Controls

import jax
import jax.numpy as jnp
from memo import memo
from memo import domain as product
from enum import IntEnum
from matplotlib import pyplot as plt
TipFurther reading

Consider the pipe motif, from the Conditional Dependence tutorial:

The joint model can be written as: P(A, B, C)

We can factorize the joint model via chain rule. One factorization of P(A, B, C) is: P(A, B, C) = P(A) \; P(B \mid A) \; P(C \mid A, B) NB this factorization is true in general, it does not depend on the causal structure.

Based on the causal structure, the a priori (in)dependencies are:

A \not\perp B \\ B \not\perp C \\ A \not\perp C \\

and the conditional (in)dependencies are: A \not\perp B \mid C \\ B \not\perp C \mid A \\ A \perp C \mid B \\

Since { A \perp C \mid B }: { P(C \mid A, B) = P(C \mid B) } and thus, P(A, B, C) = P(A) \; P(B \mid A) \; P(C \mid B)

Exercises

  1. Write the joint probability of this model.

  2. For the model in (1), factorize the joint probability using the chain rule.

  3. For the model in (1), use the causal graph to determine the conditional independencies and update the joint probability.

  4. For the model in (1), use the causal graph to write the Markov factorization.

  5. In the model below, you are interested in estimating the causal effect of X on Y. Using the causal model below, find an adjustment set that d-separates the estimand.

  1. Describe the difference between conditioning and intervention.

  2. Draw the causal graph that corresponds do(X) for the model in (5).

  3. For the model in (5), with do(X), what adjustment set d-separates the estimand?

%reset -f
import sys
import platform
import importlib.metadata

print("Python:", sys.version)
print("Platform:", platform.system(), platform.release())
print("Processor:", platform.processor())
print("Machine:", platform.machine())

print("\nPackages:")
for name, version in sorted(
    ((dist.metadata["Name"], dist.version) for dist in importlib.metadata.distributions()),
    key=lambda x: x[0].lower()  # Sort case-insensitively
):
    print(f"{name}=={version}")
Python: 3.14.3 (main, Feb  4 2026, 01:51:49) [Clang 21.1.4 ]
Platform: Darwin 24.6.0
Processor: arm
Machine: arm64

Packages:
altair==6.0.0
annotated-types==0.7.0
anyio==4.12.1
anywidget==0.9.21
appnope==0.1.4
argon2-cffi==25.1.0
argon2-cffi-bindings==25.1.0
arrow==1.4.0
astroid==4.0.4
asttokens==3.0.1
async-lru==2.1.0
attrs==25.4.0
babel==2.18.0
beautifulsoup4==4.14.3
bleach==6.3.0
certifi==2026.1.4
cffi==2.0.0
cfgv==3.5.0
charset-normalizer==3.4.4
click==8.3.1
colour-science==0.4.7
comm==0.2.3
contourpy==1.3.3
cycler==0.12.1
debugpy==1.8.20
decorator==5.2.1
defusedxml==0.7.1
dill==0.4.1
distlib==0.4.0
distro==1.9.0
docutils==0.22.4
executing==2.2.1
fastjsonschema==2.21.2
filelock==3.20.3
fonttools==4.61.1
fqdn==1.5.1
h11==0.16.0
httpcore==1.0.9
httpx==0.28.1
identify==2.6.16
idna==3.11
importlib_metadata==8.7.1
ipykernel==7.2.0
ipython==9.10.0
ipython_pygments_lexers==1.1.1
ipywidgets==8.1.8
isoduration==20.11.0
isort==7.0.0
itsdangerous==2.2.0
jax==0.9.0.1
jaxlib==0.9.0.1
jedi==0.19.2
Jinja2==3.1.6
jiter==0.13.0
joblib==1.5.3
json5==0.13.0
jsonpointer==3.0.0
jsonschema==4.26.0
jsonschema-specifications==2025.9.1
jupyter-cache==1.0.1
jupyter-events==0.12.0
jupyter-lsp==2.3.0
jupyter_client==8.8.0
jupyter_core==5.9.1
jupyter_server==2.17.0
jupyter_server_terminals==0.5.4
jupyterlab==4.5.3
jupyterlab_pygments==0.3.0
jupyterlab_server==2.28.0
jupyterlab_widgets==3.0.16
kiwisolver==1.4.9
lark==1.3.1
marimo==0.19.9
Markdown==3.10.1
MarkupSafe==3.0.3
matplotlib==3.10.8
matplotlib-inline==0.2.1
mccabe==0.7.0
memo-lang==1.2.9
mistune==3.2.0
ml_dtypes==0.5.4
msgspec==0.20.0
narwhals==2.16.0
nbclient==0.10.4
nbconvert==7.17.0
nbformat==5.10.4
nest-asyncio==1.6.0
networkx==3.6.1
nodeenv==1.10.0
notebook_shim==0.2.4
numpy==2.4.2
numpy-typing-compat==20251206.2.4
openai==2.17.0
opt_einsum==3.4.0
optype==0.15.0
packaging==26.0
pandas==3.0.0
pandas-stubs==3.0.0.260204
pandocfilters==1.5.1
parso==0.8.5
pexpect==4.9.0
pillow==12.1.0
platformdirs==4.5.1
plotly==5.24.1
pre_commit==4.5.1
prometheus_client==0.24.1
prompt_toolkit==3.0.52
psutil==7.2.2
psygnal==0.15.1
ptyprocess==0.7.0
pure_eval==0.2.3
pycparser==3.0
pydantic==2.12.5
pydantic_core==2.41.5
Pygments==2.19.2
pygraphviz==1.14
pylint==4.0.4
pymdown-extensions==10.20.1
pyparsing==3.3.2
python-dateutil==2.9.0.post0
python-dotenv==1.2.1
python-json-logger==4.0.0
PyYAML==6.0.3
pyzmq==27.1.0
referencing==0.37.0
requests==2.32.5
rfc3339-validator==0.1.4
rfc3986-validator==0.1.1
rfc3987-syntax==1.1.0
rpds-py==0.30.0
ruff==0.15.0
scikit-learn==1.8.0
scipy==1.17.0
scipy-stubs==1.17.0.2
seaborn==0.13.2
Send2Trash==2.1.0
setuptools==81.0.0
six==1.17.0
sniffio==1.3.1
soupsieve==2.8.3
SQLAlchemy==2.0.46
stack-data==0.6.3
starlette==0.52.1
tabulate==0.9.0
tenacity==9.1.4
terminado==0.18.1
threadpoolctl==3.6.0
tinycss2==1.4.0
toml==0.10.2
tomlkit==0.14.0
tornado==6.5.4
tqdm==4.67.3
traitlets==5.14.3
typing-inspection==0.4.2
typing_extensions==4.15.0
tzdata==2025.3
uri-template==1.3.0
urllib3==2.6.3
uvicorn==0.40.0
virtualenv==20.36.1
wcwidth==0.6.0
webcolors==25.10.0
webencodings==0.5.1
websocket-client==1.9.0
websockets==16.0
widgetsnbextension==4.0.15
xarray==2026.1.0
zipp==3.23.0

References

Cinelli, Carlos, Forney, Andrew, & Pearl, Judea. (2022). A Crash Course in Good and Bad Controls. Sociological Methods & Research, 00491241221099552. https://doi.org/10.1177/00491241221099552
McElreath, Richard. (2016). Statistical rethinking: A Bayesian course with examples in R and Stan. CRC Press/Taylor & Francis Group.