Title: A decision-theoretic property of conditional normalized maximum likelihood distribution
Authors: Yoshihiro Hirose - Hokkaido University (Japan) [presenting]
Abstract: Distribution prediction is considered, where we observe a data, and estimate the distribution of a future data based on the observation. The estimated distribution is called a prediction distribution. Our target is Conditional Normalized Maximum Likelihood (CNML) distribution. CNML is a generalization of Normalized Maximum Likelihood (NML) distribution. NML is the minimax distribution with respect to the regret. The regret is the difference between a candidate distribution and the best distribution for a virtually-observed value. Similarly, CNML is the minimax distribution with respect to a conditional regret. Three versions of CNMLs corresponding to three types of conditional regrets have been introduced. Based on an observed data, a conditional regret compares a prediction distribution with the best distribution for a virtually-observed value. We are interested in whether CNMLs are admissible or not under some criterion. The admissibility is a basic concern in statistical decision theory. We are also interested in the minimaxity of CNMLs. CNML is originally minimax with respect to the conditional regret. However, it is not clear that it is also minimax under other criterion, e.g., the Kullback-Leibler divergence and conditional regret risks. The result depends on statistical models we assume.