PyMC 4.0 with labeled coords and dims
Go even crazier with your virtual labelmaker!
 For the of labeled arrays
 1st example: rugby analytics
 2nd example: radon multilevel model
 There is life outside the posterior
import arviz as az
import matplotlib.pyplot as plt
import numpy as np
import pandas as pd
import pymc as pm
import aesara.tensor as at
rng = np.random.default_rng()
az.style.use("arvizdarkgrid")
For the of labeled arrays
While PyMC 3.9.0 came with some amazing news for those of us who love labeled arrays: support for using coordinate and dimension names to specify the shapes of variables in a pm.Model
; now, 2 years later, with the release of PyMC 4.0, the integration between ArviZ and PyMC has improved even more.
Therefore, I decided to post and updated version of my original blog post using version 3.x of PyMC now using version 4.0 of PyMC which was released earlier this week.
This post will focus on using PyMC coords, dims and dims without coordinate values, and explain the conversion of traces and models to InferenceData
. Here are two resources to learn about working with InferenceData
on ArviZ docs and to see InferenceData
in action in the famous radon example on PyMC examples website.
We will use an example based approach and use models from the example gallery to illustrate how to use coords and dims within PyMC 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 PyMC 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  at.mean(atts_star), dims="team")
defs = pm.Deterministic('defs', defs_star  at.mean(defs_star), dims="team")
home_theta = at.exp(intercept + home + atts[home_idx] + defs[away_idx])
away_theta = at.exp(intercept + atts[away_idx] + defs[home_idx])
# likelihood of observed data
pm.Poisson(
'points',
mu=at.stack((home_theta, away_theta)).T,
observed=df_rugby[["home_score", "away_score"]],
dims=("match", "field")
)
rugby_idata = pm.sample(1000, tune=1000, cores=4)
We have now defined the shapes of some of our variables (atts_star
and defs_star
) and annotated the dimensions of some others. This is convenient and makes the code easier to understand, but we need to remember that in PyMC, dimensions are not inherited like in xarray.
atts
is atts_star
minus a scalar value and it will be an array with the same shape as atts_star
even without the dims
argument. However, we need to annotate the dimension if we want the output to have it.
In PyMC 4.0, the default return type of all sampling functions is InferenceData
:
print(type(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 
County
: each county has its own basement, intercept:a
, and first floor, slopeb
, effects. Details are in the example notebook 
param
: one ofa
,b

param2
: same as param, used for the covariance matrix because a variable can't have repeated dimensions
We will also use a dimension called obs_id
for the n
observations. In that case however, we don't care about the coordinate values and we are fine with them being an integer id.
In PyMC 4.0, if you annotate the dimensions of a pm.Data
object (also of pm.ConstantData
or pm.MutableData
which are aliases of pm.Data
), that dimension will be automatically created as dimension without coordinate values.
coords = {
"level": ["basement", "floor"],
"county": mn_counties,
"param": ["a", "b"],
"param2": ["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.ConstantData("floor_idx", srrs_mn.floor, dims="obs_id")
county_idx = pm.ConstantData("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. We will have to indicate the dims that correspond to these variables with the idata_kwargs
argument. idata_kwargs
is used to indicate pm.sample
what arguments to pass to pm.to_inference_data
, 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 PyMC 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
.
As we saw in the previous example, pm.Deterministic
also has a dims
argument, but it can only be used to annotate the dimensions of that variable, not to define an array variable from scalar parameters.
with radon_model:
ab_county = pm.Deterministic("ab_county", at.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)
pm.Normal("y", theta, sigma=sigma, observed=srrs_mn.log_radon, dims="obs_id")
Finally we will call pm.sample
and define 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,
idata_kwargs={"dims": {"chol_stds": ["param"], "chol_corr": ["param", "param2"]}}
)
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_true = 2
b_true = 0.4
x = np.linspace(0, 10, 31)
year = np.arange(2022len(x), 2022)
y = a_true + b_true * 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");
with pm.Model() as linreg_model:
# use x_ to avoid overwriting x previously defined as np.ndarray
x_ = pm.MutableData("x", x, dims="year", coords={"year": year})
a = pm.Normal("a", 0, 3)
b = pm.Normal("b", 0, 2)
sigma = pm.HalfNormal("sigma", 2)
pm.Normal("y", a + b * x_, sigma, observed=y, dims="year")
We have now written a model in order to study our quantity of interest y
. We have used everything we have seen so far, the pm.MutableData
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 2022 and 2023.
Here however, as our goal is to generate out of sample predictions, there is an important difference. Instead of providing the coordinates when initializing the model context, we add them in the call to pm.MutableData
. By initializing the coordinates with a pm.MutableData
object we make sure that we can later modify both the length of the year
dimension and its associated coordinate values.
The dimensions defined from coords
passed while initializing the model context can't be modified. Their length is fixed. The coordinate values can be updated, but only for an array of the same length.
with linreg_model:
linreg_idata = pm.sample_prior_predictive(700)
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()
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
. sample_posterior_predictive
already returns an InferenceData, we will use the extend_inferencedata
argument to indicate PyMC to add the new groups inplace.
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:
pm.sample_posterior_predictive(linreg_idata, extend_inferencedata=True)
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 2022 and 2023. 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 and to update the coordinate values of the year
dimension.
Here we will use predictions
and extend_inferencedata
keywords. This will store these new samples in the predictions
group and the modified x
values in predictions_constant_data
and add these groups inplace to our InferenceData object
with linreg_model:
pm.set_data({"x": x[1] + x[1:3]}, coords={"year": [2022, 2023]})
pm.sample_posterior_predictive(linreg_idata, predictions=True, extend_inferencedata=True)
linreg_idata
az.plot_posterior(linreg_idata, group="predictions");
Package versions used to generate this post:
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