a novel fully Bayesian approach to elicit the shape parameters of the beta prior on $a_{0}$ based on two optimality criteria, Kullback-Leibler (KL) divergence and mean squared error (MSE). For the first criterion, we propose as optimal the beta prior whose shape parameters result in a minimized weighted average of KL divergences between the marginal posterior for $a_{0}$ and user-specified target distributions based on hypothetical scenarios where there is no discrepancy and where there is a maximum tolerable discrepancy. This class of priors on $a_{0}$ based on the KL criterion is optimal in the sense that it is the best possible beta prior at balancing the dual objectives of encouraging borrowing when the historical and current data are compatible and limiting borrowing when they are in conflict. For the second criterion, we propose as optimal the beta prior whose shape parameters result in a minimized weighted average of the MSEs based on the posterior mean of the parameter of interest when its hypothetical true value is equal to its estimate using the historical data, or when it differs from its estimate by the maximum tolerable amount. We study the properties of the proposed approaches via simulations for the i.i.d. normal and Bernoulli cases as well as for the normal linear model. Two real-world case studies of clinical trials with binary outcomes and covariates demonstrate the performance of the optimal priors compared to conventionally used priors on $a_{0}$, such as a uniform prior.

2 Asymptotic Properties of the Normalized Power Prior

Let $D$ denote the current data and $D_{0}$ denote the historical data. Let $\theta$ denote the model parameters and $L(\theta|D)$ denote a general likelihood function. The power prior (Ibrahim and Chen, 2000) is formulated as

[TABLE]

where $0\leq a_{0}\leq 1$ is the discounting parameter which discounts the historical data likelihood, and $\pi_{0}(\theta)$ is the initial prior for $\theta$. The discounting parameter $a_{0}$ can be fixed or modelled as random. Modelling $a_{0}$ as random allows researchers to account for uncertainty when discounting historical data and to adaptively learn the appropriate level of borrowing. Duan et al. (2006) propose the normalized power prior, given by

[TABLE]

where $c(a_{0})=\int L(\theta|D_{0})^{a_{0}}\pi_{0}(\theta)d\theta$ is the normalizing constant. The normalized power prior is thus composed of a conditional prior for $\theta$ given $a_{0}$ and a marginal prior for $a_{0}$.

Ideally, the posterior distribution of $a_{0}$ with the normalized power prior would asymptotically concentrate around zero when the historical and current data are in conflict, and around one when they are compatible. In this section, we study the asymptotic properties of the normalized power prior for the exponential family of distributions as well as GLMs. Specifically, we are interested in exploring the asymptotic behaviour of the posterior distribution of $a_{0}$ when the historical and current data are incompatible and when they are compatible, respectively.

2.1 Exponential Family

First, we study the asymptotic properties of the normalized power prior for the exponential family of distributions. The density of a random variable $Y$ in the one-parameter exponential family has the form

[TABLE]

where $\theta$ is the canonical parameter and $q(\cdot)$ and $b(\cdot)$ are known functions. Suppose $D=(y_{1},\dots,y_{n})$ is a sample of $n$ i.i.d. observations from an exponential family distribution in the form of (2). The likelihood is then given by

[TABLE]

where $Q(D)=\prod_{i=1}^{n}q(y_{i})$. Suppose $D_{0}=(y_{01},\dots,y_{0n_{0}})$ is a sample of $n_{0}$ i.i.d. observations from the same exponential family. The likelihood for the historical data raised to the power $a_{0}$ is

[TABLE]

where $Q(D_{0})=\prod_{i=1}^{n_{0}}q(y_{0i})$. Using the normalized power prior defined in (1), the joint posterior of $\theta$ and $a_{0}$ is given by

[TABLE]

The marginal posterior of $a_{0}$ is given by

[TABLE]

With these calculations in place, the question now arises as to what prior should be given to $a_{0}$. One commonly used class of priors on $a_{0}$ is the beta distribution (Ibrahim and Chen, 2000). Let $\alpha_{0}$ and $\beta_{0}$ denote the shape parameters of the beta distribution. We first prove that the marginal posterior of $a_{0}$ (3) with $\pi(a_{0})=\text{beta}(\alpha_{0},\beta_{0})$ converges to a point mass at zero for a fixed, non-zero discrepancy between $\bar{y}$ and $\bar{y}_{0}$.

Theorem 2.1**.**

Suppose $y_{1},\dots,y_{n}$ and $y_{01},\dots,y_{n_{0}}$ are independent observations from the same exponential family distribution (2). Suppose also that the difference in the estimates of the canonical parameter $\theta$ is fixed and equal to $\delta$, i.e., $|\dot{b}^{-1}(\bar{y})-\dot{b}^{-1}(\bar{y}_{0})|=\delta$, and $\frac{n_{0}}{n}=r$, where $\delta>0$ and $r>0$ are constants, and $\dot{b}(\cdot)=\partial_{\theta}b(\cdot)$. Then, the marginal posterior of $a_{0}$ using the normalized power prior (3) with a $\operatorname{beta}(\alpha_{0},\beta_{0})$ prior on $a_{0}$ converges to a point mass at [math]. That is, $\lim\limits_{n\rightarrow\infty}\frac{\int_{0}^{\epsilon}\pi(a_{0}|D,D_{0},\alpha_{0},\beta_{0})da_{0}}{\int_{0}^{1}\pi(a_{0}|D,D_{0},\alpha_{0},\beta_{0})da_{0}}=1$ for any $\epsilon>0$.

Proof.

See section A.2. ∎

Theorem 2.1 asserts that the normalized power prior is sensitive to any discrepancy between the sufficient statistics in large samples, as the mass of the marginal distribution of $a_{0}$ will concentrate near zero as the sample size increases for any fixed difference $\delta$. The natural question to then ask is whether Theorem 2.1 has a sort of converse in that the posterior should concentrate around one under compatibility. We derive the asymptotic marginal posterior distribution of $a_{0}$ when $\bar{y}=\bar{y}_{0}$ and show that it does not converge to a point mass at one.

Corollary 2.1**.**

Suppose $y_{1},\dots,y_{n}$ and $y_{01},\dots,y_{n_{0}}$ are independent observations from the same exponential family distribution (2). Suppose $\bar{y}=\bar{y}_{0}$ and $\frac{n_{0}}{n}=r$ where $r>0$ is a constant. The marginal posterior of $a_{0}$ using the normalized power prior, as specified in (3), converges to

[TABLE]

as $n\rightarrow\infty$.

Proof.

See section A.3. ∎

Corollary 2.1 shows that the normalized power prior fails to fully utilize the historical data when the means of the historical data and the current data are equal for a generic, non-degenerate prior on $a_{0}$. However, if $\pi(a_{0})$ is chosen to be concentrated near one, then the marginal posterior of $a_{0}$ may be concentrated near one.

