3

I am not a statistician or mathematician but am trying to learn.

My question: In Bayes Theorem, $p(C|X)=p(X|C)p(C)/p(X)$, what are the English terms for $p(X|C)$ and $p(C)/p(X)$?

In other words, is $p(C)$ the prior probability? What is the comparable word for $p(C)/p(X)$?

Is the likelihood part of the right side of this equation? If so, which part?

Sycorax
  • 76,417
  • 20
  • 189
  • 313
johnhidley
  • 81
  • 3
  • Thank you for the discussion below. It is helpful. If the liklihood is p(X|C). p(C), does 1/p(x) have a term of art? I read, in Wikipedia I think, that it may have something to do with normalizing. Is that so? If so, what is being normalized? – johnhidley Apr 23 '14 at 23:18
  • I misunderstood what you wrote. I think this is what you actually said: p(X|C) is the likelihood, and p(C) is the prior. If so, does 1/p(x) have a term of art? I read, in Wikipedia I think, that it may have something to do with normalizing. Is that so? What is being normalized? – johnhidley Apr 23 '14 at 23:25
  • $1/p(X)$ scales the quantity $p(X|C)p(C)$ so that it is a probability on the [0,1] interval. – Sycorax Apr 24 '14 at 03:28

3 Answers3

3

Assuming $X$ is your data and $C$ is the parameter, the likelihood is $p(X|C)$. $p(C)$ is the prior. I am not aware of a particular word for $p(C)/p(X)$.

Sycorax
  • 76,417
  • 20
  • 189
  • 313
  • The posterior is *proportional* to the likelihood times the prior. $\frac{p(C)}{p(X)}$ is the reason it is proportional. – Ellis Valentiner Apr 23 '14 at 18:18
  • That's true, but I think OP is looking for a term of art to describe $p(C)/p(X)$. I'm not aware of "proportional" being used in that way. – Sycorax Apr 23 '14 at 18:21
  • You're right. I don't think there is one. But it is just a constant. – Ellis Valentiner Apr 23 '14 at 18:23
  • It's not constant if $p(C)$ is a probability distribution other than the uniform. Frequentist inference *assumes* that it is constant and unknowable, but Bayesian inference does not. – Sycorax Apr 23 '14 at 18:25
1

If you write Bayes theorem as

$$ p(C|X) = \frac{ p(X|C)p(C) }{ p(X) } $$

then $p(C|X)$ is posterior distribution, $p(X|C)$ is likelihood and $p(C)$ is prior, while $p(X)$ is normalizing constant. In fact, $p(X)$ is

$$ p(X) = \int p(X|C)p(C) dC $$

and is used so that we obtain results in probability scale. However in fact if you use MCMC simulation for estimation, it is not needed and the formula reduces to

$$ p(C|X) \propto p(X|C)p(C) $$

As already mentioned by others, there is no reason to be interested in $p(C)/p(X)$ because normalizing our prior probability, i.e. our guess that we made before seeing the data, leads nowhere. I guess this is why it has no name.

Tim
  • 108,699
  • 20
  • 212
  • 390
0

There's no standard term for p(C)/p(X). The denominator p(X) is called the marginal likelihood for your model.

Kevin S. Van Horn
  • 373
  • 1
  • 5