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I am working with a Bayesian model: $T \sim exp(\theta)$ for survival data, I have chosen a gamma distribution as a prior since its conjugate by an exponential distribution. I'd like to choose a $\gamma(\alpha, \beta)$ non informative prior, since the exponential distribution has much of its probability around 0 I think it won't be convenient to choose $\gamma(0.001,0.001)$ as non informative for the exponential.

Would it be convenient to choose an improper non informative gamma prior of the form $\gamma(1/3,0)$ taking into account I want to estimate later some parameters $\phi = h(\theta)$ using simulations of $\theta^{(1)}, \theta^{(2)}, \dots, \theta^{(M)}$: $\phi^{(j)} = h(\theta^{(j)})$. I know flat priors aren't invariant under re-parametrization.

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I hope what I write is helpful.

When you don't have information about the parameter of interest, in your case the parameter $\theta$, you want to use a prior, that will let the data drive the inference. Such priors are called non informative.

For your case in particular:

Lets assume that we have $n$ iid data points, $T_{1},T_{2},...,T_{n}$ then the likelihood function of these data points, the function driven mostly by the data, is of the form:

$L(\theta|T_{1},...,T_{n})=L(T_{1},T_{2},...,T_{n};\theta)=\prod_{i=1}^{n}\theta e^{-\theta T_{i}}=\theta^{n}e^{-\theta \sum_{i=1}^{n}T_{i}}$

Also, let us consider as prior distribution for the parameter $\theta$ the $Gamma(a,b).$

The ultimate goal in Bayesian inference is to calculate the posterior distribution:

$p(\theta|T_{1},T_{2},...,T_{n})\propto L(\theta|T_{1},T_{2},...,T_{n})p(\theta) \propto\theta^{n}e^{-\theta \sum_{i=1}^{n}T_{i}} \theta^{a-1}e^{-b\theta}= \theta^{n+a-1}e^{-\theta(\sum_{i=1}^{n}T_{i}+b)}$.

If you notice in the case where you choose the parameter of the prior distribution of $\theta$ as $a=0.001$ and $b=0.001$, then $n+0.001-1\approx n$ and $\sum_{i=1}^{n}T_{i}+0.001\approx \sum_{i=1}^{n}T_{i} $. Generally speaking this means that the posterior distribution $p(\theta|T_{1},T_{2},...,T_{n})$ will be close to the likelihood function, i.e will be close to the function that is driven mostly by your data. Hence, I think that a Gamma distribution of the form $Gamma(0.001,0.001)$ is not a bad choice.

Also, not that the parameters of a Gamma distribution cannot be zero.

Lastly, if you want to can check Jeffreys Prior, which is invariant under parametrization and can be also non informative, https://en.wikipedia.org/wiki/Jeffreys_prior .

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