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I have a mixture data. I used EM to estimate the model parameters. When I calculate the log likelihood function, I found that the values is positive. So, is that ok. Can the log likelihood function be positive?

To be more clear:

$ll = \sum_{n=1}^{N}\log(\sum_{k=1}^{K} \pi_{k} f(x_n;\theta_{k}))$

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  • $\begingroup$ To my understanding, likelihood function is usually a product of probabilities and log-likelihood is a sum of logs. Because probabilities are less than 1, theirs logs should be less than 0. And because you sum those logs, you should get a negative number. $\endgroup$
    – Celdor
    Commented Dec 21, 2017 at 10:32
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    $\begingroup$ The comment above is wrong. The likelihood function of continuous parameters (such as your case, I guess) is based on probability densities, which can be greater than 1 depending on the domain and the density, so the log likelihood can occasionally be positive. Still, you should check that everything else is correct. $\endgroup$
    – lacerbi
    Commented Dec 21, 2017 at 10:35
  • $\begingroup$ yes, my density is continuous. ' $\endgroup$
    – Alice
    Commented Dec 21, 2017 at 10:37

1 Answer 1

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Simply (just summarizing the comments):

  • when using probabilities (discrete outcome), the log likelihood is the sum of logs of probabilities all smaller than 1, thus it is always negative
  • when using probability densities (continuous outcome), the log likelihood is the sum of logs of densities that can be greater than 1, thus is can be positive.
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  • $\begingroup$ If all the discrete probability is in a single outcome, log-likelihood would be 0 rather than negative. $\endgroup$
    – Glen_b
    Commented Sep 3, 2020 at 5:41

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