I have a simple Bayesian inference model involving the prior for the regression coefficient given by a Beta distribution, with mean mu=0.1 and standard deviation sigma=0.9. When I use pyMC variational inference fit with ADVI, the inferred regression coefficient looks pretty reasonable (close to 0.5, which I used to generate the input data). Everything looks as expected.

Except that the Beta distribution cannot have a standard deviation of 0.9! Its maximum standard deviation is sqrt(mu(1-mu)), which is 0.3 in this case. Prior predictive check does return an error. But the variational inference does not even seem to bother, and gives pretty fine results. How is this possible?

```
import numpy as np
import pymc as pm
import arviz as az
# Generate data
np.random.seed(100)
x=np.linspace(0, 1, 100)
y=0.5*x + np.random.normal(loc=0,scale=0.2,size=100)
# Define PyMC model
with pm.Model() as beta_model:
w = pm.Beta("w",mu = 0.1,sigma = 0.9)
y=pm.Normal('y',
mu=w*x,
sigma=1,
observed = y)
# Sampling with approximate posterior
with beta_model:
posterior_draws=pm.fit(n=20000,method="advi",
random_seed=100,progressbar=True)
inferred_w = az.summary(posterior_draws.sample(50000, random_seed=100))
# View inferred parameter
print(inferred_w)
```

Returns the following

```
mean sd hdi_3% hdi_97% mcse_mean mcse_sd ess_bulk ess_tail r_hat
w 0.508 0.255 0.087 0.94 0.001 0.001 49383.0 48409.0 NaN
```