1
$\begingroup$

In Bayes' theorem, is it necessary that the Likelihood function, say $P(X|\theta)$ must be of a discrete probability mass function? I ask because it seems like this is saying that what is the probability of observing $X$ given $\theta$, but if this was a continuous PDF, then the probability of observing exactly $X$ would always be 0. All of the examples that I've seen in my textbook always give the this function has that of a binomial distribution or some other discrete PMF, but it seems to restrictive to me.

Thanks.

$\endgroup$
  • $\begingroup$ In the (absolutely) continuous case, the likelihood can be taken to be proportional to the probability density. It gets more complicated with a mixture $\endgroup$ – Henry Feb 24 '18 at 23:21
0
$\begingroup$

Indeed it depends on the distribution model.   The likelihood function for a absolutely continuous random variable's distribution parameter corresponds to a conditioned probability densitity function.

$$\mathcal L(\theta\mid x)=f(x\mid \theta)$$

$\endgroup$
  • $\begingroup$ But, isn't then $L(\theta|x)$ the Posterior? $\endgroup$ – Thomas Moore Feb 24 '18 at 23:41

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.