If we know that:
How should I calculate the likelihood? I arrived to the next expression:
But something must be wrong because when I do this numerically, if n is big this likelihood will tend to 0... Easily deductible
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2$\begingroup$ There is nothing wrong. For example, the probability of seeing any particular string of coin flips tends to zero as you increase the number of coins filpped. $\endgroup$– jaradniemiCommented May 24, 2017 at 16:06
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1$\begingroup$ In practice, you should compute the logarithm of the likelihood, or loglikelihood. Then the numerical problems that you mention will go away. Then you will find out that when you actually use the likelihood for something, the theoretical formulas mostly use the loglikelihood ... $\endgroup$– kjetil b halvorsen ♦Commented Mar 27, 2018 at 13:41
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In practice, you would compute the logarithm of the likelihood, or loglikelihood. Then the numerical problems you mention should disappear. If the goal is to compute maximum likelihood estimates, then the loglikelihood is enough. Since $\log$ is a monotone increasing function, the likelihood and the loglikelihood is maximized at the same parameter values.
And then, if you want some indicator of the precision of those estimates, you will find that the hessian matrix of the loglikelihood at the estimate is useful, see https://en.wikipedia.org/wiki/Fisher_information or Theoretical motivation for using log-likelihood vs likelihood
If you want to choose between different models, then AIC (Akaike Information Criteria) is a possibility. Look it up, the loglikelihood again ... And soon you will discover that using loglikelihood is not merely convenient, it does also has many theoretical advantages.
answered Mar 27, 2018 at 20:39