Can't recover the scale parameter in Student's T model

Questions
1

I’m trying to model some data with the Student’s T distribution, so I generated some fake data and fitted the model to it. I’ve successfully recovered all other parameters, but the scale parameter is way off. Here is a replicable example:

import numpy as np
from scipy import stats
import pymc3 as pm
import pymc3.math as pmm

np.random.seed(123)
n_obs = 5000
n_itr = 2000
n_regressors = 3

Z = np.random.randn(n_obs, n_regressors)

α_f = 0.5
η_f = np.random.randn(n_regressors)
ϵ_f = 2
ν_f = 10

μ_f = α_f + np.dot(Z, η_f)
y_f = stats.t.rvs(loc=μ_f, scale=ϵ_f, df=ν_f)


with pm.Model() as t_model:
    α = pm.Normal('α', mu=0, sd=10)
    η = pm.Normal('η', mu=0, sd=10, shape=n_regressors)
    ϵ = pm.HalfCauchy('ϵ', 5)
    ν = pm.HalfCauchy('ν', 5)

    μ = α + pmm.dot(Z, η)
    y = pm.StudentT('y', mu=μ, lam=ϵ, nu=ν, observed=y_f)

    trace_t = pm.sample(n_itr, njob=2)

print('The fixed parameter values are:')
print('α_f = {},\nη_f = {},\nϵ_f = 2,\nν_f = 10'.format(α_f, η_f))
print('The estimated parameter values are:')
pm.df_summary(trace_t)['mean']

The output are

The fixed parameter values are:
α_f = 0.5,
η_f = [ 0.90756418  1.68521718 -1.1163093 ],
ϵ_f = 2,
ν_f = 10

The estimated parameter values are:
α        0.498621
η__0     0.900125
η__1     1.722081
η__2    -1.172383
ϵ        0.251066
ν       10.679210

All other parameters seem reasonable, but the scale parameter is way from the true value (0.25 instead 0f 2). Anyone got idea what might be the problem?

2

The scale in stats.t is the standard deviation, the equivalent in PyMC3 would be y = pm.StudentT('y', mu=μ, sd=ϵ, nu=ν, observed=y_f)

3

Of course you are right, silly question… I checked the documentation and saw lam is said to be the scale parameter so didn’t bother to check the math formula. Thanks!

4

It’s a real struggle when the same thing is called differently ¯\_(ツ)_/¯