PyMC3 with labeled coords and dims
Go crazy with your virtual labelmaker!
 For the of labeled arrays
 1st example: rugby analytics
 2nd example: radon multilevel model
 There is life outside the posterior
 Extra: generating the post image
import arviz as az
import matplotlib.pyplot as plt
import numpy as np
import pandas as pd
import pymc3 as pm
import theano.tensor as tt
rng = np.random.default_rng()
az.style.use("arvizdarkgrid")
For the of labeled arrays
For all of us who love labeled arrays, PyMC 3.9.0 came with some amazing news: support for using coordinate and dimension names to specify the shapes of variables in a pm.Model
. While this is good news by its own merit, its seamless integration with ArviZ even more impactful and relevant.
This post will focus on using PyMC3 coords and dims and the conversion of traces and models to InferenceData
using arviz.from_pymc3
. To see InferenceData
in action, refer to this example in PyMC docs.
We will use an example based approach and use models from the example gallery to illustrate how to use coords and dims within PyMC3 models.
1st example: rugby analytics
We will use an alternative parametrization of the same model used in the rugby analytics example taking advantage of dims and coords. Here, we will use as observations a 2d matrix, whose rows are the matches and whose columns are the field: home and away.
The first step after preprocessing is to define the dimensions used by the model and their coordinates. In our case, we have 3 dimensions:

team
: each team will have its own offensive and defensive power 
match
: an integer identifying the match. There are 6 teams who play twice against each other so we have6*5*2=60
matches 
field
: either home or away.
These coordinates are passed to pm.Model
as a dict whose keys are dimension names and whose values are coordinate values. The dimensions can then be used when defining PyMC3 variables to indicate their shape.
df_rugby = pd.read_csv(pm.get_data('rugby.csv'), index_col=0)
home_idx, teams = pd.factorize(df_rugby["home_team"], sort=True)
away_idx, _ = pd.factorize(df_rugby["away_team"], sort=True)
coords = {
"team": teams,
"match": np.arange(60),
"field": ["home", "away"],
}
with pm.Model(coords=coords) as rugby_model:
# global model parameters
home = pm.Flat('home')
sd_att = pm.HalfStudentT('sd_att', nu=3, sigma=2.5)
sd_def = pm.HalfStudentT('sd_def', nu=3, sigma=2.5)
intercept = pm.Flat('intercept')
# teamspecific model parameters
atts_star = pm.Normal("atts_star", mu=0, sigma=sd_att, dims="team")
defs_star = pm.Normal("defs_star", mu=0, sigma=sd_def, dims="team")
atts = pm.Deterministic('atts', atts_star  tt.mean(atts_star), dims="team")
defs = pm.Deterministic('defs', defs_star  tt.mean(defs_star), dims="team")
home_theta = tt.exp(intercept + home + atts[home_idx] + defs[away_idx])
away_theta = tt.exp(intercept + atts[away_idx] + defs[home_idx])
# likelihood of observed data
points = pm.Poisson(
'home_points',
mu=tt.stack((home_theta, away_theta)).T,
observed=df_rugby[["home_score", "away_score"]],
dims=("match", "field")
)
rugby_trace = pm.sample(1000, tune=1000, cores=4)
We have now defined the shapes or our variables, which is convenient and helps understanding the code, but the dimensions and coordinates are lost during sampling. pm.MultiTrace
objects do not store the labeled coordinates of their variables.
print(rugby_trace)
print(rugby_trace["atts"])
To also take advantage of the labeled coords and dims for exploratory analysis of the results, we have to convert the results to az.InferenceData
. This can be done with az.from_pymc3
or using the return_inferencedata=True
argument in pm.sample
. To avoid having to resample again, we will use the former and use the latter in the second example.
ArviZ is aware of the model context, and will use it to get the coords and dims automatically. If necessary however, we may also modify or add dimensions or coordinates using the dims
/coords
arguments of from_pymc3
. We'll also see an example of this afterwards.
rugby_idata = az.from_pymc3(rugby_trace, model=rugby_model)
rugby_idata
2nd example: radon multilevel model
We will now use one of the many models in the A Primer on Bayesian Methods for Multilevel Modeling notebook to dive deeper into coords and dims functionality. We won't cover the model itself, it's already explained in the example notebook, we will explain in detail how are labeled coords and dims being used.
