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

**URL:** <https://discourse.pymc.io/t/cant-recover-the-scale-parameter-in-students-t-model/396>\
**Category:** Questions\
**Created:** [October 6, 2017, 8:41am UTC](https://discourse.pymc.io/t/cant-recover-the-scale-parameter-in-students-t-model/396 "2017-10-06T08:41:17Z")\
**Posts on this page:** 4\
**Page:** 1

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**Author:** ![olivier-ma](https://yyz2.discourse-cdn.com/flex036/user_avatar/discourse.pymc.io/olivier-ma/32/189_2.png) [@olivier-ma](https://discourse.pymc.io/u/olivier-ma)\
**Post date:** [October 6, 2017, 8:41am UTC](https://discourse.pymc.io/t/cant-recover-the-scale-parameter-in-students-t-model/396/1 "2017-10-06T08:41:17Z")

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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?

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<div class="post-metadata">

**Author:** ![junpenglao](https://yyz2.discourse-cdn.com/flex036/user_avatar/discourse.pymc.io/junpenglao/32/8_2.png) [@junpenglao](https://discourse.pymc.io/u/junpenglao)\
**Post date:** [October 6, 2017, 8:51am UTC](https://discourse.pymc.io/t/cant-recover-the-scale-parameter-in-students-t-model/396/2 "2017-10-06T08:51:28Z")

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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)`

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<div class="post-metadata">

**Author:** ![olivier-ma](https://yyz2.discourse-cdn.com/flex036/user_avatar/discourse.pymc.io/olivier-ma/32/189_2.png) [@olivier-ma](https://discourse.pymc.io/u/olivier-ma)\
**Post date:** [October 6, 2017, 8:58am UTC](https://discourse.pymc.io/t/cant-recover-the-scale-parameter-in-students-t-model/396/3 "2017-10-06T08:58:54Z")

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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!

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<div class="post-metadata">

**Author:** ![junpenglao](https://yyz2.discourse-cdn.com/flex036/user_avatar/discourse.pymc.io/junpenglao/32/8_2.png) [@junpenglao](https://discourse.pymc.io/u/junpenglao)\
**Post date:** [October 6, 2017, 9:05am UTC](https://discourse.pymc.io/t/cant-recover-the-scale-parameter-in-students-t-model/396/4 "2017-10-06T09:05:59Z")

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