More detailed question about MAP hypothesis I have asked a similar question here before, I got one answer but I'm still not sure the user arrived at the answer. So I have actually changed the question I asked to help me figure out how to get the solution to this problem. 
  And to be clear, I am not looking for the final answer, only to understand how to set up the problem with the given data.

This problem is looking to find the MAP hypothesis. I understand the simpler problems but this is given threw me off a bit. 

You are given a coin that may or may not be biased. Specifically,
you have three hypotheses about the coin:

H1 = “the coin has probability 1/2 of landing heads”

H2 = “the coin has probability 1/3 of landing heads”

H3 = “the coin has probability 2/3 of landing heads”

The priors are:

$P(H_1)$ = $P(H_2)$ = $P(H_2)$ =  $\frac 13$

The event $D_1$ = " You toss the coin seven times and observe five 

heads. 

Which is the maximum a posteriori hypothesis?"


The answer given previously is $H_3$

The way I break it down is:

I have one event, thus, I am trying I believe I am to find:
$$P(D_1\mid H_1)P(H_1)$$
$$P(D_1\mid H_2)P(H_2)$$
$$P(D_1\mid H_3)P(H_3)$$

and compare all of them to see which is the postiori.

What is confusing to me is:$$P(D_1\mid H_i)$$Which value should I use for $H_i$?

And since they are independent, should I just multiply ($H_i$)($\frac57$)($\frac13$)?
You should review the lecture slides. Specifically the one on MAP

https://people.cs.umass.edu/~mcgregor/240S16/lec20.pdf

You should find it helpful.
You should review the lecture slides. Specifically the one on MAP

https://people.cs.umass.edu/~mcgregor/240S16/lec20.pdf

You should find it helpful.