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.