Non-informative prior for Exponential

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.

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.

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 .