Dropping the normalization constant in Bayesian inference Could anyone please explain to me (or provide a situation) where one would drop the normalising constant (or marginal probability term) from Bayes rule when performing Bayesian Inference?
New to Bayesian theory so any help appreciated.
Thanks in advance.
 A: I can think of a couple situations where this would be OK:


*

*You want to get the maximum a-posteriori (MAP) estimate of your parameter. In that case, the normalizing constant will not affect the value of the estimate.

*You are using MCMC or acceptance rejection to estimate the posterior.


For the first case, note that if we are performing inference on $\theta$ using data $\mathbf{x}$, then:
$$P(\theta|\mathbf{x}) \propto P(\theta)P(\mathbf{x}|\theta)$$
The MAP estimate of $\theta$ is the value of $\theta$ that maximizes $P(\theta|\mathbf{x})$. However, as you may remember from calculus, if $x^*$ maximizes a function $f(x)$, then it also maximizes $cf(x)$, where $c>0$. In this case, 
$$f:=f(\theta)=P(\theta|\mathbf{x})=c P(\theta)P(\mathbf{x}|\theta)$$
for some normalizing constance $c$. Therefore, if you find $\theta^*$ which maximizes $P(\theta)P(\mathbf{x}|\theta)$, you will have also found the value that would have maximized the actual posterior.
For the second example, I will simply state that these methods were designed to avoid the ugly integration in the denominator using advanced numerical approaches. I won't go int MCMC, but rejection methods work by creating an "envelope" above $P(\theta)P(\mathbf{x}|\theta)$ that can be used to generate draws from the target distribution. Here are some details.