2.2 Generalized Linear Models

The ability to deal with non i.i.d. data and incorporate covariates is crucial to the applicability of the normalized power prior; we thus now extend these results to generalized linear models (GLMs). We first define the GLM with a canonical link and fixed dispersion parameter. Let $y_{i}$ denote the response variable and $x_{i}$ denote a $p$-dimensional vector of covariates for subject $i=1,\dots,n$. Let $\beta=(\beta_{1},\dots,\beta_{p})^{\prime}$ be a $p$-dimensional vector of regression coefficients. The GLM with a canonical link is given by

[TABLE]

Without loss of generality, we assume $\phi=1$. Let $D=\{(y_{i},x_{i}),i=1,\dots,n\}\equiv(n,Y_{n\times 1},X_{n\times p})$ where $Y=(y_{1},\dots,y_{n})^{\prime}$ and $X=(x_{1},\dots,x_{n})^{\prime}$. Assuming the $y_{i}$’s are (conditionally) independent, the likelihood is given by

[TABLE]

where $Q(Y)=\prod_{i=1}^{n}q(y_{i},1)$. Let $\hat{\beta}$ denote posterior mode of $\beta$ obtained by solving $\partial_{\beta}\log L(\beta|D)=0$. Let $D_{0}=\{(y_{0i},x_{0i}),i=1,\dots,n_{0}\}\equiv(n_{0},Y_{0n_{0}\times 1},X_{0n_{0}\times p})$ where $Y_{0}=(y_{01},\dots,y_{0n_{0}})^{\prime}$ and $X_{0}=(x_{01},\dots,x_{0n_{0}})^{\prime}$. Assuming the $y_{0i}$’s are (conditionally) independent, the historical data likelihood raised to the power $a_{0}$ is given by

[TABLE]

where $Q(Y_{0})=\prod_{i=1}^{n_{0}}q(y_{0i},1)$. Let $c^{*}(a_{0})=\int L(\beta|y_{0})^{a_{0}}\pi_{0}(\beta)d\beta$. Using the normalized power prior defined in (1), the joint posterior of $\beta$ and $a_{0}$ is given by

[TABLE]

Let $\hat{\beta}_{0}$ denote posterior mode of $\beta$ obtained by solving $\partial_{\beta}\log\left[\frac{L(\beta|D_{0})^{a_{0}}\pi_{0}(\beta)}{c^{*}(a_{0})}\right]=0.$ The marginal posterior of $a_{0}$ is given by

[TABLE]

Now we extend Theorem 2.1 to GLMs.

Theorem 2.2**.**

Suppose $X$ is $n\times p$ of rank $p$ and $X_{0}$ is $n_{0}\times p$ of rank $p$. Suppose $\hat{\beta}-\hat{\beta}_{0}=\delta$ where $\delta\neq 0$ is a constant vector, and $\frac{n_{0}}{n}=r$ where $r>0$ is a constant scalar. Assume $n\left[\frac{\partial^{2}\log[L(\beta|D)]}{\partial\beta_{i}\partial\beta_{j}}\right]^{-1}$ and $n_{0}a_{0}\left[\frac{\partial^{2}\log[L(\beta|D_{0})^{a_{0}}\pi_{0}(\beta)]}{\partial\beta_{i}\partial\beta_{j}}\right]^{-1}$ do not depend on $n$ and $a_{0}$. Then, the marginal posterior of $a_{0}$ using the normalized power prior (5) with a $\operatorname{beta}(\alpha_{0},\beta_{0})$ prior on $a_{0}$ converges to a point mass at zero. That is, $\lim\limits_{n\rightarrow\infty}\frac{\int_{0}^{\epsilon}\pi(a_{0}|D,D_{0},\alpha_{0},\beta_{0})da_{0}}{\int_{0}^{1}\pi(a_{0}|D,D_{0},\alpha_{0},\beta_{0})da_{0}}=1$ for any $\epsilon>0$.

Proof.

See section A.4. ∎

Theorem 2.2 asserts that the normalized power prior is sensitive to discrepancies in the historical and current data in the presence of covariates. The mass of the marginal distribution of $a_{0}$ will concentrate near zero as the sample size increases for any fixed discrepancy between the historical and current data, assuming $\frac{1}{n}X^{\prime}X$ and $\frac{1}{n_{0}}X_{0}^{\prime}X_{0}$ are fixed, i.e., $n\left[\frac{\partial^{2}\log[L(\beta|D)]}{\partial\beta_{i}\partial\beta_{j}}\right]^{-1}$ and $n_{0}a_{0}\left[\frac{\partial^{2}\log[L(\beta|D_{0})^{a_{0}}\pi_{0}(\beta)]}{\partial\beta_{i}\partial\beta_{j}}\right]^{-1}$ do not depend on $n$ and $a_{0}$. Next, we derive the asymptotic marginal posterior distribution of $a_{0}$ when the sufficient statistics and covariate (design) matrices of the historical and current data equal.

Corollary 2.2**.**

Suppose $X$ is $n\times p$ of rank $p$ and $X_{0}$ is $n_{0}\times p$ of rank $p$. Let $Y=(y_{1},\dots,y_{n})^{\prime}$ and $Y_{0}=(y_{01},\dots,y_{0n_{0}})^{\prime}$. Consider the GLM in (4). If $n=n_{0}$, $X=X_{0}$, and $X^{\prime}Y=X_{0}^{\prime}Y_{0}$, then the marginal posterior of $a_{0}$ using the normalized power prior, as specified in (5), converges to

[TABLE]

as $n\rightarrow\infty$.

Proof.

See section A.5. ∎

Corollary 2.2 states that the marginal posterior of $a_{0}$ using the normalized power prior does not converge to a point mass at one when the sufficient statistics and the covariates of the historical and current data are equal. We also observe that as $p$ approaches infinity, the marginal posterior of $a_{0}$ specified above converges to a point mass at one. The form of the asymptotic marginal posterior of $a_{0}$ suggests that the normalized power prior may be sensitive to overfitting when the historical and current datasets are compatible.

In Theorem 2.3 we also relax the previous result by deriving the asymptotic marginal posterior distribution of $a_{0}$ assuming only that the sufficient statistics of the historical and current data are not equal. This means that the covariate matrices need not be equal so long as the sufficient statistics $X^{\prime}Y$ and $X_{0}^{\prime}Y_{0}$ are, increasing the applicability of the result.

Theorem 2.3**.**

Suppose $X$ is $n\times p$ of rank $p$ and $X_{0}$ is $n_{0}\times p$ of rank $p$. Let $Y=(y_{1},\dots,y_{n})^{\prime}$ and $Y_{0}=(y_{01},\dots,y_{0n_{0}})^{\prime}$. Consider the GLM in (4), where $\frac{n_{0}}{n}=r$ and $r>0$ is a constant. If $X^{\prime}Y=X_{0}^{\prime}Y_{0}$ and $X\neq X_{0}$, then the marginal posterior of $a_{0}$ using the normalized power prior, as specified in (5), is asymptotically proportional to

[TABLE]

where the definitions of $g_{n}(\beta)$, $k_{n}(\beta)$ and $\frac{|\hat{\Sigma}_{g}|^{1/2}}{|\tilde{\Sigma}_{k}|^{1/2}}$ can be found in section A.5 of the appendix.

