How Can Carl Use Maximum a Posteriori to Estimate His Book Preferences?

[marcadams267]

Here's the problem:

Suppose that Carl wants to estimate the proportion of books that he likes, denoted by 𝜃. He modeled
𝜃 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 𝜃̂ using Maximum a Posteriori method.
_____________
𝜃 | 0.8 | 0.9 |
𝑝(𝜃 )| 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(𝜃 |x) = (f(𝜃 )f(x|𝜃 ))/f(x).

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

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(𝜃 ) and the conditional probability f(x|𝜃 ).
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

 

[mmwave]

for your assistance.To start, we can define the variables as:

𝜃 - proportion of books that Carl likes
x - number of books liked by Carl in a given year

We are given the following information:

- Carl likes 17 out of 20 books in the year 2019
- 𝜃 is modeled as a probability distribution with two possible values: 0.8 and 0.9
- The corresponding probabilities for 𝜃 are given as 0.6 and 0.4, respectively

From this, we can express the probability distribution f(𝜃) as:

f(𝜃) = 0.6 if 𝜃 = 0.8
f(𝜃) = 0.4 if 𝜃 = 0.9

Next, we can express the conditional probability f(x|𝜃) as the probability of getting x successes (books liked by Carl) given a specific value of 𝜃. In this case, we can use the binomial distribution as follows:

f(x|𝜃) = (20 choose x) * 𝜃^x * (1-𝜃)^(20-x)

Now, we can substitute these values into the Bayes theorem equation:

f(𝜃|x) = (f(𝜃)*f(x|𝜃))/f(x)

We can solve for f(x) by integrating f(𝜃)*f(x|𝜃) over all possible values of 𝜃 (0.8 and 0.9):

f(x) = ∫ (f(𝜃)*f(x|𝜃)) d𝜃
= (0.6 * (20 choose x) * 0.8^x * 0.2^(20-x)) + (0.4 * (20 choose x) * 0.9^x * 0.1^(20-x))

Now, to find 𝜃̂ using Maximum a Posteriori method, we need to find the value of 𝜃 that maximizes the posterior probability f(𝜃|x). In other words, we need to find the value of 𝜃 that makes f(𝜃|x) as large as possible.

This can be achieved by taking the derivative of f(𝜃|x) with respect to 𝜃