(cf. [@dawid-1986]), whereas the Japan Meteorological Agency started probability forecasting in 1980 for Tokyo area and extended it to the whole Japan in 1986[^4]. How can we assess the quality of probability forecasting? We propose to assess probability forecasting by setting up a hypothetical betting game against forecasting agencies in the framework of game-theoretic probability by Shafer and Vovk [@ShaVov01].

We can regard the capital process of a betting strategy as a test statistic of a statistical hypothesis ([@shafer-etal-test-martingales], [@ktt-new-procedure]). Our null hypothesis is that given the probability $p_n$ announced by the agency, the outcome is indistinguishable from the Bernoulli trial with success probability $p_n$. If this hypothesis is true, then the capital process becomes a non-negative martingale and the capital process converges to a finite value almost surely. However if the announced probability $p_n$ is not good, then a clever strategy may be able to beat the forecasting agency in the betting game. In our game we construct a betting strategy based on Bayesian logistic regression modeling, which is a very standard statistical model for analyzing binary responses. We will prove some results on the strong law of large numbers in probability forecasting game with side information based on 