Proof.

See section A.6. ∎

Corollary 2.2 and Theorem 2.3 show that, for GLMs, the marginal posterior of $a_{0}$ using the normalized power prior does not converge to a point mass at one when the sufficient statistics of the historical and current data are equal. From Theorems 2.1-2.3, we conclude that, asymptotically, the normalized power prior is sensitive to discrepancies between the historical and current data, but cannot fully utilize the historical information when there are no discrepancies.

3 Optimal Beta Priors for $a_{0}$

3.1 Kullback-Leibler Divergence Criterion

In this section, we propose a prior based on minimizing the KL divergence of the marginal posterior of $a_{0}$ to two reference distributions. This prior is optimal in the sense that it is the best possible beta prior at balancing the dual objectives of encouraging borrowing when the historical and current data are compatible and limiting borrowing when they are in conflict.

Let $\bar{y}_{0}$ denote the mean of the historical data and $\bar{y}$ denote the mean of the hypothetical current data. Let $\pi_{1}(a_{0})\equiv\operatorname{beta}(c,1)$ ($c\gg 1$ is fixed) and $\pi_{2}(a_{0})\equiv\operatorname{beta}(1,c)$. The distributions $\pi_{1}(a_{0})$ and $\pi_{2}(a_{0})$ represent two ideal scenarios, where $\pi_{1}(a_{0})$ is concentrated near one and $\pi_{2}(a_{0})$ is concentrated near zero. The KL-based approach computes the hyperparameters ($\alpha_{0}$ and $\beta_{0}$) for the beta prior on $a_{0}$ that will minimize a convex combination of two KL divergences; one is the KL divergence between $\pi_{1}(a_{0})$ and the marginal posterior of $a_{0}$ when $\bar{y}=\bar{y}_{0}$, while the other is the KL divergence between $\pi_{2}(a_{0})$ and the marginal posterior of $a_{0}$ when there is a user-specified difference between $\bar{y}$ and $\bar{y}_{0}$.

Let $d=\bar{y}-\bar{y}_{0}$, representing the difference between the means of the hypothetical current data and the historical data. Our approach is centered on a user-specified maximum tolerable difference (MTD), $d_{\textrm{MTD}}$. Let $\pi^{*}(a_{0})$ denote the marginal posterior of $a_{0}$ when $d=0$. Let $\pi_{\textrm{MTD}}(a_{0})$ denote the marginal posterior of $a_{0}$ when $d=d_{\textrm{MTD}}$. For $d=0$, we want $\pi^{*}(a_{0})$ to resemble $\pi_{1}(a_{0})$ and for $d=d_{\textrm{MTD}}$, we want $\pi_{\textrm{MTD}}(a_{0})$ to resemble $\pi_{2}(a_{0})$. The distributions $\pi_{1}(a_{0})$ and $\pi_{2}(a_{0})$ have been chosen to correspond to cases with substantial and little borrowing, respectively. Therefore, our objective is to solve for $\alpha_{0}>0$ and $\beta_{0}>0$ to minimize

[TABLE]

Here $0<w<1$ is a scalar and $KL(p,q)$ for distributions $P$ and $Q$ with P as reference is defined as

[TABLE]

The scalar $w$ weights the two competing objectives. For $w>0.5$, the objective to encourage borrowing is given more weight, and for $w<0.5$, the objective to limit borrowing is given more weight.

Below we demonstrate the simulation results using this method for the i.i.d. normal case, the i.i.d. Bernoulli case and the normal linear model. We compare the marginal posterior of $a_{0}$ using the KL-based optimal prior with that using the uniform prior. For all simulations in this section, we choose $w=0.5$ so that the two competing objectives are given equal weight. We choose $c=10$ so that $\pi_{1}(a_{0})$ and $\pi_{2}(a_{0})$ represent cases with substantial and little borrowing, respectively.

3.1.1 Normal i.i.d. Case

We assume $y_{1},\dots,y_{n}$ and $y_{01},\dots,y_{0{n_{0}}}$ are i.i.d. observations from N($\mu$, $\sigma^{2}$) where $\sigma^{2}=1$. We choose $\bar{y}_{0}=1.5$ and $n=n_{0}=30$. The objective function $K(\cdot,\cdot)$ is computed using numerical integration and optimization is performed using the optim() function in (base) R (R Core Team, 2022).

In Figure 1, the first figure of each row plots the historical and current data likelihoods if the hypothetical degree of conflict is equal to $d_{\textrm{MTD}}$. For each row of the figure below, the maximum tolerable difference $d_{\textrm{MTD}}$ is chosen to be $0.5$, $1$ and $1.5$, and the corresponding optimal prior is derived for each value of $d_{\textrm{MTD}}$. For each optimal prior, we vary the observed sample mean, denoted by $\bar{y}_{\textrm{obs}}$, to evaluate the posterior based on the optimal prior for different observed current data. We use $d_{\textrm{obs}}=\bar{y}_{\textrm{obs}}-\bar{y}_{0}$ to represent the difference between the means of the observed current data and the historical data. For columns 2-4, $d_{\textrm{obs}}$ is chosen to be [math], $1$ and $1.5$, respectively. Note that the values of $d_{\textrm{MTD}}$ and $d_{\textrm{obs}}$ are relative to the choices of $\sigma^{2}$, $n$ and $n_{0}$. For example, for larger $n$, $d_{\textrm{MTD}}$ would need to be decreased to produce a similar plot to Figure 1.

From columns 2-4, we observe that when $d_{\textrm{MTD}}=0.5$, very little conflict is tolerated, and the resulting optimal prior does not strongly encourage either borrowing substantially or borrowing little. As $d_{\textrm{MTD}}$ becomes larger, larger conflict is allowed and the optimal prior shifts more towards $\pi_{1}(a_{0})$. We also observe that when $d_{\textrm{MTD}}=1$ (the optimal hyperparameters are $\alpha_{0}=1$ and $\beta_{0}=0.4$) and $d_{\textrm{MTD}}=1.5$ (the optimal hyperparameters are $\alpha_{0}=2.6$ and $\beta_{0}=0.5$), the marginal posterior of $a_{0}$ with the optimal prior more closely mimics the target distribution when $d_{\textrm{obs}}=0$, i.e., the observed current and historical data are fully compatible. As $d_{\textrm{obs}}$ increases, the marginal posterior shifts toward zero. This behaviour is highly desirable as it achieves both goals of encouraging borrowing when the datasets are compatible and limiting borrowing when they are incompatible.

