Conditioning on uncertain event: Extensions to Bayesian inference

In this paper the alternative procedure for updating probabilities (that is, to calculate the posterior distribution from the prior distribution) proposed by Richard Jeffrey is considered, which allows the addition of new information to the prior distribution under more circumstances than with the Bayesian conditioning. A predictivistic approach for the Jeffrey's rule is introduced and a definition of conjugacy according to this rule (named Jeffrey-conjugacy) is established. Results for Jeffrey-conjugacy in the exponential family are also presented. As a by-product, these results provide full predictivistic characterizations of some predictive distributions. By using both the predictivistic Jeffrey's rule and Jeffrey-conjugacy, a forecasting procedure which is applied to the Chilean stock market data is also developed. The Jeffrey's rule with the Bayesian conditioning according to their capability of incorporating unpredictable information in the forecast is compared.