What does it mean to have a pareto distribution with a shape parameter greater than .7?


I’m using az.compare to compare some traces between three different models. I get the following user warning:

UserWarning: Estimated shape parameter of Pareto distribution is greater than 0.7 for one or more samples. You should consider using a more robust model, this is because importance sampling is less likely to work well if the marginal posterior and LOO posterior are very different. This is more likely to happen with a non-robust model and highly influential observations.

What does that actually mean? How do I make a model “more robust” exactly?

I’m happy with the model results but since it will eventually be put into production, I want to make sure I’m not missing something.

For context, the model is below:

with pm.Model(coords=coords_) as full_p_hierarchy_months:
    #add data
    t_ = pm.Data('t', time_idxs / T, mutable=True, dims=['obs_id'])
    A_ = pm.Data('A', A, mutable=True, dims=['obs_idx', 'changepoints'])
    s_ = pm.Data('s', s, mutable=True, dims=['changepoints'])
    X_fourier_ = pm.Data('X_fourier', yearly_fourier, mutable=True, dims=['time', 'yearly_components'])
    item_idx_ = pm.Data('item_idx', item_idxs, mutable = True, dims=['obs_id'])
    time_idx_ = pm.Data('time_idx', time_idxs, mutable = True, dims=['obs_id'])
    #set hierarchical intercept
    intercept_mu = pm.Normal('intercept_mu', mu=7, sigma=2)
    bl_intercept = make_next_level_hierarchy_variable(name='bl_intercept', mu=intercept_mu, alpha=2, beta=1, index_=business_line_idx, dims=['business_line'])
    cat_intercept = make_next_level_hierarchy_variable(name='cat_intercept', mu=bl_intercept, alpha=2, beta=1, index_=category_idx, dims=['category'])
    subcat_intercept = make_next_level_hierarchy_variable(name='subcat_intercept', mu=cat_intercept, alpha=2, beta=1, index_=sub_category_idx, dims=['sub_category'])
    item_intercept = make_next_level_hierarchy_variable(name='item_intercept', mu=subcat_intercept, alpha=2, beta=1, index_=item_idxs, dims=['items'])
     #set hierarchical slope
    slope_mu = pm.Normal('slope_mu', mu=0, sigma=1)
    bl_slope = make_next_level_hierarchy_variable(name='bl_slope', mu=slope_mu, alpha=2, beta=1, index_=business_line_idx, dims=['business_line'])
    cat_slope = make_next_level_hierarchy_variable(name='cat_slope', mu=bl_slope, alpha=2, beta=1, index_=category_idx, dims=['category'])
    subcat_slope = make_next_level_hierarchy_variable(name='subcat_slope', mu=cat_slope, alpha=2, beta=1, index_=sub_category_idx, dims=['sub_category'])
    item_slope = make_next_level_hierarchy_variable(name='item_slope', mu=subcat_slope, alpha=2, beta=1, index_=item_idxs, dims=['items'])
    #make changepoints
    delta = make_noncentered_variable(name='delta', mu=0, sigma=1, alpha=2, beta=1/2, dims=['items', 'changepoints'])
    intercept = pm.Deterministic('intercept', item_intercept[item_idx_] + ((-s_ * A_)[time_idx_, :] * delta[item_idx_]).sum(axis=1), dims=['obs_id'])
    slope = pm.Deterministic('slope', item_slope[item_idx_] + (A_[time_idx_, :] * delta[item_idx_]).sum(axis=1), dims=['obs_id'])
    #add seasonality
    seasonal_mu = pm.Normal('seasonal_mu', mu=0, sigma=1)
    bl_seasonal = make_next_level_hierarchy_variable(name='bl_seasonal', mu=seasonal_mu, alpha=2, beta=1, index_=business_line_idx, dims=['business_line','yearly_components'])
    cat_seasonal = make_next_level_hierarchy_variable(name='cat_seasonal', mu=bl_seasonal, alpha=2, beta=1, index_=category_idx, dims=['category','yearly_components'])
    subcat_betas = make_next_level_hierarchy_variable(name='subcat_betas', mu=cat_seasonal, alpha=2, beta=1, index_=sub_category_idx, dims=['sub_category', 'yearly_components'])
    seasonal_betas = make_next_level_hierarchy_variable(name='seasonal_beta', mu=subcat_betas, alpha=2, beta=1, index_=item_idxs, dims=['items', 'yearly_components'])
    seasonal = pm.Deterministic('seasonal', (X_fourier_[time_idxs] * seasonal_betas).sum(axis=1))
    mu__ = pm.Deterministic('mu', (intercept + slope*t_ + seasonal[time_idx_]), dims='obs_id') # + months_beta[month_idxs])
    #compute likelihood
    make_likelihood(mu = mu__, beta = 1/30, observed = y)


You should start at Cross-validation FAQ plus references to get a better idea of the possible implications and how to inspect your model to find out the actual reason for this