We can compare the marginal posterior of $a_{0}$ using the optimal prior with that using a uniform prior in Figure 1. We observe that while the marginal posterior on $a_{0}$ with the uniform prior is very responsive to conflict, it does not concentrate around one even when the datasets are fully compatible. We conclude that when $d_{\textrm{MTD}}$ is chosen to be reasonably large, the optimal prior on $a_{0}$ achieves a marginal posterior that is close to the target distribution when the datasets are fully compatible, while remaining responsive to conflict in the data.

Table 1 shows the posterior mean and variance of the mean parameter $\mu$ for various combinations of $d_{\textrm{MTD}}$ and $d_{\textrm{obs}}$ values corresponding to the scenarios in Figure 1. The posterior mean and variance of $\mu$ with a normalized power prior are computed using the R package BayesPPD (Shen et al., 2022). Again, $\bar{y}_{0}$ is fixed at $1.5$. Since $\bar{y}_{\textrm{obs}}\geq\bar{y}_{0}$, within each row, the posterior mean of $\mu$ is always smaller than $\bar{y}_{\textrm{obs}}$ due to the incorporation of $\bar{y}_{0}$. We can also compare the results by column. Note that for fixed $d_{obs}$ (or equivalently $\bar{y}_{\textrm{obs}}$), if more historical information is borrowed, the posterior mean of $\mu$ will be smaller. When $d_{\textrm{obs}}=0$, the posterior mean stays constant while the variance decreases as $d_{\textrm{MTD}}$ increases. If the maximum tolerable difference is large, more historical information is borrowed, leading to reduced variance. When $d_{\textrm{obs}}=0.5$, the posterior of $\mu$ decreases as more borrowing occurs when $d_{\textrm{MTD}}$ increases. When $d_{\textrm{obs}}=1$ or $1.5$, the posterior of $\mu$ first increases and then decreases, as $d_{\textrm{MTD}}$ increases. This is a result of two competing phenomena interacting; as $d_{\textrm{MTD}}$ increases, the optimal prior gravitates towards encouraging borrowing; however, since $d_{\textrm{obs}}$ is very large, the marginal posterior of $a_{0}$ moves toward zero even though the prior moves toward one. In conclusion, we argue that the posterior estimates of $\mu$ with the optimal prior respond in a desirable fashion to changes in the data.

3.1.2 Bernoulli Model

For the Bernoulli model, we assume $y_{1},\dots,y_{n}$ and $y_{01},\dots,y_{0{n_{0}}}$ are i.i.d. observations from a Bernoulli distribution with mean $\mu$. Again, we choose $n=n_{0}=30$ and optimization is performed analogously to the normal case.

For each row of Figure 2 below, the maximum tolerable difference $d_{\textrm{MTD}}$ is chosen to be $0.2$, $0.4$ and $0.6$, and the corresponding optimal prior is derived for each value of $d_{\textrm{MTD}}$. For each optimal prior, we vary the observed $\bar{y}_{\textrm{obs}}$ to evaluate the performance of the optimal prior for different observed data. For columns 2-4, $d_{\textrm{obs}}=\bar{y}_{\textrm{obs}}-\bar{y}_{0}$ is chosen to be [math], $0.4$ and $0.6$, respectively. Values of $\bar{y}_{0}$ and $\bar{y}_{\textrm{obs}}$ are chosen so that the variance stays constant for different values of $d_{\textrm{MTD}}$ or $d_{\textrm{obs}}$.

The optimal marginal prior and posterior of $a_{0}$ for Bernoulli data are similar to those of the normal model. We observe that when the datasets are perfectly compatible, i.e., $d_{\textrm{obs}}=0$, the marginal posterior of $a_{0}$ with the optimal prior concentrates around one when $d_{\textrm{MTD}}$ is relatively large. When $d_{\textrm{obs}}$ increases to $0.4$ or $0.6$, the marginal posterior of $a_{0}$ concentrates around zero when $d_{\textrm{MTD}}$ is relatively large. The optimal prior becomes increasingly concentrated near one as $d_{\textrm{MTD}}$ increases. Compared to the marginal posterior with the uniform prior, the optimal prior on $a_{0}$ achieves a marginal posterior that closely mimics the target distribution when the datasets are fully compatible, while remaining responsive to conflict in the data.

3.1.3 Normal Linear Model

Suppose $y_{01},\dots,y_{0{n_{0}}}$ are independent observations from the historical data where $y_{0i}\sim N(\beta_{0}+\beta_{1}x_{0i},\sigma^{2})$ for the $i$-th observation and $x_{0i}$ is a single covariate. Also suppose $y_{1},\dots,y_{n}$ are independent observations from the current data where $y_{j}\sim N(\beta_{0}+\beta_{1}x_{j}+d_{\textrm{MTD}},\sigma^{2})$ for the $j$-th observation and $x_{j}$ is a single covariate. We vary $d_{\textrm{MTD}}$ to represent different degrees of departure of the intercept of the simulated current data to the intercept of the historical data. We choose $\beta_{0}=1.5$, $\beta_{1}=-1$, $\sigma^{2}=1$ and $n=n_{0}=30$. We choose $d_{\textrm{MTD}}=0.1,0.5$, and $1$ and $d_{\textrm{obs}}=0,0.5$, and $1$. The objective function $K$ is computed using Monte Carlo integration and optimization is performed using the optim() function in R.

Figure 3 shows the optimal prior and optimal posterior for $a_{0}$ as well as the posterior of $a_{0}$ with the uniform prior for various $d_{\textrm{MTD}}$ and $d_{\textrm{obs}}$ values. We observe that when the datasets are perfectly compatible, i.e., $d_{\textrm{obs}}=0$, the marginal posterior of $a_{0}$ with the optimal prior concentrates around one when $d_{\textrm{MTD}}$ is relatively large. When $d_{\textrm{obs}}$ increases to $1$, the marginal posterior of $a_{0}$ concentrates around zero. The optimal prior becomes increasingly concentrated near one as $d_{\textrm{MTD}}$ increases. Compared to the marginal posterior with the uniform prior, the optimal prior for $a_{0}$ achieves a marginal posterior that closely mimics the target distribution when the datasets are fully compatible, while remaining responsive to conflict in the data.

3.2 Mean Squared Error Criterion

In this section, we derive the optimal prior for $a_{0}$ based on minimizing the MSE. This prior is optimal in the sense that it minimizes the weighted average of the MSEs of the posterior mean of the parameter of interest when its hypothetical true value is equal to its estimate using the historical data, or when it differs from its estimate by the maximum tolerable amount. Suppose $y_{1},\dots,y_{n}$ and $y_{01},\dots,y_{0{n_{0}}}$ are observations from a distribution with mean parameter $\mu$. Let $\mu^{*}$ denote the true value of $\mu$. Let $\bar{\mu}$ denote the posterior mean of $\mu$ using the normalized power prior. Then, the MSE of $\bar{\mu}$ is

[TABLE]

In the regression setting, $\mu$ is replaced by the regresison coefficients $\beta$.

