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I think I understand maximum likelihood, but I feel like there is an inconsistency here that I'm missing.

For example, assume we're talking about coin flips, and we'll let p = probability of seeing a head, then to see the result [HHT] means that I'm calculating:

L = (p^2) * (1-p) = p^2 - p^3  

Maximizing gives:

L=2p - 3p^2 = 0

Which means that p=2/3 which is good.

Now assume that all my observations were heads [HH]. This means that my likelihood function is now:

L=p^2

Maximizing/Solving this means that p=0. Likewise, if I had said all my observations were tails [TT], then my likelihood function is:

L=(1-p)^2

Maximizing/Solving this means that p=1.

But isn't the results reversed? I would expected that seeing HH would give me p=1.

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Note that $L'(p)=0$ for a particular $p$ does not mean that $L$ achieves its maximum there: It can be a local minimum of the function. In your case, $L(p)=p^2$ achieves its minimum at $p=0$. You can verify that by computing $L''(p)=2 > 0$.

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