Is the likelihood function L ( θ | X ) equal to or proportional to p ( X | θ )?Sometimes I find that the likelihood function is written as L ( θ | X ) = p ( X | θ ) while other times L ( θ | X ) ∝ p ( X | θ ).which is correct?If L ( θ | X ) ∝ p ( X | θ ), then what is the constant?

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giosgi5 · Accepted Answer

Step 1In one sense, it doesn&#039;t matter. The absolute value of the likelihood doesn&#039;t mean much, it&#039;s the likelihood ratio that is what is actually useful.Typically, you normalize the likelihood function   L  (  θ      |    x  ) by dividing by   L  (      θ    ∗        |    x  )  =      max          θ        L  (  θ      |    x  ) to get the standard likelihood ratio.Step 2The likelihood ratio as developed above is quite useful. A very powerful/useful asymptotic result called Wilks Theorem defines the null distribution of the log-likelihood ratio. We use it to find the range   θ that cannot be rejected in favor of the MLE.The most general statement is   L  (  θ      |    x  )  ∝  p  (  θ      |    x  ), since that encompasses but doesn&#039;t require, strict equality.

Is the likelihood function L( theta |X) equal to or proportional to p(X| theta)?

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