Multiple events and using bayes theorem

[skyflashings]

I am not sure if I am doing this correctly, so please check my attempt:

I have four discrete random variables: G, H, J, L

Say, I want to find P(H|J, G, L).

Then can I write as P(H|J, G, L) = P(J, G, L|H)*P(H) = P(J|G, L, H)*P(G|L, H)*P(L|H)*P(H)

Are those equivalent? I can't find an example exactly like this one; I just want to make sure that it works this way. Thanks!

 

[Stephen Tashi]

Then can I write as P(H|J, G, L) = P(J, G, L|H)*P(H)

No. P(J,G,L | H) P(H) is equal to P(H,J,G,L), which is not the same as P(H|J,G,L)
$$P(H| J \cap G \cap L) = \frac{ P(H \cap J \cap G \cap L)}{P( J \cap G \cap L)}$$

There are various ways to rewrite the right hand side.

For example, the numerator can be written as:

$$P( H \cap J \cap G \cap L) = P(H | J \cap G \cap L) P(J \cap G \cap L)$$
$$= P(H| J \cap G \cap L) P(J | G \cap L) P(G \cap L)$$
$$= P(H| J \cap G \cap L) P(J | G \cap L) P(G | L) P(L)$$

which is similar to what you had in mind. You could also write it as you did, with P(H) as the last factor.

 

[skyflashings]

Ok, I see how it unfolds now.

One of the reasons I wanted to get my final form is that I only have a distribution for P(H), P(J|H), P(G|L,H), and P(L|H) and there is a conditional independence assumption that P(J|H)=P(J|G,L,H).

So, I need to find P(H|J, G, L) in terms of those.

If I were to follow your steps to solve for P(H|J, G, L), would it look something like:

P(H|J, G, L) = P(H, J, G, L) / ((P(H|J, G, L)*P(J|G, L)*P(G|L)*P(L))

I'm not sure how to proceed at that point..

Thanks for the help, I hope I can wrap my head around this.

 

[Stephen Tashi]

I think you need more information to calculate P(H| J, G, L). You need to be able to calculate P(J,G,L) without the condition that the event H is true. If you know the problem has an answer, think about whether there is other known information.

 

[blue_raver22]

Your attempt is correct. Bayes' theorem states that the probability of an event A given an event B can be calculated by multiplying the probability of event B given event A (P(B|A)) with the prior probability of event A (P(A)) and dividing it by the probability of event B (P(B)). In your case, the events are G, H, J, and L, and you want to find the probability of H given J, G, and L. So, your formula is P(H|J, G, L) = P(J, G, L|H)*P(H)/P(J, G, L). This is equivalent to your calculation, where you have broken down P(J, G, L|H) into P(J|G, L, H)*P(G|L, H)*P(L|H). This is known as the chain rule or product rule of probability. So, your formula is valid and you can use it to find the desired probability. Just make sure to use the correct values for each probability and you should be able to get the correct result.