Bayes rule and gaussian PDF if range of gaussian pdf is not a probability then how come it is used in Bayes rule in the same way as pmf?
 A: @Ahmed - you are definitely correct in thinking that something is not quite right here.
Conditioning on "point values" which have probability/measure 0 can be "dangerous" and can lead to what is called a Borel and Kolmogorov Paradox.  The lesson from this is that in order to be certain that we are defining a conditional probability unambiguously, it is necessary to:


*

*Specify a definite limiting process towards the null point.

*Ensure that the limit exists and is "well behaved."

*Don't change limits halfway through your calculations!


This is a rather tedious and often unnecessary process, but if find yourself getting weird answers, this is the "insurance policy" so to speak.  You can pretty much never get absurdities if you follow the above rules.  But like all insurance, there is a price - the painful process of working out a limit.
I like Edwin Jaynes in this case:
there is no right choice or wrong choice because it is for us to say which limit we want to take; i.e., which problem we want to solve.
This is Sermon #1 on mathematical limits; although it was given long ago by Kolmogorov, many who try to do probability calculations still fail to heed it and get themselves into trouble. The moral is that, unless they are defined in the statement of a problem, probabilities conditional on point values of a parameter, have no meaning until the specific limiting process is stated. More generally, probabilities conditional on any propositions of probability zero, are undefined.
A: The posterior distribution derived using continuous distributions in Bayes Theorem can always be integrated (although maybe not be hand) to give a probability. If you want to convince yourself "caveman style," run the desired probabilities through Bayes Theorem using a Gaussian CDF, then take the derivative to get the posterior PDF.  Sounds like a great homework problem to torture undergrads with.  Thanks for the idea.
If you're not masochistic, and are amenable to a "handwaving argument," just think of $f(t) dt$ as an infinitesimal probability and plug it in to the theorem.
