Hierarchical Linear Regression - Simplest Possible Example

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2

Hi,

I read everything rather quickly, please tell me if I miss something. You are using a dataset with only two groups and using two parameters to estimate it. If you want to pool towards the mean I would suggest that you only specify a global mean with a larger sd. You can see that in this way you achieve the pooled effect.

with pm.Model() as hierarchical_model:

    sigma_hierarchical = pm.HalfCauchy("sigma_hierarchical", beta=5, shape=n_sites)

    # The step that makes it hierarchical
    # We only assume beta is linked to the global
    mu_global = pm.Normal("mu_global", mu=0, sd=5)
    #sd_global = pm.HalfCauchy("sd_global", beta=0.1)
    beta_hierarchical = pm.Normal(
        "beta_hierarchical", mu=mu_global, sd=0.5, shape=n_sites
    )

    # Define likelihood
    likelihood = pm.Normal(
        "y_hierarchical",
        mu=beta_hierarchical[site_indices] * data["x"],
        sigma=sigma_hierarchical[site_indices],
        observed=data["y"],
    )

    # Inference!
    trace_hierarchical = pm.sample(draws=3000)
    
    posterior_beta_hierarchical = trace_hierarchical.get_values(
        "beta_hierarchical", burn=1000, chains=[1]
    )

image

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