Anyway, I’m curious if elpd_loo and waic_loo are the same as the conventional log-likelihood…
They produced the same value but not the value that I expected (so close to zero).
Are they the real log-likelihood or should I have to multiply somehow?

Can you clarify a bit more? The real log-likelihood values are the ones you computed in your Stan program and extracted, but I’m guessing I’m not understanding what you’re actually asking.

As datasets get large, likelihoods get really small. For a really simple example, consider the Bernoulli likelihood for p = 0.5. With one data point, that likelihood will be 0.5. With 2 data points, the likelihood will be .25. With N data points, the likelihood will be 2^{-N}. By the time you get to even modest-size datasets, the likelihood gets vanishingly small. In fact, this is part of the motivation for working with log-likelihoods instead of likelihoods; the likelihood itself tends to underflow to zero. You can expect likelihoods close to zero for models with more than a handful of data points.

Sorry for unclear explanation.
I declared vector[25] log_lik and I got [1:6000,1:25] size of log_lik after MCMC sampling.
And I extracted log-likelihood by loo package and this code.

Computed from 6668 by 25 log-likelihood matrix
Estimate SE
elpd_loo -10.5 1.4
p_loo 0.2 0.0
looic 21.0 2.9
------
Monte Carlo SE of elpd_loo is 0.0.

The expected log likelihood is above 1000, based on the prior research.
But estimated elpd_loo was 10.5. Is this appropriate value, or did I do something wrong?

elpd_loo is not the same thing as the fitted log likelihood. See here LOO package glossary — loo-glossary • loo
But also, it’s very difficult for me to imagine a dataset that yields a likelihood of e^{1000}. I’m not sure what you mean by “the prior research”, but are you sure this is a reliable number?

Edit: I guess this is possible if the likelihood functions are really strongly peaked.

Actually I was trying to analyze the open dataset, and the BIC score from the original article was around 6400. This is the summation of the BIC score of 25 participants’ data, and each of them did 180 trials. (So that being the summation of 25 * 180 = 4500 log-likelihoods.)

Anyway, if elpd_loo doesn’t actually mean the fitted log-likelihood, then how can I compute the fitted log-likelihood with loo package?

Sum over the rows of loo::extract_log_lik(..., merge_chains=TRUE). This assumes of course that the log likelihood is correctly computed in your stan model itself.

Note also that the BIC uses the maximum value of the likelihood function, whereas when fitting the model you’ll get the “typical” value of the likelihood function. These can be rather different.