Just to be clear this is a homework question. I am not asking for the answer, which is why I am leaving the specific variables out. I am more wondering how to solve this problem.
The lifetime X of a light bulb can be modeled by an exponential distribution with unknown parameter θ. Use the parameterization E(X) = 1/θ, this is,
$$p(x | \theta) = \theta e^{−\theta x} = Gamma(1;\theta)$$
Suppose we observe the following lifetimes: x1 = #1, x2 = #2, x3 = #3. Assume that an expert believes that the parameter θ should also follow an exponential distribution and should be on average #4
How would I go about answering these questions? (a) Find the MLE of θ for the observed lifetimes. (b) Give the prior distribution elicited from the expert. Is this a conjugate prior? (c) Find a Bayes estimator and compare it with the MLE from (a). (d) Give a central 95% posterior interval for θ.
Thank you for your help. The course I am taking has a habit of blurting out terms and definitions without really explaining them.
As a bonus, do you have any resources that contain an example by example walkthrough of Bayesian data analysis and/bayesian mathematics. Like a study guide or reference book?
(a) as usual, calculate the likelihood (multiplying the given density, get its log-derivative with respect to $\theta$, set it =0 and solve with respect to $\theta$. Then substitute the observed data in the estimator you found. [$\hat{\theta}_{ML}=0.5$]
(b) they said you that it is an exponential with mean 4 (a kind of Gamma):
$$\pi(\theta)=\frac{1}{4}e^{-\frac{\theta}{4}}$$
To verify if it is a conjugate prior you can calculate the posterior
$$\pi(\theta|\mathbf{x})\propto \pi(\theta)p(\mathbf{x}|\theta)$$
and verify if this posterior is still a Gamma distribution (it is!).
(c) One possible Bayes estimator is the Posterior Mean. [$\hat{\theta}_{MMSE}=0.64$]
(d) At this point you already have derived the posterior density, you have only to solve the integral set at 95%. $[0.17;1.40]$
To avoid the calculation of the integrals (that is easy but boring), remember that if $X\sim Gamma(a;b)$ then $2bX\sim \chi_{(2a)}^2$.
I wait your progresses... if you have problems I am here (the exercise is very easy)
Please, use MathJax when answering
Concluding: this is a very nice book