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conceptual stumbling block about Bayes factors.

We were recently advised by a reviewer to perform a Bayesian analysis of our data, and JASP was the recommended software to do this. Over the last couple days, I've been trying to absorb some of the key concepts (I am very new to Bayesian statistics). I've learned a lot, and find the software very easy to use (thanks for developing it!), but there is a conceptual snag that I'm trying to overcome.

Let's take the example of a paired samples t-test, with the alternative hypothesis being that the two groups are not equal. Thus, the null hypothesis is that the groups are equal, and that there is an effect size of 0 when comparing the two groups.

The Bayes factor is defined as the relative plausibility of the observed data between the alternative and null hypotheses.

Take a look at this JASP plot, which shows the posterior and prior distributions for a toy paired samples t-test example in JASP.

Here, the null hypothesis is depicted by the grey circles (effect size = 0). According to the youtube tutorial on JASP Bayesian t-tests, the Bayes factor can be visualized as the height of the prior curve at effect size 0 divided by the height of the posterior curve at effect size 0.

But to me, this seems to indicate a ratio of the probability of an effect size 0 given the prior, relative to the probability of an effect size 0 given the posterior. In other words, it seems to be weighing the probability of the null hypothesis between the posterior distribution and the prior distribution. How is this equivalent to the definition of the Bayes factor as stated above?

I have a feeling that I'm somehow confusing posterior odds with likelihood, as discussed here.

I'd really appreciate some guidance.

Comments

  • EJEJ
    edited 7:15AM

    Hi Spacediver,
    You mention: "But to me, this seems to indicate a ratio of the probability of an effect size 0 given the prior, relative to the probability of an effect size 0 given the posterior." You are exactly right! But by mathematical coincidence, this ratio is identical to the Bayes factor of H0 vs H1. This identity is not immediately obvious. You can find a full description here: http://www.ejwagenmakers.com/2010/WagenmakersEtAlCogPsy2010.pdf

    Cheers,
    E.J.

  • edited 7:15AM

    Thank you very much for the speedy and excellent response EJ!

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