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Math Help - Likelihood confirmation question

  1. #1
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    Likelihood confirmation question

    Basic question but

    p(\theta | D) = \frac{p(D|\theta)p(\theta)}{p(D)} = \frac{p(D|\theta)p(\theta)}{\int p(D|\theta)p(\theta)}

    Why is p(D) = \int p(D|\theta)p(\theta)? I thought it should be the sum? Wouldn't the integral just be 1?

    Thankyou
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  2. #2
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    Re: Likelihood confirmation question

    No. The probability of D occurring overall may be tiny across the entire range of $\theta$. It's an integral when $p(\theta)$ is a probability density function rather than a discrete probability distribution.
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  3. #3
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    Re: Likelihood confirmation question

    Quote Originally Posted by romsek View Post
    No. The probability of D occurring overall may be tiny across the entire range of $\theta$. It's an integral when $p(\theta)$ is a probability density function rather than a discrete probability distribution.
    Thankyou! So, it's just because it is a continuous distribution?
    It looks similar to marginal likelihood, can I ask how this comes into play?
    Thankyou
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  4. #4
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    Re: Likelihood confirmation question

    Quote Originally Posted by AshleyCS View Post
    Thankyou! So, it's just because it is a continuous distribution?
    It looks similar to marginal likelihood, can I ask how this comes into play?
    Thankyou
    Yes, because it's a continuous distribution.

    If the distribution depended on parameters other than $\theta$ then yes this would be a marginal distribution, but since it doesn't you end up with just a value which is the probability of D occurring at all.
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