Bayesian inference of the common standardized mean difference in meta-analysis

We develop an objective Bayesian framework for estimating the common standardized mean difference in meta-analysis. To construct noninformative priors, we derive both probability matching and reference priors. Our analysis reveals that although general second-order matching prior does not exist, a valid version can be obtained when the sample sizes of the two arms in each study are equal. Among the reference priors evaluated, we find that both one-at-a-time and two-group reference priors satisfy the first-order matching criterion, whereas Jeffreys' prior does not. Simulation studies demonstrate that the proposed matching prior and the one-at-a-time reference prior yield accurate frequentist coverage probabilities, consistently outperforming Jeffreys' prior. Finally, the practical utility of this framework is validated through two real-world meta-analytic applications, underscoring its effectiveness for robust objective Bayesian inference in standardized mean difference models.