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I have a set observations of the same distribution with an unknown parameter. My statistics textbook says for two instances of the parameters, called $\phi_1$ and $\phi_2$, $\phi_1$ is more plausible value to be the true parameter than $\phi_2$ if $p(X = x | \phi_1) > p(X = x | \phi_2)$. I have no trouble to interpret the comparison of two probabilities of a random variable, for example p(X = a) < p(X = b) because I can rely on counting on the sample space of X to make sense the comparison. From my point of view conditional probability of different conditions are computed from different (sub) sample spaces so relying on counting to interpret does not work. How do you justify the comparison of the two conditional probabilities?

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  • $\begingroup$ These are not conditional probabilities: they are probabilities that depend on a parameter. $\endgroup$ – whuber Jan 30 at 16:27
  • $\begingroup$ I see the point, but my question is still relevant. Now, we have probabilities of two different distributions, and the interpretation of the comparison does not seem more clear. $\endgroup$ – Hiep Jan 31 at 10:31
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    $\begingroup$ Please consult our threads on likelihoods. I believe any of the best ones will fully answer your questions. For a list that is even more focused, search for "likelihood ratio" $\endgroup$ – whuber Jan 31 at 15:28