# Show posterior mean can be written as a weighted average of the prior mean and MLE

Suppose $$Y_1, \dots Y_n$$ are exponentially distributed: $$Y_i | \lambda \sim Exp(\lambda)$$. Find the conjugate prior for $$\lambda$$, and the corresponding posterior distribution. Show that the posterior mean for the failure rate $$\lambda$$ can be written as a weighted average of the prior mean and the maximum likelihood estimator, $$\hat{\lambda}=\bar{y}^{-1}$$.

Given that the joint pdf as follow:-

$$P(Y_i | \lambda)=\lambda^{n}e^{-\lambda \sum_{i=1}^{n} Y_i}$$

Then the likelihood function is

$$L (\lambda | Y_i)\propto \lambda^{n}e^{-\lambda \sum_{i=1}^{n}Y_i}$$

which give Gamma distribution with $$\alpha = n+1$$ and $$\beta =\sum_{i=1}^{n} Y_i$$.

Thus, with the prior density

$$P(\lambda)\propto \lambda^{a-1} e^{- \lambda b}$$

gives the posterior distribution as follow:-

$$P(\lambda | Y_i) \propto \lambda^{n+a-1} e^{-\lambda (\sum_{i=1}^{n}Y_i + b)}$$

and resulting in Gamma distribution with $$\alpha = n+a$$ and $$\beta = \sum_{i=1}^{n} Y_i + b$$.

My problem now is how can I show the posterior mean for the failure rate λ can be written as a weighted average of the prior mean and the maximum likelihood estimator?

As far I get as the following:-

$$E[\lambda | Y] = \frac{\alpha}{\beta} = \frac{n+a}{\sum_{i=1}^{n} Y_i + b}$$

Can anyone help me to show how the posterior mean above can be written as a weighted average of the prior mean and the maximum likelihood estimator, $$\hat{\lambda}=\bar{y}^{-1}$$?

• Your $P(\lambda|Y_i)$ corresponds to a gamma distribution with $\alpha=n+a$, not $n+a+1$ – Moss Murderer Nov 14 '18 at 16:23
• @MossMurderer I've changed it. Thank you. I thought it also combination with the likelihood which give Gamma distribution for $\alpha = n+1$. – Haney Zaf Nov 14 '18 at 16:52

Continuing from where you left, \begin{align} \bar\lambda_{post}^{-1} &= \frac{\sum_{i}y_i+b}{n+a} \\ &= \frac{n}{n+a}\frac{\sum_{i}y_i}{n} + \frac{a}{n+a}\frac{b}{a} \\ &= \rho\lambda_{mle}^{-1} + (1-\rho)\bar\lambda_{prior}^{-1} \\ \end{align}
where $$\rho = \frac{n}{n+a}$$ and $$\lambda_{mle}$$, $$\bar\lambda_{prior}$$ and $$\bar\lambda_{post}$$ denote the maximum likelihood estimator, prior mean and posterior mean respectively.