# Confusion in Posterior Probability Calculation

I know posterior probability as,

$P(\theta|x)= [(P(x|\theta)*(P(\theta))/(P(x))]$,

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?

Regards, Subhabrata Banerjee.

• The term $P(y\mid x;\theta)$ occurs nowhere in the linked to article. Where did you encounter it? – Graham Kemp Jun 10 '14 at 22:36

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.
• @SUBHABRATA From Andrew Ng's notes, he first has a notion of $p(x,z;\theta)$ and $p(x;\theta)$ where $\theta$ contains all parameters governing the $x$ and $z$ (i.e., mean/variance of Gaussian, probability of success of Bernoulli, etc.). You can see in the calculation in the center of page 4 what he means by $p(z \mid x;\theta)$. In essence, you can think of the semicolon as a "conditional" bar (i.e., $p(z\mid x;\theta) \simeq p(z\mid x,\theta)$ and $p(x;\theta) \simeq p(x\mid \theta)$); the use of the semicolon just clarifies that $\theta$ is a parameter and not a random variable. – angryavian Jun 11 '14 at 15:08