Bayesian Parameter Estimation Doubt

Question

I was going through a pattern recognition book and in the chapter of Bayesian Parameter Estimation I came across this formula. I cannot understand how the 2nd line is derived from the first line. Please help me any one...D is the sample group containing n number of samples of the random variable x chosen from a pool.

Answer 1

Consider the denominator first. It is: $$ \int p(\mathbf{D}|\mu)p(\mu)d\mu=c $$ Since $\mu$ is integrated out, it's simply a constant with respect to $\mu$. Let $c^{-1}=\alpha$.

Let also $\mathbf{D}$ be a vector such that $\mathbf{D}=\begin{pmatrix}x_1,&x_2, &\dots, &x_n\end{pmatrix}$. Under the assumption that the samples are independent, we can write the density of the vector $\mathbf{D}$ (conditional on $\mu$) as:

$$ p(\mathbf{D}|\mu)=p(x_1, x_2, \dots, x_n|\mu)=p(x_1|\mu)p(x_2|\mu)\cdots p(x_n|\mu)=\prod_{k=1}^np(x_k|\mu) $$

Putting all of this together:

$$ \frac{p(\mathbf{D}|\mu)p(\mu)}{\int p(\mathbf{D}|\mu)p(\mu)d\mu}=\frac{1}{c}p(\mathbf{D}|\mu)p(\mu)=\alpha \times p(\mathbf{D}|\mu)p(\mu)=\alpha\left(\prod_{k=1}^np(x_k|\mu)\right)p(\mu)=\alpha\prod_{k=1}^np(x_k|\mu)p(\mu) $$

as desired.