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In section 6.2, in the second paragraph of p. 335 (image below) of "Probability and statistical inference 7e" by Hogg and Tanis states:

perhaps it is known that $f(x;\Theta)=(1/\Theta)e^{x/\Theta}$

where $x$ is data and $\Theta$ is a parameter.

What does "$;$" mean in this context, as opposed to "$,$" or "$|$", all three are used in different ways in the same textbook ("," and ";" are used on the same page, $|$ in the standard statement of Bayes' Theorem)?

I think I understand "," and "|" and I read $f(x,\Theta)$ as "function of data and parameters" and $f(x|\Theta)$ as "function of the data given parameters".

Here is a scan of the page for more context:

enter image description here

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f(x;θ) is the same as f(x|θ), simply meaning that θ is a fixed parameter and the function f is a function of x. f(x,Θ), OTOH, is an element of a family (set) of functions, where the elements are indexed by Θ. A subtle distinction, perhaps, but an important one, esp. when it comes time to estimate an unknown parameter θ on the basis of known data x; at that time, θ varies and x is fixed, resulting in the "likelihood function". Usage of "|" is more common among statisticians, ";" among mathematicians. – jbowman Jun 20 '12 at 19:20
Yes jbowman is correct. We sometimes call it the density of X given Θ. – Michael Chernick Jun 20 '12 at 19:51
@jbowman why not post that as an answer? My only question is - why would they use both, but I assume that it has something to do with the context (the "|" is used with "P" and the ";" with "$f$"). – Abe Jun 21 '12 at 15:11
Good thinking, Abe; that's probably it. $f$ is more generic, I suppose. – jbowman Jun 21 '12 at 15:13

2 Answers

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f(x;θ) is the density of the random variable X at the point x, with θ being the parameter of the distribution. f(x,θ) is the joint density of X and Θ at the point (x,θ) and only makes sense if Θ is a random variable. f(x|θ) is the conditional distribution of X given Θ, and again, only makes sense if Θ is a random variable. This will become much clearer when you get further into the book and look at Bayesian analysis.

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Uhhhh... $f(x|\theta)$ is the conditional distribution of $x$ given $\theta$ makes perfect sense even if $\theta$ is not a random variable. It's pretty much standard notation in classical statistics, where $\theta$ is not a random variable. – jbowman Jun 21 '12 at 17:10
Uhhhh....if you interpret that to mean that P[Θ=θ]=1 (left Θ is a random variable, right θ is a constant) then I agree. Otherwise I do not...for what then would P[Θ=θ] mean in the denominator of the definition of conditional distribution? – PeterR Jun 21 '12 at 17:30
Denominator? I can write $x \sim f(x | \mu, \sigma)$ where $f$ is a Normal distribution without reference to Bayes' Rule. $\mu$ and $\sigma$ are fixed. Others do too, for example, ll.mit.edu/mission/communications/ist/publications/…. – jbowman Jun 21 '12 at 18:28
jbowman, so what is the definition of your f(x|μ,σ) as a conditional density when μ and σ are fixed numbers (i.e. not random variables)? – PeterR Jun 21 '12 at 18:41
$x$ is distributed according to, e.g., a Normal law with mean $\mu$ and standard deviation $\sigma$. When, for example, $\mu=0, \sigma=1$, $f(0) = 0.3989...$. When $\mu=1, \sigma=1$, $f(0) = 0.2419...$. The value of $f(x)$ is conditional upon the values of $\mu$ and $\sigma$. I think, BTW, we are using the word "conditional" in two slightly different ways; you are limiting it to "conditional upon some random event occurring", and I am using it to mean that or just "given", as in "$f(x)$ given (specific values of) $\mu$ and $\sigma$". – jbowman Jun 21 '12 at 19:24
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up vote 3 down vote

Reposted from comments in response to suggestion from OP:

$f(x;\theta)$ is the same as $f(x|\theta)$, simply meaning that $\theta$ is a fixed parameter and the function $f$ is a function of $x$. $f(x,\Theta)$, OTOH, is an element of a family (set) of functions, where the elements are indexed by $\Theta$. A subtle distinction, perhaps, but an important one, esp. when it comes time to estimate an unknown parameter $\theta$ on the basis of known data $x$; at that time, $\theta$ varies and $x$ is fixed, resulting in the "likelihood function". Usage of "|" is more common among statisticians, ";" among mathematicians.

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