# Notation for conditional density

Are $$p(\mu \mid \sigma)$$ and $$p(\mu ; \sigma)$$ equivalent?

I've seen the notation $$p(b_i \mid T_i, \delta_i, y_i ; \theta)$$ used to represent the posterior distribution for $$b_i$$. I am assuming that this is the posterior distribution for $$b_i$$ conditional on $$\theta$$ being fixed.

In the past I have written this as $$p(b_i \mid T_i, \delta_i, y_i , \theta)$$. Are both okay?

Conditional probability has precise mathematical meaning: $$$$p(\mu|\sigma) = \frac{p(\mu, \sigma)}{p(\sigma)}$$$$ Semicolon in $$p(\mu; \sigma)$$ likely means the same as the comma in $$p(\mu, \sigma)$$ - just a matter of notation. In fact, this is certainly the case in your example, where $$T_i, \delta_i, y_i$$ and $$\theta$$ appear as the conditioning parameters in both expressions, although semicolon is likely to underscore that the nature of parameter $$\theta$$ is somehow different from the rest (probably this parameter is the same for all $$i$$).
• In the example given, $T_i, \delta_i$ and $y_i$ are data, and $\theta$ is a common parameter. I was thinking that a ";" was used to separate a parameter from data in the conditioning; but your comment on differentiating something common to all $i$ from something unique to each $i$ is interesting. I have never seen an example of where this has been used, but it would be interesting to see if, for data $T_i, \delta_i$ and $y_i$, parameters $\theta$ and $a_i$, would the conditioning be $p(b_i \mid T_i, \delta_i, y_i; \theta, a_i)$ or $p(b_i \mid T_i, \delta_i, y_i, a_i; \theta)$