27 votes

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

The key lies not in the absolute size of the likelihood values but in their relative comparison and the mathematical principles underlying likelihood-based methods. The smallness of the likelihood is ...
ADAM's user avatar
  • 721
12 votes

Reconciling optimisation for log-likelihood and Brier score

Although both log-loss and Brier scores provide proper scoring rules, they put emphasis on different regions of the probability distributions. Quoting from Wikipedia: the choice of a scoring rule ...
EdM's user avatar
  • 90.8k
10 votes

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

First, as others have mentioned, we usually work with the logarithm of the likelihood function, for various mathematical and computational reasons. Second, since the likelihood function depends on the ...
Durden's user avatar
  • 1,205
6 votes

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

I can think of two things that might help you. First, likelihoods are defined only to a proportionality factor and their utility comes from their use in a ratio and while they are proportional to the ...
Michael Lew's user avatar
4 votes

How to derive marginal likelihood from prior and likelihood

You have to start from first principles and work with definitions in order to produce a mathematically legit outcome. Intuition does not suffice. The marginal density of the sample $X$ under model $\...
Xi'an's user avatar
  • 104k
4 votes
Accepted

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

likelihood $\neq$ probability The likelihood function is not the same as a probability distribution and it can be defined up to a constant. Seperating likelihood from probability has always been ...
Sextus Empiricus's user avatar
4 votes

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

If you flip a coin which is known to be weighted $100$ times and it comes up heads $80$ times, then you probably have a guess as to what the weight might be. One way to formalize this intuition is to ...
Steven Gubkin's user avatar
3 votes

An example where the likelihood principle *really* matters?

You are asking for a situation where "proportional likelihoods would lead one to markedly different (and equally defensible) inferences" with such a circumstance leading to a violation of ...
Michael Lew's user avatar

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