MHB Maximize A Posteriori: Countable Hypotheses

[OhMyMarkov]

Hello everyone!

Suppose we have multiple hypothesis, $H_1, H_2,\dots ,H_N$ of equal likelihood, and we wish to choose the unobserved parameter $\theta _m$ according to the following decision rule: $m _0 = arg \max _m p(x|H_m)$.

What if there are infinitely many hypotheses? (the case is countable but infinite)

 

[CaptainBlack]

Hello everyone!

Suppose we have multiple hypothesis, $H_1, H_2,\dots ,H_N$ of equal likelihood, and we wish to choose the unobserved parameter $\theta _m$ according to the following decision rule: $m _0 = arg \max _m p(x|H_m)$.

What if there are infinitely many hypotheses? (the case is countable but infinite)

In principle there is no difference, if you want to know more you will need to be more specific.

CB

 

[OhMyMarkov]

Hello CaptainBlack,

Let's start by two hypothesis of equally likely probability ("flat normal distribution"):

$H_0: X = \theta _0 + N$
$H_1: X = \theta _1 + N$

where N is a normal random variable (lets say of variance << $\frac{a+b}{2}$)

then the solution is $\operatorname{arg\, max}_m p(x|H_m)$.

But what if there were infinitely many hypothesis, i.e. $\theta$ is a real variable. How to estimate $\theta$?

 

[CaptainBlack]

Hello CaptainBlack,

Let's start by two hypothesis of equally likely probability ("flat normal distribution"):

$H_0: X = \theta _0 + N$
$H_1: X = \theta _1 + N$

where N is a normal random variable (lets say of variance << $\frac{a+b}{2}$)

then the solution is $\operatorname{arg\, max}_m p(x|H_m)$.

But what if there were infinitely many hypothesis, i.e. $\theta$ is a real variable. How to estimate $\theta$?

I see no difference between a finite and countably infinite number of hypotheses in principle. That is other than you cannot simply pick the required hypothesis out of a list of likelihoods, that is.

But you cannot have a completely disordered collection of hypotheses there must be some logic to their order, and so there will be some logic to the order of the likelihoods and it will be that logic that will allow you to find the hypothesis with the maximum likelihood.

CB