Is the likelihood continuous?

Questions
1

Hi everybody!

I’m new to bayesian modelling and i’m currently trying to wrap my head around the core concepts.
I’m having a simple (maybe stupid) question about the likelihood:
Something i’m having trouble to understand is that a set of patchy input data (which is not continuous) leads to a continuous posterior we can sample from.
Is there some kind of fitting/interpolation process involved that fits the predefined functional form of the likelihood to the input data before the sampling of the posterior begins?
Otherwise the function wouldn’t be defined (or just return 0) between the given data points.

Thanks :slight_smile:

2

You might find my talk on this subject helpful: https://www.youtube.com/watch?v=0_oonRc7Sn8

The TLdr is that, the likelihood is continuous as long as the input parameter is also restricted to be continuous. There is no interpolation happening, what it follows is the definition of a probability distribution. For example, say you have y ~ Binomial(n, p), the likelihood function of p when you know y and n is continuous, but the likelihood function of n when you know y and p is discrete.

2 Likes