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I am trying to understand how Bayesian inference works, so this might be a very simple question. I have an experiment where I test two hypotheses predicting opposite results. Let’s say, hypothesis 1 (H1) predicts that x > 0, and hypothesis 2 (H2) predicts that x < 0.

I calculated Bayes factor with informed priors (positive and negative half-normals for H1 and H2 respectively) for two hypotheses. BF10 for the H1 was 0.04, and BF10 for the H2 was 0.13. In other words, both results indicate that I have to believe more in H0 than in any of two alternative hypotheses.

However, if I still want to make some inference on H1 and H2, can I just divide BF10(for H2) by BF10(for H1)? This ratio (it will Bayes Factor too, right?) will be 0.13 / 0.04 = 3.25. Does this result tell me that I have to increase my belief in H2 compared to H1?

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  • $\begingroup$ If I understand correctly, Bayes factor for H1 is (prior(H1) * likelihood) / (prior(H0) * likelihood). H0 (x=0) is the same for H1 and H2. $\endgroup$ – Ivan Ivanchei Oct 28 at 14:18
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    $\begingroup$ No this is not correct, the likelihood is integrated against the prior, check eg the Wikipedia page $\endgroup$ – Xi'an Oct 28 at 14:22
  • $\begingroup$ Now, after you asked, i started thinking about it, and it seems that what i did (BF(H1)/BF(H2)) is basically a Bayes factor for these two hypotheses, because the common denominator in two Bayes factors (prior(H0)*lokelihood) is cancelled in division. Would be great if somebody could confirm this. $\endgroup$ – Ivan Ivanchei Oct 28 at 14:23
  • $\begingroup$ Thanks @Xi'an ! $\endgroup$ – Ivan Ivanchei Oct 28 at 14:30
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In short, yes, but your comments indicate you might have gotten the right answer for the wrong reasons.

A Bayes Factor is the ratio of the marginal likelihoods of the data under two different hypotheses.

Let's say $L_i$ is the marginal likelihood of the data under hypothesis $H_i$, and $H_0: x = 0$, $H_1: x > 0$, and $H_2: x < 0$, where for $H_1$ and $H_2$ you have some prior over possible values of $x$ (positive and negative values, respectively).

Your Bayes Factors are then the following ratios:

  • $BF_{10} = \frac{L_1}{L_0}$
  • $BF_{20} = \frac{L_2}{L_0}$
  • $BF_{21} = \frac{L_2}{L_1}$

If you know $BF_{20}$ and $BF_{10}$, their ratio is

$$ \begin{align} \frac{BF_{20}}{BF_{10}} &= \frac{\frac{L_2}{L_0}}{\frac{L_1}{L_0}} \\ &= \frac{L_2}{L_0} \times \frac{L_0}{L_1} \\ &= \frac{L_2}{L_1} \\ &= BF_{21} \end{align} $$

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  • $\begingroup$ Indeed, I did not understand Bayes factor calculation well (i.e. marginal likelihood), but I didn't do the calculation myself (I used a script from elsewhere), therefore, the results should be right. So, it's just a BF21. Great, thank you! $\endgroup$ – Ivan Ivanchei Oct 28 at 14:29

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