The visualization shows a Bayesian two-sample t test, for simplicity the variance is assumed to be known. It illustrates both Bayesian estimation via the posterior distribution for the effect, and Bayesian hypothesis testing via Bayes factor. The frequentist p-value is also shown. The null hypothesis, H0 is that the effect δ = 0, and the alternative H1: δ ≠ 0, just like a two-tailed t test. You can use the sliders to vary the observed effect (Cohen's d), sample size (n per group) and the prior on δ.


Observed effect (d = 1)
Sample size (n = 10)
SD of prior (σδ = 0.3)

About the visualization

The prior on the effect is a scaled unit-information prior. The black, and red circle on the curves represents the likelihood of 0 under the prior and posterior. Their likelihood ratio is the Savage-Dickey density ratio, which I use here as to compute Bayes factor. The p-value is the traditional p-value for a two-sample t test with known variance (i.e. a Z test). HDI is the posterior highest density interval, which in this case is analogous a credible interval. And CI is the traditional frequentist confidence interval.

Learn more

Check out Alexander Etz's blog series "Understanding Bayes" for a really good introduction to Bayes factor. Fabian Dablander also wrote a really good post, "Bayesian statistics: why and how", which introduces Bayesian inference in general. If you're interesting in an easy way to perform a Bayesian t test check out JASP, or BayesFactor if you use R.


I've created some posters inspired by my interactive visualizations. You can find them on my Etsy shop.