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I have two normal distributions with different means and variances:

N(u1, s1)

N(u2, s2)

And I have some data points (X) that were sampled from each of them. For each data point, I want to calculate the probability that it was generated by one distribution vs. the other.

One resource I found online suggested using the likelihood ratio test. However, I am confused because they do not seem like nested models. Even if they are nested models, if I evaluate the probability density function for each distribution at a given data point (xi):

f1(xi)

f2(xi)

It's not obvious to me why you can compare these. They are not probabilities - they are just the value of the density function. So I am a little confused what you should do.

This seems like a very simple problem, so if anyone has any suggestions, please let me know.

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  • $\begingroup$ To be able to calculate such a probability you need first to specify relative probabilities for each of the two distributions. $\endgroup$
    – whuber ♦
    Commented Jan 15, 2020 at 20:42
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    $\begingroup$ This seems quite similar to stats.stackexchange.com/questions/444919/…, which has some potentially helpful comments, although no answer. $\endgroup$
    – jbowman
    Commented Jan 15, 2020 at 22:10
  • $\begingroup$ @whuber, my prior is 50% for each distribution. But I'm still confused. Can I simply divide f1(xi)/f2(xi)? If so, why does this work? My understanding is that when you evaluate the pdf, it is not a probability. It is just some number. So why can I divide these numbers by each other? $\endgroup$
    – adn bps
    Commented Jan 16, 2020 at 0:11
  • $\begingroup$ Bayes' Theorem. Also, when $f$ is a density, $f(x)\mathrm{d}x$ is a probability. In the ratio of probabilities the $\mathrm{d}x$ terms cancel. For various takes on this idea, ranging from the intuitive (such as the foregoing) to the rigorous, see stats.stackexchange.com/questions/248476. $\endgroup$
    – whuber ♦
    Commented Jan 16, 2020 at 0:16

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