In my textbook, I saw the following Log Likelihood Function written:
$$l(\tau, \gamma, \sigma^2; y) = l(\tau, \gamma, \sigma^2; y_1 | y_2) + l(\gamma, \sigma^2; y_2)$$
Where:
- $y$ is a multivariate probability distribution made of distributions $y_1$ and $y_2$
- Having parameters $\tau$, $\gamma$ and $\sigma^2$
My Question: Is there some special probability rule which allows the above likelihood to be written as:
$$l(\tau, \gamma, \sigma^2; y) = l(\tau, \gamma, \sigma^2; y_1 | y_2) + l(\gamma, \sigma^2; y_2)$$
Is this because of the rules of probability:
If $Y$ ~ $(Y_1, Y_2)$, then $P(Y) = P(Y_1 | Y_2) * P(Y_2)$
Thus, when the logarithm is taken, we get an expression in the above form.
Is my analysis correct?
Thanks!