The code used to load and clean the data is hidden, click the button below to see it.
srrs2 = pd.read_csv(pm.get_data("srrs2.dat"))
srrs2.columns = srrs2.columns.map(str.strip)
srrs_mn = srrs2[srrs2.state == "MN"].copy()
srrs_mn["fips"] = srrs_mn.stfips * 1000 + srrs_mn.cntyfips
cty = pd.read_csv(pm.get_data("cty.dat"))
cty_mn = cty[cty.st == "MN"].copy()
cty_mn["fips"] = 1000 * cty_mn.stfips + cty_mn.ctfips
srrs_mn = srrs_mn.merge(cty_mn[["fips", "Uppm"]], on="fips")
srrs_mn = srrs_mn.drop_duplicates(subset="idnum")
u = np.log(srrs_mn.Uppm).unique()
n = len(srrs_mn)
srrs_mn.county = srrs_mn.county.map(str.strip)
srrs_mn["county_code"], mn_counties = pd.factorize(srrs_mn.county)
srrs_mn["log_radon"] = np.log(srrs_mn.activity + 0.1)
The first step is again defining the dimensions and their coordinate values:

Level
: observations can correspond to the basement or the first floor 
obs_id
: unique integer identifying each observation 
County
: each county has its own basement, intercept:a
, and first floor, slopeb
, effects. Details are in the example notebook 
param
: one ofa
,b

param_bis
: same as param, used for the covariance matrix because a variable can't have repeated dimensions
coords = {
"Level": ["Basement", "Floor"],
"obs_id": np.arange(n),
"County": mn_counties,
"param": ["a", "b"],
"param_bis": ["a", "b"],
}
We'll begin to define the model creating the indexing arrays that will implement the hierarchical model. We are using the pm.Data
container to tell ArviZ to store the variables in the constant_data
group. Moreover, pm.Data
defines a theano shared variable, so its values can be modified in order to call pm.sample_posterior_predictive
using different data. This is particularly interesting for regressions for example in order to generate predictions for the model.
with pm.Model(coords=coords) as radon_model:
floor_idx = pm.Data("floor_idx", srrs_mn.floor, dims="obs_id")
county_idx = pm.Data("county_idx", srrs_mn.county_code, dims="obs_id")
We'll also use a LKJCholeskyCov
as prior for the covariance matrix. As you can see, it has no dims
argument. Given that we are going to use return_inferencedata=True
here in order to get an InferenceData directly as a result of pm.sample
, we will have to indicate the dims that correspond to these variables as idata_kwargs
. idata_kwargs
is used to indicate pm.sample
what arguments to pass to az.from_pymc3, which is called internally to convert the trace to InferenceData.
with radon_model:
sd_dist = pm.Exponential.dist(0.5)
a = pm.Normal("a", mu=0.0, sigma=5.0)
b = pm.Normal("b", mu=0.0, sigma=1.0)
z = pm.Normal("z", 0.0, 1.0, dims=("param", "County"))
chol, corr, stds = pm.LKJCholeskyCov(
"chol", n=2, eta=2.0, sd_dist=sd_dist, compute_corr=True
)
We now will store two intermediate results as variables. However, one is wrapped inside a pm.Deterministic
whereas the other is not. Both are equally valid. pm.Deterministic
tells PyMC3 to store that variable in the trace. Thus pm.Deterministic
should only be used when we actively want to store the intermediate result. In our case, we want to store ab_county
but not theta
.
with radon_model:
ab_county = pm.Deterministic("ab_county", tt.dot(chol, z).T, dims=("County", "param"))
theta = a + ab_county[county_idx, 0] + (b + ab_county[county_idx, 1]) * floor_idx
sigma = pm.Exponential("sigma", 1.0)
y = pm.Normal("y", theta, sigma=sigma, observed=srrs_mn.log_radon, dims="obs_id")
Finally we will call pm.sample
with return_inferencedata=True
and defining the dimensions of the covariance matrix as idata_kwargs
.
with radon_model:
radon_idata = pm.sample(
2000, tune=2000, target_accept=0.99, random_seed=75625, return_inferencedata=True,
idata_kwargs={"dims": {"chol_stds": ["param"], "chol_corr": ["param", "param_bis"]}}
)
There is life outside the posterior
The posterior is the center of Bayesian analysis but other quantities such as the prior or the posterior predictive are also crucial to an analysis workflow. We'll use a linear regression to quickly showcase some of the key steps in a Bayesian workflow: prior predictive checks, posterior sampling, posterior predictive checks (using LOOPIT) and out of sample predictions.
