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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

marked as duplicate by whuber Sep 2 '13 at 15:19

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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
up vote 17 down vote accepted

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

  • 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

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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    How is $f(x;θ)$ spoken verbally? Do you say " f of x given θ"? – stackoverflowuser2010 Oct 23 '14 at 22:09
  • @stackoverflowuser2010 - yes, exactly so. – jbowman Oct 27 '14 at 20:47
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    I found in some Coursera videos that Stanford professor Andrew Ng verbalizes the semicolon as "parameterized by." See: class.coursera.org/ml-005/lecture/34 . So the example would be spoken as "f of x parameterized by theta". – stackoverflowuser2010 Nov 10 '14 at 2:50
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    Saying "given" or "conditional" is very different (in general) from "parameterized." I'd hate if someone saw this and thought the two were equivalent. Saying "parameterized" is only appropriate when the quantity being conditioned on is a parameter indexing the pdf of the variable in the first term. For two variables (e.g., f(x;y)), using that term would be wrong. – ATJ Jun 17 '16 at 17:07
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    @MikeWilliamson - Sure, pick a notation where you know what everything means and stick with it! That way when you go back to something you did earlier, like 4 hours earlier in my experience, you don't have to figure out what you meant when you used that "|". I agree, it is annoying, but after a while you just observe the first use of the notation and remember it for the rest of the paper / book; the distinctions are not usually what's important, anyway. – jbowman Aug 18 at 18:42

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