Wednesday, June 29, 2022

BETA CONJUGATE PRIOR for GEOMETRIC LIKELIHOOD | including 1st success vs NOT including 1st success

conjugate prior for a geometric likelihood two scenarios covered when the number of trials are up to and including the first success and when the number of trials are up to but not including the first success let's look at the first one.

We can write our likelihood the geometric distribution as the probability of x given theta is equal to 1 minus theta to the power of x minus 1 times theta where theta is the probability of success and x represents the number of trials the beta distribution is the conjugate.

Prior for geometric likelihood because the geometric distribution times a beta distribution is proportional to a beta distribution a likelihood times conjugate prior is proportional to the posterior the beta distribution can be written as the probability of x given alpha and beta is equal to gamma of alpha plus.

Beta over gamma of alpha times gamma beta times x to the power of alpha minus 1 times 1 minus x to the power of beta minus 1. we need to swap the axes with theta because a sample parameter is represented by theta because we are only looking at the proportionality we can ignore the.

Constants simplifying this we will have theta to the power of alpha plus 1 minus one times one minus theta to the power of beta plus x minus one minus one this is proportional to a beta distribution with parameters alpha plus one and beta plus x minus one replacing these parameters into a beta.

Distribution we have our complete posterior distribution so now we know that the beta is a conjugate prior to the geometric distribution now let's look at scenario 2 when the number of trials are up to but not including the first success we can represent this as the probability.

Of x given theta is equal to 1 minus theta to the power of x times theta we will consider the same beta prior the posterior is proportional to the likelihood times prior replacing the likelihood we have 1 minus theta to the power of x times theta since we only need the terms involving theta from the beta prior we will have.

Theta to the power of alpha minus 1 times 1 minus theta to the power of beta minus 1. simplifying this just as we've done before this is proportional to beta distribution with parameters alpha plus 1 and beta plus x replacing these parameters will give us the complete posterior distribution so.

Now we have proven that the beta is a conjugate prior for a geometric likelihood you

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