How is randomness quantified in Bayesian Statistics? How is randomness quantified in Bayesian Statistics? In the finite case of N items, it is simple, since I can assign a probability of 1/N to each of the item. However, I wonder what happens if I want to quantify the case when the number of items is not random, for example choosing a random natural numbers. Of course, the easy way out is to just say that the probability is 0 because otherwise, it leads to contradiction, but if that is the case, how can one calculate the statistics of an experiment that is based on such picking of random numbers?
 A: Suppose you are trying to figure out how many fish of a
particular kind are in a small lake. It must be between 0 and
a billion. Would you assign equal probabilities to the numbers
o through a billion. Probably not. Maybe you or someone you
know has previous experience fishing in this particular lake. 
Maybe you know a lot about fish and lakes of various sizes.
You probably won't be able to assign exact probabilities,
but you can probably do better than assigning equal probabilities
to a billion integers. (A method called 'capture-recapture' or
'mark-recapture' is one reasonable way to do an experiment
that should give a better estimate than even an educated guess.
You might want to read about it on Wikipedia or elsewhere
online.)
Suppose you are a newly hired consultant for a proposition
to raise taxes to pay for school repairs. You want to know
the chances that the proposition will pass the way things
stand today. In the US, almost all elections end up
more closely divided than a 30-70 split, many are really
close to 50-50. So before taking a poll you might pick
a 'prior' distribution from the beta family to represent
your initial guess. Unless you are from Mars, you would
probably not use a uniform distribution on (0, 1).
Often, Bayesian statisticians pick prior distributions
based on expert opinion, past history, or personal experience.
The procedures of Bayesian inference meld the information
in the prior distribution with the data from a survey or
experiment to obtain a posterior distribution that is
then used to make a point or interval estimate. Results
from several prior distributions are sometimes obtained
in order to see how much influence various prior distributions
have on final results.
Your question is very general and my answer is correspondingly
vague. If you have a specific problem in mind, we might
be able to have a more concrete discussion.
