# Mean of mu and mean of predictive posterior distribution

**URL:** <https://discourse.pymc.io/t/mean-of-mu-and-mean-of-predictive-posterior-distribution/8432>\
**Category:** Questions\
**Tags:** linear\_model\
**Created:** [December 10, 2021, 5:35pm UTC](https://discourse.pymc.io/t/mean-of-mu-and-mean-of-predictive-posterior-distribution/8432 "2021-12-10T17:35:20Z")\
**Posts on this page:** 3\
**Page:** 1

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**Author:** ![FHTE](https://avatars.discourse-cdn.com/v4/letter/f/46a35a/32.png) [@FHTE](https://discourse.pymc.io/u/FHTE)\
**Post date:** [December 10, 2021, 5:35pm UTC](https://discourse.pymc.io/t/mean-of-mu-and-mean-of-predictive-posterior-distribution/8432/1 "2021-12-10T17:35:20Z")

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Hi everyone,  
question updated:

I am new to pymc3 and trying to understand the inner workings on an example of linear regression:

```auto
with pm.Model() as m_5_1:

    marriage_age_s = pm.Data("marriage_age_s", df['MedianAgeMarriage_s'].values)
    divorce_rate_s = pm.Data("divorce_rate_s", df['Divorce_s'])

    alpha = pm.Normal("alpha", mu=0, sd=0.2)
    beta_a = pm.Normal("beta_a", mu=0, sd=0.5)
    sigma = pm.Exponential("sigma", 1)
    
    mu = pm.Deterministic("mu", alpha + beta_a*marriage_age_s)
    
    divorce_rate_hat = pm.Normal("divorce_rate_hat", mu=mu, sd=sigma, observed=divorce_rate_s)
    
    m_5_1_trace = pm.sample(1000, tune=1000,return_inferencedata=False)
    posterior_pred = pm.sample_posterior_predictive(m_5_1_trace)

```

I am trying to grasp the difference between the mu values from the trace and the average of the posterior distribution.

In particular:  
The following code gets me the means of the mu values for the input datapoints:

```auto
age_std_seq = df['MedianAgeMarriage_s']
mu_pred = np.zeros((len(age_std_seq), len(m_5_1_trace) * m_5_1_trace_.nchains))

for i, age_std in enumerate(age_std_seq):
    mu_pred[i] = m_5_1_trace["alpha"] + m_5_1_trace["beta_a"] * age_std

mu_pred.mean(axis=1)

```

I get exactly the same values by taking mean of the deterministic mu:  
`m_5_1_trace['mu'].mean(axis=0)`

But for the same model, when I sample posterior predictive distribution and average the predicted values like this:

`posterior_pred['divorce_rate_hat'].mean(axis=0)`

I get very similar, but slightly different values.

What causes the difference? I was thinking that the values should be the same - what am I missing?

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**Author:** ![cluhmann](https://yyz2.discourse-cdn.com/flex036/user_avatar/discourse.pymc.io/cluhmann/32/3083_2.png) [@cluhmann](https://discourse.pymc.io/u/cluhmann)\
**Post date:** [December 11, 2021, 2:57am UTC](https://discourse.pymc.io/t/mean-of-mu-and-mean-of-predictive-posterior-distribution/8432/2 "2021-12-11T02:57:18Z")

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Welcome!

The observed `divorce_rate_hat`, despite being _centered_ at `mu`, is distributed around `mu` with `sd=sigma`. Your calculation of `mu_pred` recreates the “best fitting regression line” (e.g., `mu`) but doesn’t incorporate `sigma` whereas the posterior predictions do. To recreate the posterior predictions, you would need to take `mu_pred`, plug in as the location (mean) in a normal distribution, grab the corresponding value of `sigma` from the trace and plug it in as the scale (SD) and draw a random value from _that_ distribution.

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**Author:** ![FHTE](https://avatars.discourse-cdn.com/v4/letter/f/46a35a/32.png) [@FHTE](https://discourse.pymc.io/u/FHTE)\
**Post date:** [December 11, 2021, 12:27pm UTC](https://discourse.pymc.io/t/mean-of-mu-and-mean-of-predictive-posterior-distribution/8432/3 "2021-12-11T12:27:10Z")

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Thank you very much for the detailed explanation!