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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?

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Conditional probability has precise mathematical meaning: \begin{equation} p(\mu|\sigma) = \frac{p(\mu, \sigma)}{p(\sigma)} \end{equation} 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 $i$).

A bit of search led me to this question, where the answer seems to concur with mine: what is the semicolon notation in joint probability?

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  • $\begingroup$ 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 $p(b_i \mid T_i, \delta_i, y_i, a_i; \theta)$ $\endgroup$
    – JLee
    Commented Nov 20, 2019 at 15:14
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    $\begingroup$ What I wanted to stress is that this is not a strict mathematical notation, but the author's subjective choice to improve readability. $\endgroup$
    – Roger V.
    Commented Nov 20, 2019 at 15:28

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