Improving speed with large dimensional problem


I previously asked about improving model sampling speed fitting about 14K observations to about 200 groups.

I am fitting a hierarchical model with Negative Binomial(alpha, mu) distributed likelihood (observations underneath level_2). The priors for the parameters alpha and mu (level_2) are Gamma(hyper_alpha_mu, hyper_alpha_sd) and Gamma(hyper_mu_mu, hyper_mu_sd) and the priors for the two levels above (level_1, level_0) are weakly informative HalfNormal distributions.

I am now fitting approximately 120K observations with 2000 classes at level_2 and 7 classes at level_1. The observations are count data on the order of magnitude of 10^5 - 10^7.

On a modestly powered laptop, the model goes through about 40 iterations a second during the initialization and it takes 3 hours to achieve 3000 samples using NUTS. When I try the sampling on a more powerful AWS instance, the sampling would take even longer.

Is there anything I can change in the model or the data to improve sampling speed?

See below for model specification.

Here is the dataset.

anonymized_dataset.csv (3.0 MB)

import pandas as pd
import pymc3 as pm

def create_association(df, parent_col, child_col):
    renamed = df.rename(
        columns={parent_col: 'parent', child_col: 'child'}
    association = renamed[['parent', 'child']].drop_duplicates()
    association.set_index('child', inplace=True)

    return association

data = pd.read_csv('anonymized_dataset.csv')

associations = {
    'level_0': {'parent_col': 'level_0', 'child_col': 'level_1'},
    'level_1': {'parent_col': 'level_1', 'child_col': 'level_2'},
    'level_2': {'parent_col': 'level_2', 'child_col': 'observation'}

for level in associations:
    associations[level] = create_association(data, **associations[level])

print('number of groups at level_0: ', len(associations['level_0']['parent'].unique()))
print('number of associations of level_1 to level_0: ', len(associations['level_0']['parent'].values))
print('number of groups at level_1: ', len(associations['level_1']['parent'].unique()))
print('number of associations of level_2 to level_1: ', len(associations['level_1']['parent'].values))
print('number of groups at level_2: ', len(associations['level_2']['parent'].unique()))
print('number of observations associated to level_2:  ', len(associations['level_2']['parent'].values))

observed = pd.merge(
    associations['level_2'], data[['observation', 'counts']],
    left_index=True, right_on='observation'

with pm.Model() as model:
    hyper_hyper_alpha_mu = pm.HalfNormal(
    hyper_hyper_alpha_sd = pm.HalfNormal(

    hyper_hyper_mu_mu = pm.HalfNormal(
    hyper_hyper_mu_sd = pm.HalfNormal(

    hyper_alpha_mu = pm.HalfNormal(
    hyper_alpha_sd = pm.HalfNormal(
    hyper_mu_mu = pm.HalfNormal(
    hyper_mu_sd = pm.HalfNormal(

    mu = pm.Gamma(
    alpha = pm.Gamma(

    y_est = pm.NegativeBinomial(

with model:
    hierarchical_trace = pm.sample(3000, progressbar=True)


The first thing I would try is changing the Gamma to a HalfNormal and use a non-centered parameterization.

Do you observed any speed up of NUTS before and after the tuning sample?


By tuning sample, do you mean the initialization sample for ADVI? If so, for ADVI, it was about 30 it/s for the initialization and 1.5 s/it for the sampling from NUTS.

Is there any literature on how to use a non-centered parameterization?


By tuning I meant the speed of NUTS before and after 500 samples (ie, the default number of tuning).
I am asking because if there is a speed difference, it indicates the initialization from ADVI is not good, PyMC3 ends up tuning the mass matrix in NUTS anyway, which sometimes could be quite long.

There are quite a few blog posts on parameterization in Hierarchical models:

This discussion on Stan forum is pretty great too:


Hmm, I do recall that the first 500 samples appeared faster under the NUTS sampling. Should I specify more initial samples during ADVI?

Thank you for the blog posts on parameterization; I’ll check those out and see if they help.


If you have the first 500 samples faster, it could be the case that the sampler fall into some kind of funnel. You should run multiple chain (e.g., njobs=4) and check the trace. Did you see any divergent warning?


I did not see any divergent warning. I’ll retry with njobs=4, but doesn’t that default to the number of cores automatically which would be 4?


Nope, you need to specify the number of chains. :slightly_smiling_face:


FYI, have a look at this post for speed related diagnostic.


Ok got it. Thank you for your advice. I’ll reply after trying the reparametrization and checking divergence.


Hello, none of the links posted demonstrate how to reparametrize distributions outside of the location-scale family. Is it still possible to reparametrize the half-normal to achieve the same thing?


Which post? it seems that a link is missing :slight_smile:


Oops, here is the link Frequently Asked Questions