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 pltControls
blocked / unblocked paths
d-separation
frontdoor criterion
backdoor criterion
backdoor adjustment
backdoor criterion as d-separation
do-calculus
counterfactuals
Important point: while assessing if A \perp B \mid Z in general requires assessing every state of Z, blocking a path in an analysis only requires stratifying by the observed states of Z (assuming no missing data).
Joint model: P(A, B, C)
One factorization via chain rule: 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
Write the joint probability of the model above.
Factorize the joint probability using the chain rule.
Use the causal graph to determine the conditional independencies and update the joint probability.
Use the causal graph to write the Markov factorization.
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.
Render env
%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: 3.13.2 (main, Feb 5 2025, 18:58:04) [Clang 19.1.6 ]
Platform: Darwin 23.6.0
Processor: arm
Machine: arm64
Packages:
annotated-types==0.7.0
anyio==4.8.0
appnope==0.1.4
argon2-cffi==23.1.0
argon2-cffi-bindings==21.2.0
arrow==1.3.0
asttokens==3.0.0
async-lru==2.0.4
attrs==25.1.0
babel==2.17.0
beautifulsoup4==4.13.3
bleach==6.2.0
certifi==2025.1.31
cffi==1.17.1
cfgv==3.4.0
charset-normalizer==3.4.1
click==8.1.8
comm==0.2.2
contourpy==1.3.1
cycler==0.12.1
debugpy==1.8.13
decorator==5.2.1
defusedxml==0.7.1
distlib==0.3.9
executing==2.2.0
fastjsonschema==2.21.1
filelock==3.17.0
fonttools==4.56.0
fqdn==1.5.1
h11==0.14.0
httpcore==1.0.7
httpx==0.28.1
identify==2.6.8
idna==3.10
importlib_metadata==8.6.1
ipykernel==6.29.5
ipython==9.0.1
ipython_pygments_lexers==1.1.1
ipywidgets==8.1.5
isoduration==20.11.0
jax==0.5.2
jaxlib==0.5.1
jedi==0.19.2
Jinja2==3.1.6
joblib==1.4.2
json5==0.10.0
jsonpointer==3.0.0
jsonschema==4.23.0
jsonschema-specifications==2024.10.1
jupyter-cache==1.0.1
jupyter-events==0.12.0
jupyter-lsp==2.2.5
jupyter_client==8.6.3
jupyter_core==5.7.2
jupyter_server==2.15.0
jupyter_server_terminals==0.5.3
jupyterlab==4.3.5
jupyterlab_pygments==0.3.0
jupyterlab_server==2.27.3
jupyterlab_widgets==3.0.13
kiwisolver==1.4.8
MarkupSafe==3.0.2
matplotlib==3.10.1
matplotlib-inline==0.1.7
memo-lang==1.1.0
mistune==3.1.2
ml_dtypes==0.5.1
nbclient==0.10.2
nbconvert==7.16.6
nbformat==5.10.4
nest-asyncio==1.6.0
networkx==3.4.2
nodeenv==1.9.1
notebook_shim==0.2.4
numpy==2.2.3
opt_einsum==3.4.0
optype==0.9.1
overrides==7.7.0
packaging==24.2
pandas==2.2.3
pandas-stubs==2.2.3.241126
pandocfilters==1.5.1
parso==0.8.4
pexpect==4.9.0
pillow==11.1.0
platformdirs==4.3.6
plotly==5.24.1
pre_commit==4.1.0
prometheus_client==0.21.1
prompt_toolkit==3.0.50
psutil==7.0.0
ptyprocess==0.7.0
pure_eval==0.2.3
pycparser==2.22
pydantic==2.10.6
pydantic_core==2.27.2
Pygments==2.19.1
pygraphviz==1.14
pyparsing==3.2.1
python-dateutil==2.9.0.post0
python-dotenv==1.0.1
python-json-logger==3.3.0
pytz==2025.1
PyYAML==6.0.2
pyzmq==26.2.1
referencing==0.36.2
requests==2.32.3
rfc3339-validator==0.1.4
rfc3986-validator==0.1.1
rpds-py==0.23.1
ruff==0.9.10
scikit-learn==1.6.1
scipy==1.15.2
scipy-stubs==1.15.2.0
seaborn==0.13.2
Send2Trash==1.8.3
setuptools==75.8.2
six==1.17.0
sniffio==1.3.1
soupsieve==2.6
SQLAlchemy==2.0.38
stack-data==0.6.3
tabulate==0.9.0
tenacity==9.0.0
terminado==0.18.1
threadpoolctl==3.5.0
tinycss2==1.4.0
toml==0.10.2
tornado==6.4.2
tqdm==4.67.1
traitlets==5.14.3
types-python-dateutil==2.9.0.20241206
types-pytz==2025.1.0.20250204
typing_extensions==4.12.2
tzdata==2025.1
uri-template==1.3.0
urllib3==2.3.0
virtualenv==20.29.3
wcwidth==0.2.13
webcolors==24.11.1
webencodings==0.5.1
websocket-client==1.8.0
widgetsnbextension==4.0.13
xarray==2025.1.2
zipp==3.21.0