Bayes point estimate using mode of a Gamma posterior distribution Let's say the posterior distribution of $\theta$ is Gamma with $$\alpha = 40, \qquad \beta = \frac{1}{0.5 + \sum_{i = 1}^{10} X

Bayes point estimate using mode of a Gamma posterior distribution Let's say the posterior distribution of $\theta$ is Gamma with $$\alpha = 40, \qquad \beta = \frac{1}{0.5 + \sum_{i = 1}^{10} X_i}$$ > What is the Bayes point estimate using the mode of the posterior distribution?

Wikipedia lists the mode of a gamma distribution of the mode to be $\frac{\alpha-1}{\beta}$, so your point estimate would be $39(0.5+\sum_{i=1}^{10}X_i)$.

You can also find the mode of a gamma distribution by computing the derivative of $x^{\alpha-1}e^{-\beta x}$ which is $x^{\alpha-2}e^{-\beta x}(\alpha-1-\beta x)$, setting it to zero and solving (then checking that the result is a maximum).

Bayes point estimate using mode of a Gamma posterior distribution Let's say the posterior distribution of $\theta$ is Gamma with $$\alpha = 40, \qquad \beta = \frac{1}{0.5 + \sum_{i = 1}^{10} X_i}$$ > What is the Bayes point estimate using the **mode** of the posterior distribution?

Bayes point estimate using mode of a Gamma posterior distribution Let's say the posterior distribution of $\theta$ is Gamma with $$\alpha = 40, \qquad \beta = \frac{1}{0.5 + \sum_{i = 1}^{10} X_i}$$ > What is the Bayes point estimate using the mode of the posterior distribution?