Geometric Prior Distribution

Question

What is an appropriate posterior distribution given a geometric prior distribution? I think a Bernoulli posterior would be appropriate, but am unsure if this is correct. Can anyone verify my idea, or give an alternative posterior?

Thank you!

Answer 1

You do not choose the posterior distribution. A statistical model is (in simple words) a mathematical object which takes as input :

• observations $\mathbf{y} \in \mathcal{Y}$,
• parameters $\boldsymbol\theta \in \Theta$.

In a Bayesian framework, you consider the parameters $\boldsymbol\theta$ as random variable. The probability distribution $p$ of $\boldsymbol\theta$ is called prior distribution ($\boldsymbol\theta \sim p(\boldsymbol\theta)$) and is choosen by the statistician. The prior somehow inputs into the model your knowledge about the parameters, how certain you are that the "true" parameters are in a specific region of the parameters space.

Given a statistical model and a prior distribution, you can define the posterior distribution of your parameters given the observations. This probability distribution is $p(\boldsymbol\theta \mid \mathbf{y})$, the conditional distribution of $\boldsymbol\theta$ given the observations $\mathbf{y}$. Using the Bayes rule, you have : $$ p(\boldsymbol\theta \mid \mathbf{y}) \propto p(\mathbf{y} \mid \boldsymbol\theta)p(\boldsymbol\theta) \tag{$\star$}$$ where $p(\mathbf{y} \mid \boldsymbol\theta)$ is the likelihood of your observations $\mathbf{y}$. The likelihood function $\boldsymbol\theta \in \Theta \mapsto p(\mathbf{y} \mid \boldsymbol\theta)$ is given by your statistical model. From $(\star$), you see that the posterior distribution is not something you choose. It is the result of "statistical model + prior".