Some reasons for reconciling confidence intervals and bayesian intervals

Abstract

It is well known that one of the conflicts between Bayesian and frequentist approach to inference lies in the determination and interpretation of "interval estimates". In particular, Bayesians deny any probabilistic meaning to realised confidence intervals. On the contrary, the author shows that a confidence coefficient must inevitably be interpreted as a subjective probability, at least when there is a large uncertainty in the assignment of a prior distribution. Besides, this viewpoint is operationally sustained by some formal relationships between confidence and Bayesian intervals: exact equality between the respective coverage probabilities in some special cases (Lindley, amongst others), exact equality again between mean values of posterior probabilities and mean values of confidence coefficients in all cases (Pratt), and approximate equality between coverage probabilities in most cases, especially if the interval is calculated on a sample with a not too small size (Welch and Peers and others).