Let $\bar{y}_{0}$ denote the mean of the historical data. We aim to find the hyperparameters, $\alpha_{0}$ and $\beta_{0}$, for the beta prior for $a_{0}$ that will minimize

[TABLE]

where $d_{\textrm{MTD}}$ is the maximum tolerable difference. Again, we use $d_{\textrm{obs}}=\bar{y}_{\textrm{obs}}-\bar{y}_{0}$ to represent the difference between the means of the observed current data and the historical data.

3.2.1 Normal i.i.d. Case

We demonstrate the use of this criterion for the normal i.i.d. case. Suppose $y_{1},\dots,y_{n}$ and $y_{01},\dots,y_{0{n_{0}}}$ are i.i.d. observations from N($\mu$, $\sigma^{2}$) where $\sigma^{2}=1$ and $n=n_{0}=30$. In this example, we fix $\mu^{*}$ and $y_{MTD}$ at $1.5$, and define $d_{\textrm{MTD}}=\bar{y}_{0}-\bar{y}$ and $d_{\textrm{obs}}=\bar{y}_{0}-\bar{y}$. The posterior mean of $\mu$ is computed using Monte Carlo integration and optimization is performed using a grid search. The optimal prior, optimal posterior, and the posterior using the uniform prior for $a_{0}$ are plotted in Figure 4. When $d_{\textrm{MTD}}=0.5$, the optimal prior is unimodal with mode around $0.3$. When $d_{\textrm{MTD}}=1$, the optimal prior is concentrated near zero. When $d_{\textrm{MTD}}=1.5$, the optimal prior is U-shaped and favouring either strong or weak borrowing. When $d_{\textrm{MTD}}$ is small, the algorithm cannot distinguish between the two competing scenarios in the objective function and the resulting optimal prior concentrates around $0.5$. When $d_{\textrm{MTD}}$ is large, the optimal prior will favour the two scenarios equally. For columns 2-4, $d_{\textrm{obs}}$ is chosen to be [math], $1$ and $1.5$. The marginal posterior using the optimal prior concentrates more around zero as $d_{\textrm{obs}}$ increases for a given $d_{\textrm{MTD}}$. Comparing Figures 4 and 1, we observe that the optimal prior derived using the MSE criterion is more conservative in the sense that it tends to discourage borrowing than that derived using the KL criterion.

Table 2 shows the MSE for the optimal prior, $\operatorname{beta}(1,1)$ (uniform) and $\operatorname{beta}(2,2)$ as well as the percent reduction of MSE of the optimal prior compared to the uniform prior. We can see that the percent reduction of MSE increases as $d_{\textrm{MTD}}$ increases. Table 3 displays the decomposition of MSE into bias squared and variance for the three choices of priors. When $d_{\textrm{MTD}}=0.5$ or $1$, the prior discourages borrowing which results in smaller bias and larger variance. When $d_{\textrm{MTD}}=1.5$, the model can distinguish easily between the two contrasting objectives, leading to smaller bias and smaller variance. Additional simulation results for varying choices of $n$ and $n_{0}$ are provided in section B of the appendix.

4 Case Studies

We now illustrate the proposed methodologies by analysing two clinical trial case studies. First, we study an important application in a pediatric trial where historical data on adults is available. This constitutes a situation of increased importance due to the difficulty in enrolling pediatric patients in clinical trials (U.S. Food and Drug Administration, 2016). Then, we study a classical problem in the analysis clinical trials: using information from a previous study. This is illustrated with data on trials of interferon treatment for melanoma.

4.1 Pediatric Lupus Trial

Enrolling patients for pediatric trials is often difficult due to the small number of available patients, parental concern regarding safety and technical limitations (Psioda and Xue, 2020). For many pediatric trials, additional information must be incorporated for any possibility of establishing efficacy (Psioda and Xue, 2020). The use of Bayesian methods is natural for extrapolating adult data in pediatric trials through the use of informative priors, and is demonstrated in FDA guidance on complex innovative designs (U.S. Food and Drug Administration, 2019).

Belimumab (Benlysta) is a biologic for the treatment of adults with active, autoantibody-positive systemic lupus erythematosus (SLE). It was proposed that the indication for Belimumab can be expanded to include the treatment of children (Psioda and Xue, 2020). The clinical trial PLUTO (Brunner et al., 2020) has been conducted to examine the effect of Belimumab on children 5 to 17 years of age with active, seropositive SLE who are receiving standard therapy. The PLUTO study has a small sample size due to the rarity of childhood-onset SLE. There have been two previous phase 3 trials, BLISS-52 and BLISS-76 (Furie et al., 2011; Navarra et al., 2011), which established efficacy of belimumab plus standard therapy for adults. The FDA review of the PLUTO trial submission used data from the adult trials to inform the approval decision (Psioda and Xue, 2020). All three trials employ the same composite primary outcome, the SLE Responder Index (SRI-4).

We conduct a Bayesian analysis of the PLUTO study incorporating information from the adult studies, BLISS-52 and BLISS-76, using a normalized power prior. We derive the optimal priors on $a_{0}$ based on the KL criterion and the MSE criterion.

Our parameter of interest is the treatment effect of Belimumab for children, denoted by $\beta$. The total sample size of the pooled adult data (BLISS-52 and BLISS-76) is $n_{0}=1125$ and the treatment effect is $0.481$. We choose $d_{\textrm{MTD}}=0.481$ to be the maximum tolerable difference. The pediatric data has a sample size of $92$ and the estimated treatment effect is $0.371$. We use the asymptotic normal approximation to the logistic regression model with one covariate (the treatment indicator). We choose $w=0.5$ and $n=100$ (sample size of the simulated current dataset). For the KL criterion, the objective function $K$ is computed using Monte Carlo integration and optimization is performed using the optim() function in R. For the MSE criterion, the posterior mean of $\beta$ is computed using the R package hdbayes (Alt, 2022) and optimization is performed using a grid search where the values of $\alpha_{0}$ and $\beta_{0}$ range from $0.5$ to $6$ with an increment of $0.5$. The optimal priors derived using the KL criterion and MSE criterion are displayed in Figure 5. The optimal prior derived using the KL criterion is $\operatorname{beta}(5.5,5.5)$, which is a unimodal and symmetric distribution centred at $0.5$. The optimal prior derived using the MSE criterion is $\operatorname{beta}(2,5)$, which is a unimodal distribution with mode around $0.2$. Table 4 provides the posterior mean, standard deviation and 95% credible interval for $\beta$ using the optimal priors and several other beta priors for comparison. We observe that while the posterior means remain the same, the posterior standard deviation is lowest with the optimal prior derived using the KL criterion. In this case, the optimal prior using the KL criterion borrowed the most historical information out of the five prior choices being considered.

