Why are density functions sometimes written with conditional notation?

I keep seeing density functions that don’t explicitly arise from conditioning written with the conditional sign:
For example for the density of the Gaussian N(μ,σ) why write:
f(x|μ,σ)=12πσ2exp(xμ)22σ2

instead of

f(x)=12πσ2exp(xμ)22σ2

Is this done purely to be explicit as to what the parameter values are or(what I’m hoping for) is there some meaning related to conditional probability?

Answer

  • In a Bayesian context, the parameters are random variables, so in that context the density is actually the conditional density of X(μ,σ). In that setting, the notation is very natural.
  • Outside of a Bayesian context, it is just a way to make it clear that the density depends (here I am using this word colloquially, not probabilistically) on the parameters. Some people use fμ,σ(x) or f(x;μ,σ) to the same effect.
  • This latter point can be important in the context of likelihood functions. A likelihood function is a function of the parameters θ, given some data x. The likelihood is sometimes written as L(θx) or L(θ;x), or sometimes as L(θ) when the data x is understood to be given. What is confusing is that in the case of a continuous distribution, the likelihood function is defined as the value of the density corresponding to the parameter θ, evaluated at the data x, i.e. L(θ;x):=fθ(x). Writing L(θ;x)=f(x) would be confusing, since the left-hand side is a function of θ, while the right-hand side ostensibly does not appear to depend on θ. While I prefer writing L(θ;x):=fθ(x), some might write L(θ;x):=f(xθ).
  • I have not really seen much consistency in notation across different authors, although someone more well-read than I can correct me if I am wrong.

Attribution
Source : Link , Question Author : stochasticmrfox , Answer Author : angryavian

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