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.


  • $\begingroup$ 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

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.

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

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

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