4.2 Melanoma Trial

Interferon Alpha-2b (IFN) is an adjuvant chemotherapy for deep primary or regionally metastatic melanoma. IFN was used in two phase 3 randomized controlled clinical trials, E1684 and E1690 (Kirkwood et al., 1996). In this example, we choose overall survival (indicator for death) as the primary outcome. We conduct a Bayesian analysis of the E1690 trial incorporating information from the E1684 trial, using a normalized power prior. We include three covariates in the analysis, the treatment indicator, sex and the logarithm of age. As before, we obtain the optimal priors for $a_{0}$ based on both the KL criterion and the MSE criterion.

Our parameter of interest is the treatment effect of IFN, denoted by $\beta$. The total sample size of the E1684 trial is $n_{0}=285$ and the treatment effect is $-0.423$. We choose $d_{\textrm{MTD}}=0.423$ to be the maximum tolerable difference. The E1690 trial has a sample size of $427$ and the treatment effect is $0.098$. We use the asymptotic normal approximation to the logistic regression model with three covariates. We choose $w=0.5$, $d_{\textrm{MTD}}=0.423$ and $n=400$ (sample size of the simulated current dataset). For the KL criterion, the objective function $K$ is computed using Monte Carlo integration and optimization is performed using the optim() function in R. For the MSE criterion, the posterior mean of $\beta$ is computed using the R package hdbayes (Alt, 2022) and optimization is performed using a grid search where the values of $\alpha_{0}$ and $\beta_{0}$ range from $0.5$ to $6$ with an increment of $0.5$. The optimal priors derived using the KL criterion and MSE criterion are displayed in Figure 6. The optimal prior derived using the KL criterion is $\operatorname{beta}(0.7,1.5)$, which has mass around zero. For the MSE criterion, the optimal prior derived is $\operatorname{beta}(5.5,3)$, which is unimodal with mode around $0.7$. This is likely due to the fact that $d_{\textrm{MTD}}$ is small relative to the total sample size of $712$ – see also simulations in section B of the appendix. Because the observed difference is larger than $d_{\textrm{MTD}}$, the marginal posterior of $a_{0}$ has mode around $0.4$, which discourages more strongly than the prior. Table 5 provides the posterior mean, standard deviation and 95% credible interval for $\beta$ using the optimal priors and several other beta priors for comparison. The posterior mean is the largest for the optimal prior derived by the KL criterion and the $\operatorname{beta}(0.5,0.5)$ prior, indicating that the least historical information is borrowed. The optimal prior derived using the MSE criterion borrows the most, resulting in the smallest posterior mean and smallest variance.

5 Discussion

In this paper, we have explored the asymptotic properties of the normalized power prior when the historical and current data are compatible and when they are incompatible. We have proposed two criteria based on which the optimal hyperparameters of the prior for $a_{0}$ can be derived. While the exact values of the hyperparameters can be obtained using our objective functions, we suggest the following rules of thumb for estimating the optimal prior given different choices of the maximum tolerable difference. When the KL criterion is used, a beta distribution centered around $0.5$, such as the $\operatorname{beta}(2,2)$, is optimal for small values (when plots of the current and historical data likelihoods substantially overlap) of maximum tolerable difference , while a beta distribution with mean close to $1$, such as the $\operatorname{beta}(2,0.5)$, should be used for large values of maximum tolerable difference. When the MSE criterion is used, a beta distribution with mean less than $0.5$, such as the $\operatorname{beta}(3,6)$, is optimal for small values of maximum tolerable difference, while a beta distribution with modes at zero and one, as for example a $\operatorname{beta}(0.5,0.5)$, should be used for large values of maximum tolerable difference. The MSE criterion is a more conservative criterion, in the sense that it tends to discourage borrowing, than the KL criterion. Potential future work includes extending our method to survival and longitudinal outcomes, as well as accommodating dependent discounting parameters when multiple historical datasets are available.

Appendix A Proofs from Section 2

A.1 Technical conditions

We start our presentation by stating technical conditions under which the limiting theorems presented in section 2 hold. Then, we state an important result (Theorem A.1) in Chen (1985) which gives support to many of the proofs herein. In what follows, we will follow Chen (1985) in establishing the necessary conditions for the limiting posterior density to be normal. Let the parameter space of interest be $\Theta$ and a $p$-dimensional Euclidean space and let $B_{r}(a)=\{\theta\in\Theta:|\theta-a|\leq r\}$ be a neighbourhood of size $r$ of the point $a\in\Theta$. Also, write $L_{n}(\theta):=\sum_{i=1}^{n}\log f(x\mid\theta)$.

Theorem A.1**.**

[Bayes Central Limit Theorem (Chen (1985))] Suppose that for each $n>N$ with $N>0$, $L_{n}$ attains a strict local maximum $\hat{\theta}_{n}$ such that $L_{n}^{\prime}(\theta):=\frac{\partial}{\partial\theta}L_{n}(\theta)=0$ and the Hessian $L_{n}^{\prime\prime}(\theta):=\frac{\partial^{2}}{\partial\theta^{2}}L_{n}(\theta)$ is negative-definite for all $\theta\in\Theta$.

Moreover, suppose $\hat{\theta}_{n}$ converges almost surely to $\theta_{0}\in\Theta$ as $n\to\infty$ and the prior density $\pi(\theta)$ is positive and continuous at $\theta_{0}$. Assume that the following conditions hold:

1. C1

The largest eigenvalue of $\left[L^{\prime\prime}(\theta)\right]^{-1}\to 0$ a.s. as $n\to\infty$; 2. C2

For $\varepsilon>0$ there exists (a.s.) $N_{\varepsilon}>0$ and $r>0$ such that for all $n>\max\left\{N,N_{\varepsilon}\right\}$ and $\theta\in B_{r}(\hat{\theta}_{n})$, $L^{\prime\prime}(\theta)$ is well-defined and

[TABLE]

where $I_{p}$ is the $p$-dimensional identity matrix and $A(\varepsilon$ is a $p\times p$ positive semidefinite matrix whose largest eigenvalue goes to zero as $\varepsilon\to 0$. 3. C3

The sequence of posterior distributions $p_{n}(\theta\mid x)$ satistifies, as $n\to\infty$,

[TABLE]

for $r>0$, i.e., the sequence of posteriors is consistent at $\hat{\theta}_{n}$. Here we have assumed that the support of the posterior distributions is $\Theta$, but this could be replaced by a sequence $\Theta_{n}$.

Then we say that the posteriors converge in distribution to a normal with parameters $\hat{\theta}_{n}$ and $\left[-L^{\prime\prime}(\hat{\theta}_{n})\right]^{-1}$. For notational convenience we will (somewhat informally) write

[TABLE]

as $n\rightarrow\infty$.

A.2 Proof of Theorem 2.1

Now we move on to present a proof for Theorem 2.1 in section 2, which discusses the concentration of the posterior of $a_{0}$ at zero as the sample sizes increase in the case when there is some discrepancy between the historical and current data sets.

