Suppose I have which are i.i.d as Bernoulli random variables with and the prior is normal with mean 0 and variance 100. How can I find the posterior distribution if the number of equal to one is 5 and n is 16?
What do I use for the likelihood here? Do I differentiate since it's CDF of logistic distribution or do I use the pdf of a bernoulli?
The question confuses me
Last edited by MrAnon9; 12-13-2011 at 01:55 PM.
so I just have to find the product of the 1 over 1 + e ?
What? The likelihood would be a binomial distribution: (16 choose 5) * p^5 (1-p)^11 and replace p with the quantity above.
Also are you sure the prior has a variance of 0?
I meant 100 sorry
Just want to say that a prior with variance 0 is very stubborn - it will not change.
If you have a point mass prior, you are "not Bayesian" :P
Anyway back to OP problem, since they are not in a conjugate class, so after using the Bayes theorem to write down the posterior, not much can be simplified.
Why is it a binomial by the way? Also, can I not get rid of the 16 choose 5 as a constant of proportionality~?
The combinatoric coefficient in the front is not important. You will cancel it anyway. You can also think the sample condition on the parameter is just Bernoulli.
Because you don't know the order that the successes came. There is only 1 way to order having no successes. There are 16 ways to have 1 success... it's the same argument as in the development of the binomial distribution. Mainly because it is a binomial distribution.
Last edited by MrAnon9; 12-14-2011 at 04:16 PM.
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