Is there a difference between posterior denisty and posterior distribution?

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March 24th 2011, 01:39 PM
sirellwood

Is there a difference between posterior denisty and posterior distribution?

Is there a difference between posterior denisty and posterior distribution? Or are they 2 names for the same thing?
March 24th 2011, 02:57 PM
harish21

Posterior Density refers to the pdf of the posterior distribtuion.
March 24th 2011, 03:14 PM
sirellwood

ok so does that mean that if iv got a posterior distribution of $x^{101}$ $e^{-29x/2}$ , thats fine is it? so if i need it to be a pdf do i need to change it? because if i integrate it over 0....infinity it doesnt sum to 1?

Is this where i would need a multiplier "k" in the posterior density?
March 24th 2011, 03:27 PM
harish21

can you write down your question or show how you got till there? What you have mentioned as your posterior distribution looks to me like the kernel of a gamma distribution.
March 24th 2011, 04:06 PM
sirellwood

Suppose that a random sample of size n = 100 from an exponential distribution with parameter http://www.mathhelpforum.com/math-he...5dc8912759.png is about to be collected. It is thought that a Ga(2, 2) prior distribution for http://www.mathhelpforum.com/math-he...5dc8912759.png is appropriate but, since it is believed that values of http://www.mathhelpforum.com/math-he...5dc8912759.png exceeding 6 are impossible, the prior distribution is truncated above 6.

Calculate the prior probability that http://www.mathhelpforum.com/math-he...5dc8912759.png exceeds 6 assuming the truncated prior distribution. What is the probability when the prior distribution is not truncated?

Suppose the data yield ¯x = 1/8. Calculate the posterior probability that http://www.mathhelpforum.com/math-he...5dc8912759.png
exceeds 6 assuming the truncated prior distribution. What is the probability
when the prior distribution is not truncated?
March 24th 2011, 04:13 PM
sirellwood

so i integrated my prior function over the range 0....6 and set it equal to 1 to find my constant "k" to make it a valid distribution.

i found "k" to be 4/(1-13exp{-12})

then from here i started getting confused as to how to calculate the prior probabilities. I guess thats when i started getting mixed up between densities and distributions?

Would you be able to walk me through what you did, I have attained answers but really not sure whether iv used the correct integrals or not because of my confusion! :-s
March 24th 2011, 05:57 PM
theodds

The probability distribution is the function that maps sets to probabilities. The density is the function that you integrate over. So, for example a distribution would be a function $\phi$ such that $\phi(A) = P(X \in A)$ whereas the density is the function $f$ such that $\int_A f(x) \ dx = P(X \in A)$ .

If the question asks to find the posterior distribution, reporting the density should be enough to satisfy whoever is asking the question.