Proof.

We first utilise Theorem A.1 to rewrite the limiting marginal posterior distribution of $a_{0}$. Under the regularity conditions, as $n\rightarrow\infty$,

[TABLE]

where asymptotically $\hat{\theta}=\dot{b}^{-1}(\bar{y})$, $\hat{\theta}_{0}=\dot{b}^{-1}(\bar{y}_{0})$, $v=(n\ddot{b}(\hat{\theta}))^{-1}$, and $v_{0}(a_{0})=(a_{0}n_{0}\ddot{b}(\hat{\theta}_{0}))^{-1}$. For simplicity of notation, let $v_{0}=v_{0}(a_{0})$, $\ddot{b}^{-1}=\ddot{b}^{-1}(\bar{y})$ and $\ddot{b}_{0}^{-1}=\ddot{b}^{-1}(\bar{y}_{0})$. Then the kernel of the marginal posterior of $a_{0}$ becomes

[TABLE]

Then the marginal posterior of $a_{0}$ becomes

[TABLE]

Let $h(a_{0})=\frac{a_{0}r\delta^{2}}{2\ddot{b}_{0}^{-1}\left(1+a_{0}r\frac{\ddot{b}^{-1}}{\ddot{b}_{0}^{-1}}\right)}$ and $f(a_{0})=\left[\frac{a_{0}r}{1+a_{0}r\frac{\ddot{b}^{-1}}{\ddot{b}_{0}^{-1}}}\right]^{\frac{1}{2}}$. Then the denominator is

[TABLE]

Let $A=A_{1}+A_{2}$ where

[TABLE]

and

[TABLE]

We want to show $\lim\limits_{n\rightarrow\infty}\frac{A_{2}}{A_{1}}=0$.

First, we can see that

[TABLE]

Then $\inf_{x\in[\epsilon,1]}h(x)=h(\epsilon).$ We can also see that $h^{\prime}(a_{0})$ is continuous since $1+a_{0}r\frac{\ddot{b}^{-1}}{\ddot{b}_{0}^{-1}}$ is nonzero on $(0,1)$.

We then observe that

[TABLE]

Thus $\sup_{x\in[\epsilon,1]}f(x)=f(1)$.

Now we are ready to find the upper bound of $A_{2}$. Since, for any $a_{0}\in[\epsilon,1]$, $f(a_{0})\leq f(1)$ and $\exp(-nh(a_{0}))\leq\exp(-nh(\epsilon))$ , we have

[TABLE]

where $C_{1}>0$ is an integration constant. Now we find the lower bound of $A_{1}$. We know that

[TABLE]

Further, $a_{0}^{\alpha_{0}-1}\geq\min((\frac{\epsilon}{2})^{\alpha_{0}-1},\epsilon^{\alpha_{0}-1})$, corresponding to $\alpha_{0}\geq 1$ and $\alpha_{0}<1$, respectively. Similarly, $(1-a_{0})^{\beta_{0}-1}\leq\min((1-\epsilon)^{\beta_{0}-1},(1-\frac{\epsilon}{2})^{\beta_{0}-1})$, corresponding to $\beta_{0}\geq 1$ and $\beta_{0}<1$, respectively. Since $h^{\prime\prime}(a_{0})<0$, $\sup_{x\in[\frac{\epsilon}{2},\epsilon]}h^{\prime}(x)=h^{\prime}(\frac{\epsilon}{2})$. In addition, $\inf_{x\in[\frac{\epsilon}{2},\epsilon]}f(x)=f(\frac{\epsilon}{2})$. Then we have

[TABLE]

where $C_{2}>0$ is again an integration constant. Therefore,

[TABLE]

Thus, $\lim\limits_{n\rightarrow\infty}\frac{A_{2}}{A_{1}}=0$ by L’Hopital’s rule. Since $\frac{A_{2}}{A_{1}}\geq\frac{A_{2}}{A}$, $\lim\limits_{n\rightarrow\infty}\frac{A_{2}}{A}=0$. Hence, $\lim\limits_{n\rightarrow\infty}\frac{A_{1}}{A}=1$. ∎

A.3 Proof for Corollary 2.1

Proof.

The result follows by setting $\delta=0$ and $\ddot{b}^{-1}=\ddot{b}_{0}^{-1}$ into 8. ∎

A.4 Proof for Theorem 2.2

Proof.

By Theorem A.1, we know that

[TABLE]

where $\Sigma(\beta)=-\left[\frac{\partial^{2}\log[L(\beta|D)]}{\partial\beta_{i}\partial\beta_{j}}\right]^{-1}$, and also

[TABLE]

where $\Sigma_{0}(a_{0},\beta)=-\left[\frac{\partial^{2}\log[L(\beta|D_{0})^{a_{0}}\pi_{0}(\beta)]}{\partial\beta_{i}\partial\beta_{j}}\right]^{-1}$. For simplicity of notation, let $\Sigma=\Sigma(\hat{\beta})$ and $\Sigma_{0}=\Sigma_{0}(a_{0},\hat{\beta})$. Then the marginal posterior of $a_{0}$ becomes

[TABLE]

and

[TABLE]

where $\Sigma_{n}=(\Sigma^{-1}+\Sigma_{0}^{-1})^{-1}$. Then

[TABLE]

We want to show that, if $\Sigma$ and $\Sigma_{0}$ are $p\times p$ positive definite matrices,

[TABLE]

for $\delta\neq 0$ and $\epsilon>0$.

We can write $\Sigma=n^{-1}P$ and $\Sigma_{0}=(nra_{0})^{-1}P_{0}$ (Fahrmeir and Kaufmann, 1985), where $P$ and $P_{0}$ are positive definite and independent of $a_{0}$ and $n$. Then $\Sigma_{n}=(\Sigma^{-1}+\Sigma_{0}^{-1})^{-1}=n^{-1}(P^{-1}+ra_{0}P_{0}^{-1})^{-1}$.

Now,

[TABLE]

and

[TABLE]

In addition,

[TABLE]

Let

[TABLE]

and

[TABLE]

Then the denominator is

[TABLE]

First, we show $h(a_{0})$ is differentiable.

Lemma A.2**.**

Let $A$ and $B$ be positive definite matrices of the same dimension. Then, the eigenvalues of $AB$ are positive.

Proof.

