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Maximum a posteriori

marcadams267

New member
Aug 26, 2019
8
Here's the problem:

Suppose that Carl wants to estimate the proportion of books that he likes, denoted by πœƒ. He modeled
πœƒ as a probability distribution given in the following table. In the year 2019, he likes 17 books out of a
total of 20 books that he read. Using this information, determine πœƒΜ‚ using Maximum a Posteriori method.
_____________
πœƒ | 0.8 | 0.9 |
𝑝(πœƒ )| 0.6 |0.4 |
_____________

My attempt at a solution:
I know I have to use Bayes theorem to solve this, so the equation is:
f(πœƒ |x) = (f(πœƒ )f(x|πœƒ ))/f(x).

So next, I have to find f(πœƒ ) and f(x|πœƒ ) and realize that f(x) is the marginal pdf of x - which I can solve by
integrating f(πœƒ )f(x|πœƒ )dπœƒ

However, I'm stuck on the first step as I'm not entirely sure how to express the data on the table as the pdf f(πœƒ ) and the conditional probability f(x|πœƒ ).
While I can reasonably attempt the math, I would like help translating the words of this problem into actual equations that I can use to solve the problem. Thank you