I’m having a little trouble parsing what you say here, so I might be interpreting your question wrong.
The basic thing to keep in mind is that the prior odds multiplied by the likelihood ratio (what you call the “odds ratio multiplier”) give you the posterior odds. Your problem appears to stem from the fact that you are working directly with prior and posterior probabilities, without converting them to odds. To convert P(A) into the odds for A, divide it by 1 - P(A).
In the first case, numStarts is 4, and your prior is 1/numStarts, so your prior odds are 1:3. Since the posterior probability is 0.9, the posterior odds are 9:1. So your likelihood ratio is:
LR = (9:1) / (1:3) = 27
In the second case your prior is 1⁄2, so prior odds are 1:1. You are assuming the likelihood ratio is the same, so LR = 27. Your posterior odds, then, are 27 * (1:1) = 27:1. This means your posterior probability is 27⁄28, or 0.96.
I’m having a little trouble parsing what you say here, so I might be interpreting your question wrong.
The basic thing to keep in mind is that the prior odds multiplied by the likelihood ratio (what you call the “odds ratio multiplier”) give you the posterior odds. Your problem appears to stem from the fact that you are working directly with prior and posterior probabilities, without converting them to odds. To convert P(A) into the odds for A, divide it by 1 - P(A).
In the first case, numStarts is 4, and your prior is 1/numStarts, so your prior odds are 1:3. Since the posterior probability is 0.9, the posterior odds are 9:1. So your likelihood ratio is:
LR = (9:1) / (1:3) = 27
In the second case your prior is 1⁄2, so prior odds are 1:1. You are assuming the likelihood ratio is the same, so LR = 27. Your posterior odds, then, are 27 * (1:1) = 27:1. This means your posterior probability is 27⁄28, or 0.96.
I hope I understood your question correctly.
Yes, thanks! Silly me; sorry. I will rewrite the post.