While working on a particular problem from Chapter 7 of Statistical Rethinking, I observed a pretty striking difference in PyMC posterior distributions compared to Numpyro and the original R rethinking package. The original problem is the same one described by roesta07 here which has the original R code. It’s a simple linear model which can be instantiated this way in PyMC3:

```
with pm.Model() as m7_1:
a = pm.Normal("a", mu=0.5, sd=1)
b = pm.Normal("b", mu=0, sd=10)
log_sigma = pm.Normal("log_sigma", mu=0, sd=1)
mu = a + b * df_brains["mass_std"]
brain_std = pm.Normal(
"brain_std", mu=mu, sd=np.exp(log_sigma), observed=df_brains["brain_std"]
)
sample = pm.sample(1000, tune=1000, return_inferencedata=True)
```

**The posterior distribution of all three parameters (a, b, log_sigma) that PyMC3 outputted was different than the Numpyro and R rethinking package.** The most obvious difference is when looking at the log_sigma value (or transformed to sigma). Here is looking at PyMC3 vs. running code from the Numpyro repo of Chapter 7.

The difference is clear by looking at the mean for log-sigma:

PyMC3 mean for log-sigma: -1.3848

Numpyro mean for log-sigma: -1.7089

R rethinking mean for log-sigma: -1.7048

The similarity between Nympyro and R’s output leads me to believe that PyMC3 is “wrong”.

As mentioned, the difference is also seen in other parameters (means are the same but variances are higher in both a and b parameters).

Is there a rational reason for PyMC3 showing these differences compared to Numpyro and R rethinking?

(New user restrictions prevent me from adding more links but hope this explanation is sufficient. I’m aware of the PyMC3 repo for Statistical Rethinking, but it does not setup this problem the way the book does.)