# What does "parameterized by" mean?

Sometimes I have seen likelihood written as $$L(\mu,\sigma |y)$$ and sometimes as $$L(y|\mu,\sigma)$$.

I have been told that in the first case it means that there is a pre-assumed model depicting the probability density of $$\mu$$ and $$\sigma$$ parameterized by $$y$$. In the second case it means the pre-assumed model depicts the probability density of $$y$$, parameterized by $$\mu$$ and $$\sigma$$

I have seen the $$|$$ symbol used before in Bayes theory and read it as "given that"

I am from a programming background (C#) and tend to think of parameters as inputs to a function.

• I am not aware of the latter usage of likelihood. But let others comment on this. Oct 10, 2022 at 19:21
• $L(y \mid \mu,\sigma)$ looks unconventional as a likelihood, and you would more usually see a parametrised density as $f(y \mid \mu,\sigma)$ or $p(y \mid \mu,\sigma)$ Oct 10, 2022 at 20:01
• Related question $L(\theta;x)=f(x;\theta)$ vs. $L(\theta;x)\propto f(x|\theta)$ Oct 11, 2022 at 17:17

“Parametrized by” means that the function $$f$$ of $$x$$ has additional parameters $$\Theta$$, so we can write it as $$f(x;\Theta)$$. In such a case, we call the function by $$x$$ given a fixed value of $$\Theta$$.

Likelihood has two meanings, the traditional on, and Bayesian. Traditionally, likelihood is written as

$$L(\theta|x) \propto P(x|\theta)$$

The vertical bar $$|$$ is used on the right-hand side to denote conditional probability and is a slight abuse of notation on the left-hand side. People write it as $$L(\theta|x)$$ to show that we keep the data $$x$$ fixed, but we evaluate the function for different parameters $$\theta$$. In a Bayesian setting, you usually would not see the left-hand side notation, just the right-hand side will be called the likelihood (here you might have seen $$L$$ used instead of $$P$$). See Wikipedia entry on likelihood seems ambiguous for more discussion.

• I suspect many Bayesians are happy with $L(\theta \mid x)$ though some would say $L(\theta \mid x) \propto P(x \mid \theta)$. A key point of such a likelihood is that it is function of $\theta$ but does not need to sum or integrate to $1$ over $\theta$. Oct 10, 2022 at 19:22
• Hm. That is right, Tim. I never encountered likelihood to be written in that expression. Oct 10, 2022 at 19:23
• @Henry the multiplicative constant. That sums it up. Oct 10, 2022 at 19:23
• @User1865345 If I flip a biased coin with parameter $p$ of heads and see $H,T,H,T,H,H$ then the likelihood for a particular value of $p$ is the same as if I record $4$ heads and $2$ tails in total. It does not matter whether I write this as $p^4(1-p)^2$ or $15 p^4(1-p)^2$ Oct 10, 2022 at 19:35
• Exactly @Henry. I agreed with your initial as well this comment. Nothing of any disagreement here. I just mentioned the multiplicative constant which doesn't matter. Oct 10, 2022 at 19:37

What is parameter? What is a parametric model?

Definition $$1.$$ Let $$(\Omega,\mathfrak F)$$ be a probability space. The set of probability measures $$\{ \mathbb P_\theta:~{\boldsymbol \theta\in\Theta}\}$$ indexed by a parameter $$\boldsymbol\theta$$ is a parametric family if and only if $$\Theta\in\mathbb R^n,~n\in\mathbb Z^{>0}$$, and each probability measure is known when $$\boldsymbol\theta$$ is known. Here $$\Theta$$ is parameter space.

Remark $$1.$$ A parametric model assumes the population comes from a parametric family.

Example $$1.$$ A parametric family of $$n\in\mathbb Z^{>0}$$ dimensional normal distributions indexed by $$( \boldsymbol\mu, \mathbf \Sigma)$$ is

$$\{\mathcal N_n(\boldsymbol \mu,\mathbf \Sigma): \boldsymbol\mu\in\mathbb R^n,\mathbf \Sigma\in\mathcal M_n\}.\tag 1$$

So, basically the probability measure is labeled or indexed by parameter(s) and the primary objective of inference is to draw information about the parameter $$\theta$$ in order to know the probability measure (if it is $$\sigma$$-finite, then the set is identified by the densities; so the problem is to infer about the parameter to draw information about the associated pdf).

$$\bullet$$ $$\mathcal L(\boldsymbol\theta|\mathbf y)$$ denotes, given that $$\mathbf Y= \mathbf y$$ is realised or observed, the likelihood of parameter $$\boldsymbol\theta.$$

$$\rm [I]$$ Mathematical Statistics, Jun Shao, Springer Science$$+$$Business Media, $$2003,$$ section $$2.1.2,$$ p. $$94.$$
$$\rm [II]$$ Testing Statistical Hypotheses, E. L. Lehmann, Joseph P. Romano, Springer Science$$+$$Business Media, $$2005,$$ section $$1.1,$$ p. $$3.$$
$$\rm [III]$$ Statistical Inference, George Casella, Roger L. Berger, Wadsworth, $$2002,$$ section $$6.3,$$ p. $$290.$$