I use to think that any marginal distribution on a random variable can be called as prior probability distribution.

Is it true?

Or is it false due to the reason that the random variable under consideration may not be a latent random variable and the term prior should be used strictly with the marginal distributions over a latent random variable only?


2 Answers 2


The term prior (as well as posterior) is usually reserved for distributions defined in a Bayesian framework on objects that are not considered as random variables by other inferential approaches, namely parameters. Latent variable models are most often defined outside the Bayesian/non-Bayesian dichotomy and the distribution of the latent variable generally depends on parameters, so is also conditional on the realisation of these parameters in a Bayesian framework. Since it is a matter of terminology, there is no true or false (or right or wrong) to call a marginal distribution a prior, but this may prove confusing to Bayesians and non-Bayesians alike.


Before answering your question, let's first explain some basic Bayesian mindset.

In Bayesian statistics, everything is a random variable, the only difference between these random variables is whether they are observed or hidden. Say for example if you believe $X$ follows a distribution defined by $\theta$, denote $$ X \sim P(X|\theta) $$ Where $\theta$ is the parameter of the distribution, from Bayesian perspective it's also a random variable. Usually in this case random variable $X$ is observed and $\theta$ is not, and you want to infer/learn/esitmate $\theta$ based on your observations. In such situations there's no matter of "prior", "marginal" or "posterior"

The term "prior", "marginal" or "posterior" matters when you believe $\theta$ follows some other distribution $$ \theta \sim P(\theta|\gamma) $$ Then we call this "other distribution" the prior, more specifically it's the piror distribution for $\theta$. Among all three random variables $X$, $\theta$ and $\gamma$, usually $X$ and $\gamma$ are observed, $\theta$ is not, and you want to estimate $\theta$ based on the observed $X$ and $\gamma$. So yes the term "prior" is usually on hidden random variables, of course you can believe there's a prior distribution for $\theta$ even when it is observed, but usually nobody do so(why would anyone esitimate something that is already observed?). And, if you can't observe $\gamma$, you can even assume $\gamma$ follows a distribution defined by another random variable $\eta$, then $P(\gamma | \eta)$ will be the prior for $\gamma$. Hope this answers your question regarding to "prior".

Now let's talk about "marginal". In previous example people usually interested in the distribution of $X$ (while $\theta$ is hidden), given $\gamma$, the distribution $$ X \sim P(X|\gamma) $$ is called the "marginal distribution". The term "marginal" came from the fact that $P(X|\gamma)$ is acquired by marginalizing out $\theta$ from the joint distribution: $$ p(X|\gamma) = \int_\theta p(X|\theta)p(\theta|\gamma) $$


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