I have always had the following question about Bayesian Probabilities.
Suppose you observe the weather for 90 days. You observe that:
- 65 days it was Sunny
- 25 days it Rained
This means that there is a 0.72 probability of it being Sunny (65/90) and a 0.28 probability of it Raining (25/90) - to me, these probability formulas seem to be equivalent to the Maximum Likelihood Estimates.
My Question: Is there a way to "influence" these probabilities using a Bayesian Prior? Just like regression coefficients in a Bayesian Linear Regression model can be "influenced" using a Bayesian Prior - suppose we have reasons to believe that the true probabilities of these weather states have a Normal Distribution with mean = mu and variance = sigma - would it be possible to "factor" this information in and adjust our estimates of these weather frequencies?
For instance - could this perhaps result in P(Sunny) = 0.71 and P(Rain) = 0.29 ?
Note: I think a Binomial Distribution might be more suitable for a Prior Distribution in this example.