Bayesian t-test is more easy, you can try to convert a Stan implementation of John Kruschke’s Bayesian Estimation Supersedes the t-test (BEST), in John K. Kruschke, “Bayesian Estimation Supersedes the t test,” *Journal of Experimental Psychology* 142, no. 2 (May 2013): 573-603, doi:10.1037/a0029146.

This should be easy to do in Turing

```
// Adapted from code by Michael Clark
// https://github.com/m-clark/Miscellaneous-R-Code/blob/master/ModelFitting/Bayesian/rstant_testBEST.R
// Stuff coming in from R
data {
int<lower=1> N; // Sample size
int<lower=2> n_groups; // Number of groups
vector[N] y; // Outcome variable
int<lower=1, upper=n_groups> group_id[N]; // Group variable
}
// Stuff to transform in Stan
transformed data {
real mean_y;
mean_y = mean(y);
}
// Stuff to estimate
parameters {
vector[2] mu; // Estimated group means
vector<lower=0>[2] sigma; // Estimated group sd
real<lower=0, upper=100> nu; // df for t distribution
}
// Models and distributions
model {
// Priors
// curve(expr = dnorm(mean_y, 2), from = -5, to = 5)
mu ~ normal(mean_y, 2);
// curve(expr = dcauchy(x, location = 0, scale = 1), from = 0, to = 40)
sigma ~ cauchy(0, 1);
// Kruschke uses a nu of exponential(1/29)
// curve(expr = dexp(x, 1/29), from = 0, to = 200)
nu ~ exponential(1.0/29);
// Likelihood
for (n in 1:N){
y[n] ~ student_t(nu, mu[group_id[n]], sigma[group_id[n]]);
}
}
// Stuff to calculate with Stan
generated quantities {
// Mean difference
real mu_diff;
// Effect size; see footnote 1 in Kruschke:2013
// Standardized difference between two means
// See https://en.wikipedia.org/wiki/Effect_size#Cohen's_d
real cohen_d;
// Common language effect size
// The probability that a score sampled at random from one distribution will
// be greater than a score sampled from some other distribution
// See https://janhove.github.io/reporting/2016/11/16/common-language-effect-sizes
real cles;
mu_diff = mu[1] - mu[2];
cohen_d = mu_diff / sqrt(sum(sigma)/2);
cles = normal_cdf(mu_diff / sqrt(sum(sigma)), 0, 1);
}
```

Stan Code from https://www.andrewheiss.com/blog/2019/01/29/diff-means-half-dozen-ways/imdb_best.stan.

One more thing? Why not fit two models that represents the competing hypotheses and do a LOO-CV using Pareto Smoothing Importance Sampling (PSIS-LOO)? The LOO calculates the Leave-one-out Cross-Validation of the elpd (*expected log pointwise predictive density*):

and see which model performs better? You wouldn’t even be restricted to only comparing two competing hypotheses at once. You could compare 10, 20, … all together.

It’s hard to convert frequentist approximations shortcuts (as E.T. Jaynes says “adhockeries”) to Bayesian Equivalent. As Allen Downey says (Bayesian and frequentist results are not the same, ever – Probably Overthinking It):

The posterior distribution represents everything you know about the parameters; if you reduce it to a single number, an interval, or a probability, you lose useful information. In fact, you lose exactly the information that makes the posterior distribution useful in the first place.

It’s like comparing a car and an airplane by driving the airplane on the road. You would conclude that the airplane is complicated, expensive, and not particularly good as a car. But that would be a silly conclusion because it’s a silly comparison. The whole point of an airplane is that it can fly.