Purpose

Just testing my intuition w.r.t. degrees of freedom in the students T distribution.

• Cauchy: df = 1.
• Normal: df = infinity (or at least some really high number)

This should be reflected when using PyMC3.

import pymc3 as pm
import numpy as np
import matplotlib.pyplot as plt
import arviz as az

%load_ext autoreload
%autoreload 2
%matplotlib inline
%config InlineBackend.figure_format = 'retina'

The autoreload extension is already loaded. To reload it, use:
  %reload_ext autoreload

normal = np.random.normal(size=20000)
cauchy = np.random.standard_cauchy(size=20000)

with pm.Model() as normal_model:
    mu = pm.Normal("mu", mu=0, sd=100)
    sd = pm.HalfNormal("sd", sd=100)
    nu = pm.Exponential("nu", lam=0.5)
    like = pm.StudentT("like", mu=mu, sd=sd, nu=nu, observed=normal)

    trace = pm.sample(2000, tune=2000)

Auto-assigning NUTS sampler...
Initializing NUTS using jitter+adapt_diag...
Multiprocess sampling (4 chains in 4 jobs)
NUTS: [nu, sd, mu]
Sampling 4 chains, 0 divergences: 100%|██████████| 16000/16000 [00:20<00:00, 790.93draws/s] 

axes = az.plot_trace(trace)

Many degrees of freedom for normal distribution. Makes sense.

with pm.Model() as cauchy_model:
    mu = pm.Normal("mu", mu=0, sd=100)
    sd = pm.HalfNormal("sd", sd=100)
    nu = pm.Exponential("nu", lam=1)
    like = pm.StudentT("like", mu=mu, sd=sd, nu=nu, observed=cauchy)

    trace = pm.sample(2000, tune=2000)

Auto-assigning NUTS sampler...
Initializing NUTS using jitter+adapt_diag...
Multiprocess sampling (4 chains in 4 jobs)
NUTS: [nu, sd, mu]
Sampling 4 chains, 0 divergences: 100%|██████████| 16000/16000 [00:22<00:00, 709.93draws/s]

axes = az.plot_trace(trace)

Basically 1 degree of freedom when inferring $\nu$ from Cauchy-distributed data. Yes :).