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I need to do a bayesian test for a simple random sample with exponential distribution and N(2,3) as a prior distribution conditioned by: Null hypothesis=> $\theta$ less or equal to 1 ; Alternative hypothesis=> $\theta$>3

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  • $\begingroup$ I am catching up on the lingo in stats so let me make sure I understand your problem. You have a set of data that you know is sampled from an exponential distribution. You have prior belief on the parameter $\lambda$ being drawn from a normal distribution with mean $\mu=2$ and standard deviation $\sigma=3$. You would like to compare the probability $\theta = \frac{1}{\lambda}<1$ with an alternative hypothesis $\theta = \frac{1}{\lambda} <3$. Is that correct? $\endgroup$ May 21, 2016 at 17:23
  • $\begingroup$ I only have a mistake, the alternative hypothesis is $\theta$ > 3, rest of problem is correct. $\endgroup$
    – GeorgeF
    May 21, 2016 at 17:35
  • $\begingroup$ Is the problem that you can't find the posterior distribution? It should be pretty easy to derive, or you can probably just Google it. $\endgroup$
    – Jon
    May 21, 2016 at 17:49
  • $\begingroup$ I'm trying to find the posterior distribution using the parametric formulation: en.wikipedia.org/wiki/Bayesian_inference#Parametric_formulation ; but I don't know how to solve that integral. I didn't find it in Google. $\endgroup$
    – GeorgeF
    May 21, 2016 at 18:05
  • $\begingroup$ Are you talking about the denominator? If so that should be a constant independent of your $\lambda$ parameter. You might be better off considering the ratio of probabilities. Then you don't need to take that integral. It looks like you get something rather ugly with erf everywhere. $\endgroup$ May 21, 2016 at 18:37

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So in order to solve this problem you find the posterior distribution $p(\theta | \bf{x})$, where $x_i \sim Exp(\theta)$. The Likelihood is simple, $\theta ^n exp(-\theta \sum_{i=1}^n x_i)$ and the prior is $p(\theta) \propto exp(-\frac{(\theta - \mu)^2}{2 \sigma^2})$.

The posterior: $p(\theta | \bf{x}) \propto \theta ^n exp(-\theta \sum_{i=1}^n x_i) exp(-\frac{(\theta - \mu)^2}{2 \sigma^2})$

Somehow you want to make all of that look like the kernel of an exponential, gamma, or Normal. From my quick attempt, I didn't see an easy way of doing this. I would suggest changing priors if you can, to something like a reference prior or conjugate (i.e. gamma).

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