comparing 2 likelihood values

Question

Are likelihood values (density values) comparable across different types of distributions?

For example, if you have a data point that has a likelihood value of .05 under a normal distribution and .025 under a student T distribution, the .05 and .025 are comparable?

In that case, that data point is twice as likely to come from the normal distribution given its parameter values vs. the T distribution given its parameters?

Welcome to Cross Validated! Could you please say what exactly you mean that a point is twice as “likely”? For better or for worse, “likelihood” in formal probability theory and mathematical statistics has a specific meaning that isn’t perfectly aligned with the colloquial English use of the word. — Dave, Aug 20 '23 at 02:56
i just mean if they are on the same scale then .05 is 2 times .025 and since the values indicate the likelihood of observing the data it’s 2times as likely — jhn5v78, Aug 20 '23 at 03:01
the probability of the the observed data given the parameters P(X = x | θ) — jhn5v78, Aug 20 '23 at 03:26
This is a good example of why this kind of back-and-forth clarification in the comments is useful, as likelihood is not quite a probability and can exceed one. Could you please clarify further what you want to know and the context where it has arisen? — Dave, Aug 20 '23 at 03:29
“The likelihood is the probability that a particular outcome x is observed when the true value of the parameter is θ” https://en.m.wikipedia.org/wiki/Likelihood_function — jhn5v78, Aug 20 '23 at 03:31
That quote is an incorrect statement. If nothing else, likelihood can exceed one, while probability cannot. — Dave, Aug 20 '23 at 03:32
https://hea-www.harvard.edu/AstroStat/aas227_2016/lecture1_Robinson.pdf see slide 8 “it’s the probability the parameter value would yield the observed data” what is your definition? — jhn5v78, Aug 20 '23 at 03:38
Likelihood can exceed one. Thus, likelihood is not probability. — Dave, Aug 20 '23 at 03:43
https://www.cs.cmu.edu/~10315/notes/10315_S23_Notes_MLE_draft.pdf pg 3 — jhn5v78, Aug 20 '23 at 04:00
it’s commonly understood to be density which the question noted “(density values)” — jhn5v78, Aug 20 '23 at 04:08
Jhn5v78, the example in the notes you linked to refers to discrete distributions (binomial, etc). For continuous distributions such as normal, you can turn densities into probabilities by taking the integral. This gives the probability that the data fall into the interval you integrated over; A density twice as large for distribution 1 as for distribution 2 on that interval indeed means data being twice as likely. By likelihood in statistics we understand a function of parameters given the data. This is not to be understood as probability of the parameter. — Ute, Aug 20 '23 at 11:32
hi ute, you can turn densities into probability if you take the integral of the pdf since in a pdf the parameter is not fixed but here i am asking about likelihoods and if you take the integral of likelihood density values it will likely not equal 1 so it is not a probability. see slide 7 of harvard. the likelihood needs a normalization term. i ask are 2 (unnormalized) density values from different distributions comparable — jhn5v78, Aug 20 '23 at 13:10

score 1 · Answer 1 · answered Dec 17 '23 at 16:36

You seem to be asking two different questions. The first one, about comparability, is a duplicate of Likelihood comparable across different distributions

Then you say:

In that case, that data point is twice as likely to come from the normal distribution given its parameter values vs. the T distribution given its parameters?

You cannot conclude that! Likelihood values come from a probability model, yes, but they cannot therefore itself be interpreted as probabilities (neither as probability densities). The wording of your conclusion has a bayesian flavour, and such a conclusion would require some full bayesian model, with priors.

comparing 2 likelihood values

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