Automatic Re-centering and Re-scaling


Brandon T. Willard



Using symbolic_pymc we can automate the PyMC3 model transformation in “Why hierarchical models are awesome, tricky, and Bayesian” and improve sample chain quality.

import numpy as np
import pandas as pd

import pymc3 as pm

import theano
import theano.tensor as tt

from functools import partial

from unification import var

from kanren import run
from kanren.graph import reduceo

from symbolic_pymc.theano.meta import mt
from symbolic_pymc.theano.pymc3 import model_graph, graph_model
from symbolic_pymc.theano.utils import canonicalize

from symbolic_pymc.relations.theano import non_obs_walko
from symbolic_pymc.relations.theano.distributions import scale_loc_transform

tt.config.compute_test_value = 'ignore'

data = pd.read_csv('')
data['log_radon'] = data['log_radon'].astype(theano.config.floatX)
county_names = data.county.unique()
county_idx = data.county_code.values

n_counties = len(data.county.unique())

with pm.Model() as model_centered:
    mu_a = pm.Normal('mu_a', mu=0., sd=100**2)
    sigma_a = pm.HalfCauchy('sigma_a', 5)
    mu_b = pm.Normal('mu_b', mu=0., sd=100**2)
    sigma_b = pm.HalfCauchy('sigma_b', 5)
    a = pm.Normal('a', mu=mu_a, sd=sigma_a, shape=n_counties)
    b = pm.Normal('b', mu=mu_b, sd=sigma_b, shape=n_counties)
    eps = pm.HalfCauchy('eps', 5)
    radon_est = a[county_idx] + b[county_idx] * data.floor.values
    radon_like = pm.Normal('radon_like', mu=radon_est, sd=eps,

# Convert the PyMC3 graph into a symbolic-pymc graph
fgraph = model_graph(model_centered)
# Perform a set of standard algebraic simplifications
fgraph = canonicalize(fgraph, in_place=False)

def reparam_graph(graph):
    """Apply re-parameterization relations throughout a graph."""

    graph_mt = mt(graph)

    def scale_loc_fixedp_applyo(x, y):
        return reduceo(partial(non_obs_walko, scale_loc_transform), x, y)

    q = var()
    expr_graph = run(0, q,
                     # Apply our transforms to unobserved RVs only
                     scale_loc_fixedp_applyo(graph_mt, q))

    expr_graph = expr_graph[0]
    opt_graph_tt = expr_graph.reify()

    # PyMC3 needs names for each RV
    opt_graph_tt.owner.inputs[1].name = 'Y_new'

    return opt_graph_tt

fgraph_reparam = reparam_graph(fgraph.outputs[0])

# Convert the symbolic-pymc graph into a PyMC3 graph so that we can sample it
model_recentered = graph_model(fgraph_reparam)


with model_centered:
    centered_trace = pm.sample(draws=5000, tune=1000, cores=4)[1000:]

with model_recentered:
    recentered_trace = pm.sample(draws=5000, tune=1000, cores=4)[1000:]


>>> pm.traceplot(centered_trace, varnames=['sigma_b'])

Fig. 1 Original model trace results.


>>> pm.traceplot(recentered_trace, varnames=['sigma_b'])

Fig. 2 Transformed model trace results.