[lukego]

Likelihoods aren’t fundamentally small.

The center of a normal distribution has high likelihood (e.g. 1000000) if the standard deviation is small or low likelihood if the standard deviation is large (e.g. 1/1000000.)

This effect is amplified when you are working with products of likelihoods. They can be infinitesimal or astronomical.

Giant likelihoods really surprised me the first time I experienced them but they’re not uncommon when you work with synthetic test data in high dimensions and/or small scales.

They still integrate to the same magnitude because the higher likelihood values are spread over shorter spans.

[jdhwosnhw]

Another issue is that likelihoods associated with continuous distributions very often have units. You can’t meaningfully assign “magnitude” to a quantity with units. You can always change your unit system to make the likelihood of, say, a particular height in a population of humans to be arbitrarily large.