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So the likelihood function for a binomial distribution is:

enter image description here

Why is the likelihood function above not multiplied by a combinatorics term: n! / (x! * (n - x)!)

If the likelihood function is interpreted as the probability of an outcome occurring x times out of n trials as the parameter, θ, varies, shouldn't there be a combinatorics term as that would provide the actual probability?

Thanks!

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We often don’t care about the likelihood, just the value for which the likelihood is maximized.

When you use the likelihood function to find a maximum likelihood estimator, you get the same point giving the maximum whether you include constants out front or not. Sure, that maximum value will be different, but that is not our concern.

So let’s make it convenient for ourselves and drop constants out in front, especially bulky combinatorics terms!

While we’re at it, we usually take the logarithm of the likelihood function since its derivative is easier to calculate, and log doesn’t change the point at which the maximum occurs.

Edit

This is in my comment but ought to be in the main post. We typically care about the argmax of a likelihood function, not the max itself.

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    $\begingroup$ Thanks! I'm guessing there isn't a case where we actually need the likelihood itself? I know when we do Bayes Analysis the constant would cancel out when dividing the joint by the marginal. $\endgroup$
    – confused
    Commented Oct 14, 2019 at 2:10
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    $\begingroup$ I don’t want to go as far as saying that we’d never care about the likelihood, but we often can get away with dropping coefficients because they cancel or because we care about the argmax instead of the max itself. $\endgroup$
    – Dave
    Commented Oct 14, 2019 at 2:25

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