I know posterior probability as,
$P(\theta|x)= [(P(x|\theta)*(P(\theta))/(P(x))]$,
as given in http://en.wikipedia.org/wiki/Posterior_probability
I am slightly confused with the term $P(y|x;\theta)$,
should I interpret it as, posterior of y given x on the parameter of $\theta$, and how may I interpret
$P(y|x;\theta)=[(P(x|y)*(P(y))/(P(x))]$ under the parameter theta,
or anything else?
If any one of the esteemed members may kindly suggest?
Thanks in Advance,
Regards, Subhabrata Banerjee.
I think you will need to provide more context for where you found the second term $P(y \mid x ; \theta)$, as far as I know, it has not standard definition and its meaning is usually inferred from the context.
My guess is that $\theta$ governs the distribution of $y$. As an example, maybe $y$ is Bernoulli with parameter $\theta$, so $P(y)$ (which might be more properly written $P(y;\theta)$) is defined by $P(y=1):=\theta$ and $P(y=0):=1-\theta$. Again, this is just my guess; you might get better answers if you provide more context. Note that $\theta$ does not appear in your second equation explicitly.
Also, note that $\theta$, $x$, and $y$ play different roles in your two equations. You just need to permute them around, but I wanted to mention this in case you did not notice.