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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?

user10024395
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  • I think it depends on what you're trying to model. If you describe your experiment in more detail, it may become more clear what modeling decisions are appropriate. – littleO Jul 06 '15 at 07:14
  • http://math.stackexchange.com/questions/14777/why-isnt-there-a-uniform-probability-distribution-over-the-positive-real-number

    Does this mean that it is impossible to randomly pick a number from a pool of infinite numbers? @littleO

     – user10024395 Jul 06 '15 at 08:48
  • No, it is possible, just not in such a way that each number is expected to come up the same number of times. Basically, when your outcome space is not finite, you need to "concentrate" your probability around some set of outcomes and let the probably trail off to zero for the rest. –  Jul 06 '15 at 12:23
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    You could use a any unbounded discrete distribution. Geometric and Poisson immediately come to mind. –  Jul 06 '15 at 12:24

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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.

BruceET
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