Posterior mode, posterior mean and posterior variance of a posterior distribution of dirichlet form What is the significance of finding the posterior  mean, posterior mode and posterior variance in Dirichlet - multinomial conjugate pair Bayesian estimation?   Are all of them equally important while estimating a posterior probability? Which should be chosen out of the three?
 A: I will try to explain this in general not focusing on the Dirichlet-Multinomial case and based on some Decision theory notions. The Bayes estimator $\hat{\theta}$ is the rule that minimizes the expected posterior loss 
$$
\mathbb{E}_{\theta|x}[L(\theta,\hat{\theta})]= \int_{-\infty}^{\infty}L(\theta,\hat{\theta}) \pi(\theta|x)d\theta
$$
where $\pi(\theta|x)$ is the posterior distribution and $L(\theta,\hat{\theta})$ is the loss function. The loss function measures how much we "pay" when we choose an "action" $\hat{\theta}$ and the true value is $\theta.$ For example the quadratic loss function is given by
$$
L(\theta,\hat{\theta}) = (\hat{\theta}-\theta)^2.
$$ 
So if you choose as a point estimator the posterior mean $\hat{\theta}=\mathbb{E}_{\theta|x}\theta$ this minimizes the expected posterior loss when the quadratic loss function is used. 
The median of the posterior distribution minimizes the expected posterior loss when 
$$
L(\theta,\hat{\theta})=c|\hat{\theta}-\theta|, \quad c>0,
$$
the absolute loss function. 
The choice of the estimator depends on the application. For example if one has a multimodal posterior distribution it is not reasonable to assume that the posterior mean is an appropriate estimate and should take the posterior mode. That is the assumption is that the $0-1$ loss function has been used
$$
L(\theta,\hat{\theta}) = \begin{cases} 0,\quad |\hat{\theta}-\theta|<0\\
1,\quad |\hat{\theta}-\theta|\geq 0\end{cases}.
$$
