  Clinical trials usually involve sequential patient entry. When designing a
clinical trial, it is often desirable to include a provision for interim
analyses of accumulating data with the potential for stopping the trial early.
We review Bayesian sequential clinical trial designs based on posterior
probabilities, posterior predictive probabilities, and decision-theoretic
frameworks. A pertinent question is whether Bayesian sequential designs need to
be adjusted for the planning of interim analyses. We answer this question from
three perspectives: a frequentist-oriented perspective, a calibrated Bayesian
perspective, and a subjective Bayesian perspective. We also provide new
insights into the likelihood principle, which is commonly tied to statistical
inference and decision making in sequential clinical trials. Some theoretical
results are derived, and numerical studies are conducted to illustrate and
assess these designs.
