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Math Help - Bayesian statistics

  1. #1
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    Bayesian statistics

    I have a discrete variable, let's call it X, and a continuous variable called Z.

    I know the posterior f(Z|X=xi) has Gaussian Distribution with expected value ui and variance sigma2i (and I know these for all possible xi).

    Also I know that, according to the Bayes' theorem:

    f(Z|X=xi) = c * f(Z) * L(X=xi|Z), where the constant is there to normalize.

    The problem is, I have access to Z and want to find the xi that maximizes the likelihood L(X=xi|Z).

    L(X=xi|Z) = f(Z|X=xi) / (c * f(Z))

    but that constant could be different for every xi... How should I go about it?
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  2. #2
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    Quote Originally Posted by pedrosorio View Post
    I have a discrete variable, let's call it X, and a continuous variable called Z.

    I know the posterior f(Z|X=xi) has Gaussian Distribution with expected value ui and variance sigma2i (and I know these for all possible xi).

    Also I know that, according to the Bayes' theorem:

    f(Z|X=xi) = c * f(Z) * L(X=xi|Z), where the constant is there to normalize.

    The problem is, I have access to Z and want to find the xi that maximizes the likelihood L(X=xi|Z).

    L(X=xi|Z) = f(Z|X=xi) / (c * f(Z))

    but that constant could be different for every xi... How should I go about it?
    What exactly are you trying to do?

    CB
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  3. #3
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    Basically I want to use this to discretize values.

    For example, I measure the speed of a ball in m/s. I have 3 possible discrete speeds (slow, medium, fast). And I assume that for each discrete speed, the measured speed will have a gaussian distribution with a given mean and variance.

    What I want is, given the measured speed, find the most likely discrete speed. The "intuitive" is just to take the one with the highest posterior, but I'm not sure it's correct.
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  4. #4
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    I know that c is a normalizing constant, that is:

    1/c = Integral wrt Z of f(z) * L(X=xi|z)

    I think integral equals P(X=xi)?
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