Posterior progress.

I has been 25 years since Medical Decision Making published its first edition. This journal, and its sponsor, the Society for Medical Decision Making, have been the driving force behind the use of Bayesian thinking in clinical medicine and health care. Although other journals have published applications of decision analysis or cost-effectiveness analysis, and the march of methodological progress is led primarily by journals targeted at professional statisticians, MDM serves the full spectrum of producers and consumers of Bayesian statistics in health care. Its readership encompasses both clinicians looking for application and scientists looking to advance Bayesian methodology in their fields. In the early days, the mathematics was very simple. One started with a prior probability of disease, applied (or not! ) a test with dichotomous outcome having a known sensitivity and specificity, and calculated the posterior probability of disease. These calculations could be captured in a 2 × 2 contingency table, avoiding the need to use even the simplest mathematical formulations of Bayes’s theorem. As with many other aspects of statistics, the statistical world was far ahead of the medical one. Other fields, especially operations research, grappled with Bayesian models with latent variables having complicated distributions. The calculation of a Bayesian posterior distribution in full generality was recognized to be a very difficult problem. Even statisticians who regarded the Bayesian approach to statistics as superior to frequentist techniques on theoretical and philosophical grounds acknowledged that the intractability of calculating posterior distributions in realistic situations was a serious barrier to widespread adoption of the Bayesian paradigm in science. There were, of course, simple situations that lent themselves to easy calculation. When the prior distribution is “conjugate” to the conditional (e.g., normal-normal, beta-binomial, to mention just two), then the posterior distribution is from the same family as the prior, and its parameters are related to those of the prior and conditional by simple algebraic formulas. Conjugate prior analysis appeared in MDM in 1989 but never achieved great popularity. The number of clinical situations that could be adequately modeled using conjugate distributions was small, and although the formulas for the posterior distributions were simple and accessible to a wide audience, the derivation of those formulas was not. The conjugacy relationship between distributions is not intuitive, and understanding, though not applying, the formulas requires a working knowledge of calculus. As a technique with limited application whose rationale is not transparent to the target audience, its place in the health care literature has been limited. Another early approach to calculating posteriors in difficult situations is the use of approximations. Under certain conditions, it can be proved that a posterior distribution is reasonably approximated by one from a parametric family, such as a normal distribution. The article in this issue by Hazen and Huang uses and explains this approach nicely. They provide the necessary formulas, as well as sample size conditions for adequacy of approximation. By showing how the technique can be used to conduct a probabilistic sensitivity analysis for Mrus and Tsevat’s costeffectiveness analysis, they demonstrate that the technique is applicable to a broad variety of situations that arise commonly in epidemiology, medicine, and health policy. The method is both intuitive and computationally simple—it can even be carried out in a spreadsheet. A great deal of serious Bayesian analysis can be done through this approach. In fact, most costeffectiveness analyses conducted today could apply this technique with considerable reduction of computational burden compared to the usual approach to sensitivity analysis. It is worth pointing out that some situations dealt with by Hazen and Huang, those involving only estimation of a single parameter, might also have been dealt with using the conjugate priors approach. But the conjugate priors approach may not apply when the model parameters are not independent From the Department of Family & Social Medicine, Albert Einstein College of Medicine, Bronx, NY.