Equality in conditional probability

[Avatrin]

Hi

In Dudas Pattern Classification, he Writes that $P(x,\theta|D)$ can always be written as $P(x|\theta,D)P(\theta|D)$. However, I cannot find any justification for this. So, why are these Equal?

 

[Mark44]

Hi

In Dudas Pattern Classification, he Writes that $P(x,\theta|D)$ can always be written as $P(x|\theta,D)P(\theta|D)$. However, I cannot find any justification for this. So, why are these Equal?

How are $P(x,\theta|D)$ and $P(x|\theta,D)$ defined? I took a lot of classes in probability and mathematical statistics, but it has been many years ago. I don't recall seeing this notation.

 

[Avatrin]

How are $P(x,\theta|D)$ and $P(x|\theta,D)$ defined? I took a lot of classes in probability and mathematical statistics, but it has been many years ago. I don't recall seeing this notation.

$P(x,\theta|D)$ is the probability of $x$ and $\theta$ given $D$ which is our sample set. $P(x|\theta,D)$ is the probability of $x$ given $D$ and $\theta$. $x$ is a random variable, and $\theta$ is a parameter which we are estimating by considering it to be a random variable.

 

[EngWiPy]

Start by writing $P(x,\theta/D)$ using Bayes rule.