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I have been working on an assignment lately wherein the object is to implement an MCMC approach to simulate from a generated posterior distribution.

The posterior distribution is generated from a likelihood given by $p(x|\lambda) \sim \text{Poisson}(\lambda)$ and a prior for $\lambda$ given by $p(\lambda) \sim \text{Exponential}(\lambda_{0})$.

Hence, $$ p(\lambda) = \lambda_{0}e^{-\lambda_{0} \lambda} $$

In the assignment, we are we are not provided with a value for $\lambda_{0}$, but we are told that there is a $90\%$ belief that $\lambda < 7$.

Given the above, I have inferred that:

$$ \int_0^7 \lambda_{0}e^{-\lambda_{0} \lambda}d\lambda = 0.90$$

And, solving for $\lambda_{0}$ in the integral above, I arrived at $\lambda_{0} \approx 0.329$.

  • My first question: Is this a valid way to determine the a value for $\lambda_{0}$ given the limited information issued in the assignment?

Also, in the same assignment, we are instructed to use a $\text{Gamma}$ prior as an alternative to the $\text{Exponential}$ prior mentioned above. We are given no additional information about the $\text{Gamma}$ prior (i.e. parameter values or distributions on those priors) we should use.

I have inferred that if I compute the $\mathbf{E}[\lambda] = \lambda_{0}^{-1}$ and $\text{Var}[\lambda] = \lambda_{0}^{-2}$ for the $\text{Exponential}$ prior mentioned above, I can use this to solve for the parameters of the alternative $\text{Gamma}$ prior. More mathematically:

$\mathbf{E}[\lambda] = \frac{\alpha}{\beta} = \lambda_{0}^{-1}$ and $\text{Var}[\lambda] = \frac{\alpha}{\beta^{2}} = \lambda_{0}^{-2}$

Solving for $\alpha$ and $\beta$ yields:

$\alpha = \lambda_{0}^{0} = 1 = 0.329^{0} = 1$ and $\beta = \lambda_{0}^{1} = \lambda_{0} = 0.329^{1} = 0.329$

  • My second question: Is this a valid way to derive the parameter values for $\alpha$ and $\beta$ for the $\text{Gamma}$ mentioned above given the limited information provided in the assignment? In other words, can I use the expected value and variance derived from the $\text{Exponential}$ prior to develop an analogous $\text{Gamma}$ prior? Or is it more important to preserve the $90\%$ belief that $\lambda < 7$?

EDIT:

Perhaps I should mention that using the $\text{Gamma}$ parameters specified above, yields the same prior distribution (and hence identical posterior distribution) as the $\text{Exponential}$ prior distribution (specific case of $\text{Gamma}$ distribution), but perhaps this is the point.

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based on your description it seems like you don't necessarily need the exponential to be analogous. –  jerad Nov 22 '12 at 22:46
    
@jerad : I guess it doesn't need to be analogous, but I was intending to use analogy (or some likeness to the exponential prior distribution) as a platform for forming a sensible gamma prior. Perhaps this isn't a good idea though. –  user9171 Nov 22 '12 at 23:24
1  
well re: your second question, does your derived distribution put 90% of the probability mass below 7? If so, I would think the more important thing is to convince yourself that your mcmc algorithm is behaving correctly. –  jerad Nov 23 '12 at 0:41
    
@Stat : I didn't use the Poisson at all to derive the form of the prior. The Poisson distribution is the likelihood distribution. It is used in the computation of the posterior distribution, but not the prior. What I have elected to do is: firstly, demonstrate the the $\text{Exponential}$ distribution is a special case of the $\text{Gamma}$ distribution through the means given above, and, then, apply a $\text{Gamma}$ distribution with parameters $\alpha$ and $\beta$ such that $90\%$ of the mass of the distribution is below $\lambda = 7$. –  user9171 Nov 23 '12 at 3:25
    
If the assignment specifically says to use a prior with 90% of the mass on $\lambda < 7$, then I'd say that's the most important part of constructing your prior, regardless of which distribution is used. But you never got specific about the goals of your assignment, so it's difficult for us to give advice about what is or isn't important. –  jerad Nov 23 '12 at 3:32

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