Why is everything based on likelihoods even though likelihoods are so small?

Re:Why is everything based on likelihoods even though likeli... by b10nic on Sunday February 18, 2024 @11:47AM (#39420544)

Not quite; the probability of n/2 successes in n trials is given as Binomial(n,p) not p^n. p^n is correct for a single sequence but there are many possible sequences that result half heads, half tails and so you have a factor of "N choose X" or the so called "Binomial Coefficient".

> (0.4)^20 × (0.6)^20

and I think you mean (0.4)^10 × (0.6)^10 or more generally p^x*(1-p)^n-x.

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Re:Why is everything based on likelihoods even though likeli... by l_e_o_n on Sunday February 18, 2024 @12:21PM (#39420921)

I'm talking about the whole sequence; you're talking about the number of heads (or) tails in the sequence.

The number of heads is a sufficient statistic, so we'll get the same likelihood ratios out, but the likelihood values themselves will be larger.

You could make a similar point about the original CrossValidated Normal(0, 1)^N example by summarizing the data with the mean and sum of squares.

This doesn't work if the data were Cauchy(0, 1)^N instead.

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