I am trying to apply the Bayes theorem in a realistic example so that I can understand it better. I need your help to make sure that my logic is correct.
I am applying for jobs and I have a series of theta probabilities for getting invited for an interview (theta = 100). The probabilities assigned to theta values are higher for low values. I expect that each posting has about 100 applicants and so if I am one of them I get 1% chance (it does not work this way but let's keep it simple for now). So for theta = 0.01, I assign the highest p(theta). As theta decreases so does the probability.
I've applied to 60 jobs so far and got invited for an interview twice. So p(D|theta) = theta^2 * (1/theta)^58. And so my p(D) = sum ( p(D|theta)*p(theta)).
It seems to work efficiently and indeed the posterior has an increased probability for values of theta higher than 0.01, which makes sense.
Is this model alright?
Also what if I wanted to pinpoint the exact most probable theta value? Currently the range between 0.01 and 0.99 is lacking depth. I could make the test in the range of 1000 values but wouldn't that complicate the model? Is there a more elegant way to determine the exact theta?
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