I am reading Statistical Rethinking (Section 6.2.1.2).
The topic of this section is measuring accuracy for a Bayesian model, i.e. accuracy of the model of predicting correctly an outcome.
The example compares the predictions in {sunny, rainy} of two weatherpersons along 10 days. The two weatherpersons are the two models to compare. The first model assigns a probability for each day of being rainy. The second model predicts all days to be sunny.
The author compares the two models using two different measures of accuracy: hit rate (number of days the model predicts correctly) and joint probability.
Given the following data:
Day 1 2 3 4 5 6 7 8 9 10
PredictedRainy_A 1 1 1 .6 .6 .6 .6 .6 .6 .6
PredictedRainy_B 0 0 0 0 0 0 0 0 0 0
Observed R R R S S S S S S S
The second model (B) has a higher hit rate (7 vs 5.8), because by predicting all days to be sunny it gets them right 7 times.
However the first model (A) excels when we consider the joint probability of getting all the weather forecasts right (0.005 vs 0).
I understand why the joint probability is a better measure of accuracy in this case, however I don't understand this passage quoted from the author:
And the joint probability is the measure we want. Why? Because it appears in Bayes' theorem as the likelihood.
How does the Bayes' theorem likelihood relate to the choice of this measure of accuracy?