By the spectral decomposition, $A=P\Lambda P^{T}$ where $\Lambda=diag(\lambda_{1},\dots,\lambda_{p})$ and $\lambda_{1},\dots,\lambda_{p}$ are the eigenvalues of $A$. Then $A^{\frac{1}{2}}=P\Lambda^{\frac{1}{2}}P^{T}$ is symmetric $\Rightarrow v^{T}A^{\frac{1}{2}}BA^{\frac{1}{2}}v=(A^{\frac{1}{2}}v)^{T}B(A^{\frac{1}{2}}v)>0$. So $A^{\frac{1}{2}}BA^{\frac{1}{2}}$ is positive definite. Since $A^{\frac{1}{2}}(A^{\frac{1}{2}}BA^{\frac{1}{2}})A^{-\frac{1}{2}}=AB$, $A^{\frac{1}{2}}BA^{\frac{1}{2}}$ and $AB$ are similar. Then they have the same eigenvalues and the eigenvalues of $AB$ are positive. ∎

Let $B=a_{0}I-(-r^{-1}P_{0}P^{-1})$. Then

[TABLE]

where $adj(B)$ is the cofactor matrix of $B$. The entries of $adj(B)$ are polynomials in $a_{0}$, so $\frac{1}{2}\delta^{T}a_{0}P^{-1}adj(B)\delta$ is a polynomial in $a_{0}$ and thus differentiable. Then we show that $\det(B)^{-1}$ is differentiable on $(0,1)$. Since $\det(B)$ is a polynomial of $a_{0}$, it suffices to show that it is nonzero on $(0,1)$. Note that $\det(B)$ is the characteristic polynomial of $-r^{-1}P_{0}P^{-1}$. Since $P_{0}$ and $P^{-1}$ are positive definite, $-r^{-1}P_{0}P^{-1}$ has negative eigenvalues by Lemma A.2. So $\det(B)$ is nonzero on $(0,1)$. Thus, we have shown $h(a_{0})$ is differentiable.

We then proceed to show that $h^{\prime}(a)>0$.

Let $E=P_{0}+a_{0}rP$. Then $h(a_{0})=\frac{1}{2}a_{0}r\delta^{T}E^{-1}\delta$. Therefore,

[TABLE]

We know that $(E^{-1})^{\prime}=E^{-1}E^{\prime}E^{-1}=E^{-1}PE^{-1}$. Since $P$ is positive definite and $E$ is symmetric, $v^{T}E^{-1}PE^{-1}v=(E^{-1}v)^{T}PE^{-1}v>0.$ Thus, $E^{-1}PE^{-1}$ is positive definite. Then $a_{0}r\frac{1}{2}\delta^{T}(E^{-1})^{\prime}\delta>0$. Since $E^{-1}$ is positive definite, $\frac{1}{2}r\delta^{T}E^{-1}\delta>0.$ So $h^{\prime}(a_{0})>0$.

We also show $h^{\prime}(a_{0})$ is continuous. It suffices to show that $\det(E)$ is nonzero on $[0,1]$. Since $E=rBP$ where $P$ is full rank, $\det(E)=c\det(B)$ where $c\neq 0$. Since $\det(B)$ is nonzero, $\det(E)$ is also nonzero.

Next, we will show that $f(a_{0})=a_{0}^{\frac{p}{2}}\det(a_{0}I-(-r^{-1}P_{0}P^{-1}))^{-\frac{1}{2}}=a_{0}^{\frac{p}{2}}\det(B)^{-\frac{1}{2}}$ is continuous on $[0,1]$. We have previously proven that $\det(B)$ is nonzero on $[0,1]$. Then $f(a_{0})$ is continuous on $[0,1]$, and it will attain its minima and maxima on the closed interval. Let $t_{1}=\max_{[\epsilon,1]}(f(a_{0}))$ and $t_{2}=\min_{[\frac{\epsilon}{2},\epsilon]}(f(a_{0}))$. Since $a_{0}^{\alpha_{0}-1}(1-a_{0})^{\beta_{0}-1}$ is continuous on $[\frac{\epsilon}{2},\epsilon]$, denote its minimum by $t_{3}$.

We write $A=A_{1}+A_{2}$ where

[TABLE]

Now we want to show that $\lim_{n\rightarrow\infty}\frac{A_{2}}{A_{1}}=0.$

First, we find the upper bound of $A_{2}$. Since $h(a_{0})$ is monotone increasing, $\exp(-nh(a_{0}))\leq\exp(-nh(\epsilon))$. Since $f(a_{0})\leq t_{1}$, we have

[TABLE]

Next, we find the lower bound of $A_{1}$. We have previously shown that $h^{\prime}(a_{0})$ is continuous on $(0,1)$. Then $h^{\prime}(a_{0})$ attains its maximum on $[\frac{\epsilon}{2},\epsilon]$. Let $t_{4}=\max_{[\frac{\epsilon}{2},\epsilon]}(h^{\prime}(a_{0}))$. We can write

[TABLE]

Therefore,

[TABLE]

and $\lim\limits_{n\rightarrow\infty}\frac{A_{2}}{A_{1}}\rightarrow 0$ by L’Hopital’s rule. Since $\frac{A_{2}}{A_{1}}\geq\frac{A_{2}}{A}$, $\lim\limits_{n\rightarrow\infty}\frac{A_{2}}{A}\rightarrow 0$. Then $\lim\limits_{n\rightarrow\infty}\frac{A_{1}}{A}\rightarrow 1$. ∎

A.5 Proof of Corollary 2.2

Proof.

Based on the assumptions, we have $\Sigma=a_{0}\Sigma_{0}$. The result follows if we plug $\delta=0$ into 10. ∎

A.6 Proof for Theorem 2.3

Proof.

The Laplace approximation for multiple parameters has the form

[TABLE]

where $\hat{\beta}$ is maximizes $-f(\beta)$, and $\hat{\Sigma}_{p\times p}=\left[\frac{\partial^{2}f(\hat{\beta})}{\partial{\beta_{j}}\partial{\beta_{k}}}\right]^{-1}$.

When $X^{\prime}Y=X_{0}^{\prime}Y_{0}$ and $X\neq X_{0}$,

[TABLE]

Define

[TABLE]

Then we have

[TABLE]

where $\hat{\beta}$ maximizes $-g_{n}(\beta)$. Similarly, define

[TABLE]

Then we have

[TABLE]

where $\tilde{\beta}$ maximizes $-k_{n}(\beta)$.

We compute the gradients of $g_{n}(\beta)$ and $k_{n}(\beta)$ and get

[TABLE]

We can see that asymptotically, $\hat{\beta}\neq\tilde{\beta}$. Then we have

[TABLE]

where

[TABLE]

The marginal posterior of $a_{0}$ is then proportional to 11 multiplied by $\pi(a_{0})$. ∎

Appendix B Additional Simulations for MSE Criterion

Figures 7 and 8 show the MSE as a function of the prior mean of $a_{0}$ for increasing ratios of $n/n_{0}$ when the total sample size is fixed. We observe that as $n/n_{0}$ increases, the model will increasingly benefit, i.e. the MSE is reduced, from borrowing more, but this trend is less prominent when the total sample size is larger.

The total sample size of the PLUTO trials in section 4.1 is about twice the total sample size of the melanoma trials in section 4.2. The total sample size of the melanoma trials is not large enough for the model to criticize the maximal tolerable difference that we chose. Therefore, the optimal prior derived using the MSE criterion encourages borrowing for the melanoma trial.