[l_e_o_n]

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.