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?

  • $\begingroup$ 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. $\endgroup$ Oct 28 '20 at 14:18
  • 2
    $\begingroup$ No this is not correct, the likelihood is integrated against the prior, check eg the Wikipedia page $\endgroup$
    – Xi'an
    Oct 28 '20 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. $\endgroup$ Oct 28 '20 at 14:23
  • $\begingroup$ Thanks @Xi'an ! $\endgroup$ Oct 28 '20 at 14:30

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

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

  • $\begingroup$ 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! $\endgroup$ Oct 28 '20 at 14:29

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.