We will start generating some simulated data (code hidden, click to expand) and defining the model. As it's a simple linear regression we'll only have scalar parameters, a
, b
and sigma
.
a_ = 2
b_ = 0.4
x_ = np.linspace(0, 10, 31)
year_ = np.arange(2021len(x_), 2021)
y_ = a_ + b_ * x_ + rng.normal(size=len(x_))
fig, ax = plt.subplots()
ax.plot(x_, y_, "o")
ax.text(
0.93, 0.9, r"$y_i = a + bx_i + \mathcal{N}(0,1)$", ha='right', va='top', transform=ax.transAxes, fontsize=18
)
ax.set_xticks(x_[::3])
ax.set_xticklabels(year_[::3])
ax.set_yticks([])
ax.set_xlabel("Year")
ax.set_ylabel("Quantity of interest");
coords = {"year": year_}
with pm.Model(coords=coords) as linreg_model:
x = pm.Data("x", x_, dims="year")
a = pm.Normal("a", 0, 3)
b = pm.Normal("b", 0, 2)
sigma = pm.HalfNormal("sigma", 2)
y = pm.Normal("y", a + b * x, sigma, observed=y_, dims="year")
We have now written a model in order to study our super interesting quantity y
. We have used everything we have seen so far, the pm.Data
container and the labeled dims and coords. We will now simulate a workflow starting from prior predictive checks and finishing with predicting the values of our quantity of interest in 2021 and 2022.
Priors
We start by sampling both prior and prior predictive with pm.sample_prior_predictive
. This will generate a dictionary whose keys are variable names and whose values are numpy arrays. We can then pass this dictionary to az.from_pymc3
as the prior
argument. ArviZ will then use the information in the pm.Model
instance to 1) split the variables between prior
and prior_predictive
groups, 2) fill the observed_data
and constant_data
groups and 3) get the dims
and coords
if present.
with linreg_model:
prior = pm.sample_prior_predictive(700)
linreg_idata = az.from_pymc3(prior=prior)
linreg_idata
We can now use plot_ppc
to perform prior predictive checks for our model.
az.plot_ppc(linreg_idata, group="prior");
with linreg_model:
idata_aux = pm.sample(return_inferencedata=True)
linreg_idata.extend(idata_aux)
linreg_idata
az.plot_pair(linreg_idata);
Posterior predictive
Our third step will be to evaluate the posterior predictive at the observations so we can perform model checking with functions such as plot_ppc
or plot_loo_pit
. Here again we are using the extend
trick to keep all our data as part of the same InferenceData
. This has two main advantages. plot_loo_pit
requires both the posterior_predictive
group, generated here and the log_likelihood
group which was created in pm.sample
. In addition, keeping all our data in a single InferenceData
means we can store it as a netCDF and share a single file to allow reproducing the whole exploratory analysis of our model.
with linreg_model:
post_pred = pm.sample_posterior_predictive(linreg_idata)
idata_aux = az.from_pymc3(posterior_predictive=post_pred)
linreg_idata.extend(idata_aux)
linreg_idata
We will now get to use plot_loo_pit
, which as expected does not show any issues. To learn how to interpret those plots, you can read the LOOPIT tutorial.
az.plot_loo_pit(linreg_idata, y="y");
Predictions
Finally, our last step will be to get some predictions, which in this case is evaluating the posterior predictive at positions different than the observations. In the example below, we are evaluating our predictions at 2021 and 2020. To do so, we are using pm.set_data
to modify the values of x
to the ones that correspond to these two future years.
Here we will use from_pymc3_predictions
instead of from_pymc3
+extend
. from_pymc3_predictions
combines functionality from both of these functions and let's the user choose how to handle predictions depending on the goal at hand: if idata_orig
is not present, the returned object will be an InferenceData
containing only the predictions groups; if idata_orig
is present and inplace=False
the returned InferenceData
will be a copy of idata_orig
with the predictions groups added, and with inplace=True
there is no returned object, the preditcions groups are added to idata_orig
which is not returned.
