# Thread: Calculate Posterior from flat prior (?)

1. ## Calculate Posterior from flat prior (?)

Here's the problem:

Suppose that the number of defects on a roll of magnetic recording tape has a Poisson distribution for which the mean u is either 1.0 or 1.5 and the prior p.f. of u is L(1.0)=0.4 and L(1.5)=0.6. If a roll of tape selected at random is found to have three defects, what is the posterior pf of u.

I just started messing with Bayesian statistics, so I apologize in advance if my logic is way off. I know that the posterior is proportional to the product of the likelihood and prior, but simply multiplying 0.4 by the Poisson distribution p(3;1) doesn't result in the correct answer. I doubt using the conjugate Poisson-gamma would work since I wasn't given an alpha or beta. Also, for a uniform distribution, I could have used alpha=1 and beta=1 in a gamma distribution to model [0,1], but I'm not quite sure if anything similar to that applies here.

Any help would be awesome!

2. ## Re: Calculate Posterior from flat prior (?)

Awesome! You have no problems! That's good to hear

3. ## Re: Calculate Posterior from flat prior (?)

Originally Posted by Dason
Awesome! You have no problems! That's good to hear
I hit tab then enter accidentally! I fixed it.

4. ## Re: Calculate Posterior from flat prior (?)

Note that the posterior is proportional to the prior times the likelihood. So 0.4 times the likelihood will give you something proportional to the posterior. Since we're only dealing with two values for u you can just do the same thing for the other value and then divide both by the sum of the two. That was very handwavy and not very mathematical but does that make sense to you?

I don't know what you're trying to do in the second paragraph - it sounds like you were already handed your prior distribution. Also a Gamma(1,1) is most definitely not a uniform distribution. I think you might have meant Beta(1,1). But I don't see what that has to do with the problem at hand?

5. ## Re: Calculate Posterior from flat prior (?)

Originally Posted by Dason
Note that the posterior is proportional to the prior times the likelihood. So 0.4 times the likelihood will give you something proportional to the posterior. Since we're only dealing with two values for u you can just do the same thing for the other value and then divide both by the sum of the two. That was very handwavy and not very mathematical but does that make sense to you?

I don't know what you're trying to do in the second paragraph - it sounds like you were already handed your prior distribution. Also a Gamma(1,1) is most definitely not a uniform distribution. I think you might have meant Beta(1,1). But I don't see what that has to do with the problem at hand?

Whoops. Yeah; I meant beta for the uniform. The last paragraph was, if anything, me just trying anything since my initial attempts just didn't work out.

And wow. The whole time I was treating both values as separate problems -- never putting the two together. As a result, I never had a denominator for the equation that equals the posterior. Since it's discrete, I just had to sum the two answer and what i had is no longer just proportional. That was really easy.

For some reason, had it been a continuous distribution and the prior was some continuous function, I would have done it correctly. Odd this didn't occur to me.

Thanks!

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