Posterior Density vs. Posterior Distribution

[gajohnson]

Homework Statement

Explain the difference between posterior density and posterior distribution

Homework Equations

NA

The Attempt at a Solution

This isn't a homework question per se, but it will help with something I'm working on. Anyway, my textbook defines posterior distribution as:

Likelihood * Prior Density/ ∫Likelihood X Prior Density

However, it goes on to talk about posterior density without explicitly discussing the differences, and I can't tell if those two terms are interchangeable or not. For instance, one question asks me to find the posterior density of something, and another the posterior distribution. Any help with these concepts would be greatly appreciated!

 

[Ray Vickson]

Homework Statement

Explain the difference between posterior density and posterior distribution

Homework Equations

NA

The Attempt at a Solution

This isn't a homework question per se, but it will help with something I'm working on. Anyway, my textbook defines posterior distribution as:

Likelihood * Prior Density/ ∫Likelihood X Prior Density

However, it goes on to talk about posterior density without explicitly discussing the differences, and I can't tell if those two terms are interchangeable or not. For instance, one question asks me to find the posterior density of something, and another the posterior distribution. Any help with these concepts would be greatly appreciated!

Sometimes (not too often) the words "density" and "distribution" are used almost interchangeably although *usually* the word distribution is used more as a descriptor of "type"---as, for example, normal distribution or gamma distribution or Poisson distribution. Nowadays, the term 'distribution function' is being used increasingly in place of the term 'cumulative distribution function'.

The thing you wrote above looks to me like a posterior *density* function, assuming you are speaking of a continuous random variable.

 

[gajohnson]

Sometimes (not too often) the words "density" and "distribution" are used almost interchangeably although *usually* the word distribution is used more as a descriptor of "type"---as, for example, normal distribution or gamma distribution or Poisson distribution. Nowadays, the term 'distribution function' is being used increasingly in place of the term 'cumulative distribution function'.

The thing you wrote above looks to me like a posterior *density* function, assuming you are speaking of a continuous random variable.

Thanks! I do get the impression that my book is using them interchangeably here, and I am talking about a continuous random variable. If they weren't being used interchangeably, what would the difference be?

In addition, maybe you can answer another qualitative question for me. In finding the mode of a posterior distribution (the MAP), why do I not need to consider the denominator of the equation that I mentioned earlier? I understand that, in practice, I only need to maximize the numerator, but I'm not exactly sure why. Thank you!