Derive Posterior Distribution from Prior Distribution

Question

Suppose that a random vector $(X_1,...,X_k)$ has a multinomial distribution $MULT(n; p_1,...,p_k)$.

Now consider the Dirichlet prior for $(p_1,...,p_k)$:

$$g(p_1,...,p_k)=\frac{\Gamma(\beta_1+...+\beta_k)}{\Gamma(\beta_1)\times...\times\Gamma(\beta_k)}\prod_{j=1}^kp_j^{\beta_j-1}$$ where $\beta_j>0$, $\sum_{j=1}^kp_j=1$ and $\Gamma(\alpha)$ is the gamma function.

Under this prior, derive the posterior distribution of $(p_1,...,p_k)$.

I'm sorry, but I'm at a total loss here. I'm very overwhelmed by this and don't even know how to get started. Would you please point me in the right direction? Thanks.

Answer 1

The posterior distribution can be found by Bayes rule: the postrior pdf of $\vec p=(p_1,...,p_k)$ given one sample $\vec X=(X_1,...,X_k)$ from multinomial distribution $MULT(n; p_1,...,p_k)$ is $$ f(\vec p|\vec X=\vec x) \propto g(\vec p)\cdot\mathbb P(\vec X=\vec x|\vec p) $$ Here the sign $\propto$ means ''proportional to'', and the constant in r.h.s. is chosen to make the r.h.s. a probability density function. This constant should depend on $n$, $x_1,\ldots,x_k$ and $\beta_1,\ldots,\beta_k$. $$ f(\vec p|\vec X=\vec x) \propto \frac{\Gamma(\beta_1+...+\beta_k)}{\Gamma(\beta_1)\dots\Gamma(\beta_k)}\prod_{j=1}^kp_j^{\beta_j-1} \cdot \dfrac{n!}{x_1!\dots x_k!}p_1^{x_1}\dots p_k^{x_k}$$ $$ \propto \prod_{j=1}^kp_j^{x_j+\beta_j-1}, \text{ where }p_j>0 \text{ and } \sum_{j=1}^k p_j=1. $$ As you can see, the posterior distribution is again Dirichlet distribution.