with linreg_model:
pm.set_data({"x": x_[1] + x_[1:3]})
predictions = pm.sample_posterior_predictive(linreg_idata)
az.from_pymc3_predictions(
predictions, coords={"year": [2021, 2022]}, idata_orig=linreg_idata, inplace=True
)
linreg_idata
az.plot_posterior(linreg_idata, group="predictions");
from mpl_toolkits.mplot3d import Axes3D
points = np.array([
[1, 1, 1],
[1, 1, 1 ],
[1, 1, 1],
[1, 1, 1],
[1, 1, 1],
[1, 1, 1 ],
[1, 1, 1],
[1, 1, 1]
])
fig = plt.figure(dpi=300)
ax = fig.add_axes([0, .05, 1, .8], projection='3d')
side = 3
r = side * np.array([1,1])
one_2d = side * np.ones((1,1))
one_1d = side * np.ones(1)
X, Y = np.meshgrid(r, r)
################
### theta ###
################
ax.plot_surface(X,Y,one_2d, color="C0", zorder=1)
ax.plot_surface(X,one_2d,Y, color="C0", zorder=1)
ax.plot_surface(one_2d,X,Y, color="C0", zorder=1)
ax.text2D(0.3, 0.14, r"$\theta$", transform=ax.transAxes)
# school dim
school_grid = np.linspace(1, 1, 8) * side
school_one = np.ones_like(school_grid) * side * 1.1
schools = np.array(["Choate", "Deerfield", "Phillips Andover", "Phillips Exeter",
"Hotchkiss", "Lawrenceville", "St. Paul's", "Mt. Hermon"])
ax.plot(school_grid, school_one, school_one, marker="", color="k")
for school, pos in zip(schools, school_grid):
ax.text(pos, side*1.1, side*1.15, f"{school}", zdir="y", clip_on=False)
# chain dim
chain_grid = np.linspace(1, 1, 6) * side
chain_one = np.ones_like(chain_grid) * side * 1.1
ax.plot(chain_one, chain_grid, chain_one, marker="", color="k")
for chain, pos in enumerate(chain_grid):
ax.text(side*1.1, pos, side*1.15, f"{chain}", zdir="y", va="bottom", ha="center")
ax.text(side*1.3, 0, side*1.3, "Chain", zdir=(1, 1, .2), va="bottom", ha="center")
# draw dim
draw_grid = np.linspace(1, 1, 50) * side
draw_one = np.ones_like(draw_grid) * side * 1.1
draws = np.arange(0, 50, 10)
ax.plot(draw_one, draw_one, draw_grid, marker="_", color="k")
for draw, pos in zip(draws, draw_grid[draws]):
ax.text(side*1.1, side*1.15, pos, f"{draw}", zdir=None, ha="right")
ax.text2D(0.12, 0.4, "Draw", transform=ax.transAxes, rotation="vertical")
################
### tau ###
################
ax.plot_surface(4+one_2d,X,Y, color="C1", zorder=1)
ax.text2D(0.525, 0.1, r"$\tau$", transform=ax.transAxes)
################
### mu ###
################
ax.plot_surface(8+one_2d,X,Y, color="C2", zorder=1)
ax.text2D(0.66, 0.08, r"$\mu$", transform=ax.transAxes)
################
### grids ###
################
grid = np.linspace(1, 1, 10) * side
grid_one = np.ones(10)
lw = .3; alpha = .7
for chain_pos in chain_grid:
ax.plot(grid, grid_one * chain_pos, grid_one * side, color="k", alpha=alpha, zorder=3, lw=lw)
ax.plot(grid_one * side, grid_one * chain_pos, grid, color="k", alpha=alpha, zorder=5, lw=lw)
ax.plot(grid_one * side + 4, grid_one * chain_pos, grid, color="k", alpha=alpha, zorder=6, lw=lw)
ax.plot(grid_one * side + 8, grid_one * chain_pos, grid, color="k", alpha=alpha, zorder=7, lw=lw)
for draw_pos in draw_grid:
ax.plot(grid, grid_one * side, grid_one * draw_pos, color="k", alpha=alpha, zorder=4, lw=lw)
ax.plot(grid_one * side, grid, grid_one * draw_pos, color="k", alpha=alpha, zorder=5, lw=lw)
ax.plot(grid_one * side + 4, grid, grid_one * draw_pos, color="k", alpha=alpha, zorder=6, lw=lw)
ax.plot(grid_one * side + 8, grid, grid_one * draw_pos, color="k", alpha=alpha, zorder=7, lw=lw)
for school_pos in school_grid:
ax.plot(grid_one * school_pos, grid, grid_one * side, color="k", alpha=alpha, zorder=4, lw=lw)
ax.plot(grid_one * school_pos, grid_one * side, grid, color="k", alpha=alpha, zorder=4, lw=lw)
ax.axis("off");
ax.view_init(azim=69)
#fig.savefig("labeled_arys.png", dpi=300)