[bscphil]

> It is fair to ask why the likelihoods are useful if they are so small

The way the question demonstrates "smallness" is wrong, however. They quote the product of the likelihoods of 50 randomly sampled values - 9.183016e-65 - as if the smallness of this value is significant or meant anything at all. Forget the issue of continuous sampling from a normal distribution, and just consider the simple discrete case of flipping a coin. The combined probability of any permutation of 50 flips is 0.5 ^ 50, a really small number. That's because the probability is, in fact, really small!

[knightoffaith]

Right - and so the more appropriate thing to do is not look at the raw likelihood of any one particular value but instead look at relative likelihoods to understand what values are more likely than other values.

[adiM]

Therefore, likelihood ratios! (Or log likelihood ratios)

[anon946]

For the discrete case, it seems that a better thing to do is consider the likelihood of getting that number of heads, rather than the likelihood of getting that exact sequence.

I am not sure how to handle the continuous case, however.

[lupire]

Of course you ignore irrelevant ordering of data points. That's not the issue.

The issue, for discrete or continuous (which are mathematically approximations of each other), is that the value at a point is less important than the integral over a range. That's why standard deviation is useful. The argmax is a convenient average over a weightable range of values. The larger your range, the greater the likelihood that the "truth" is in that range.

If you only need to be correct up to 1% tolerance, the likelihood of a range of values that have $SAMPLING_PRECISION tolerance is not importance. Only the argmax is, to give you a center of the range.