Causal Discovery from
Observational Data
Brady Nealcausalcourse.com

What if we don’t have the causal graph?

Brady Neal

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What if we don’t have the causal graph?
Causal discovery: data

Brady Nealcausal graph

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What if we don’t have the causal graph?
Causal discovery: datacausal graph

Structure identification: identifying the causal graph

Brady Neal

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Independence-Based Causal Discovery
Assumptions
Markov Equivalence and Main Theorem
The PC Algorithm
Can We Do Better?

Semi-Parametric Causal Discovery
No Identifiability Without Parametric Assumptions
Linear Non-Gaussian Setting
Nonlinear Additive Noise Setting

Brady Neal

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Independence-Based Causal Discovery
Assumptions
Markov Equivalence and Main Theorem
The PC Algorithm
Can We Do Better?

Semi-Parametric Causal Discovery
No Identifiability Without Parametric Assumptions
Linear Non-Gaussian Setting
Nonlinear Additive Noise Setting

Brady Neal

Assumptions

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Faithfulness Assumption

Brady Neal

Assumptions

5 / 45

Faithfulness Assumption
Recall the Markov assumption: X ??G Y | Z =) X ??P Y | Z<latexit sha1_base64="zlPcHd38t40hUna9e0GNDw+9jeg=">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</latexit>

Brady Neal

Assumptions

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Faithfulness Assumption
Recall the Markov assumption: X ??G Y | Z =) X ??P Y | Z<latexit sha1_base64="zlPcHd38t40hUna9e0GNDw+9jeg=">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</latexit>

Causal graph

Brady Neal

Assumptions

Data

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Faithfulness Assumption
Recall the Markov assumption: X ??G Y | Z =) X ??P Y | Z<latexit sha1_base64="zlPcHd38t40hUna9e0GNDw+9jeg=">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</latexit>

Brady Neal

Causal graph

Data

Causal graph

Data

Assumptions

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Faithfulness Assumption
Recall the Markov assumption: X ??G Y | Z =) X ??P Y | Z<latexit sha1_base64="zlPcHd38t40hUna9e0GNDw+9jeg=">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</latexit>

Faithfulness:

Brady Neal

Causal graph

Data

Causal graphData<latexit sha1_base64="hcvMBKxaM6ynzjkNMhCc6VBJEEg=">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</latexit>

X ??G Y | Z (= X ??P Y | Z

Assumptions

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Violation of Faithfulness

Brady Neal

Assumptions

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Violation of Faithfulness
Faithfulness:

Brady Neal<latexit sha1_base64="hcvMBKxaM6ynzjkNMhCc6VBJEEg=">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</latexit>

X ??G Y | Z (= X ??P Y | Z

Assumptions

6 / 45

Violation of Faithfulness
Faithfulness:<latexit sha1_base64="hcvMBKxaM6ynzjkNMhCc6VBJEEg=">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</latexit>

X ??G Y | Z (= X ??P Y | Z<latexit sha1_base64="TZPsAeWD3LFNlhH1rFZ4zmsmB1o=">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</latexit>

A

B

C

D

Brady Neal

Assumptions

6 / 45

Violation of Faithfulness
Faithfulness:<latexit sha1_base64="hcvMBKxaM6ynzjkNMhCc6VBJEEg=">AAAUs3icrVhLb9tGEN6kr8R9OalvvbBxCvQgK5KbwLkYDWAnLYoGcAC/kigwJJKyCPNVkrLqCPo5vfaP9Noe+m/6zbdLUdSLcmAJIlezM9/Mzmu57MS+l2aNxn+3bn/08Seffnbn7trnX3z51dfr9+4fp1E/sd0jO/Kj5LTTTl3fC92jzMt89zRO3HbQ8d2TzsWezJ9cuknqReFhdhW774L2eeh1PbudgXS2/tOp1fJCx41dXMLs7GfrtdUKPMd6A3oAC1ync2VNMbXiMdfZ+maj3uDHmh00zWBTmc9BdO/uX6qlHBUpW/VVoFwVqgxjX7VViu9b1VQNFYP2Tg1BSzDyOO+qkVqDbB9cLjjaoF7geo5/bw01xH/BTCltQ4uPXwJJS31veByMu6Tqu+i3JngX6RgSW2y8wr1jMANQM9UDtUou51xdroNvULHqDGt4ytV6WElMivjBLq25i7uP/xlWKNcrcLoYOZBKMLJB80HVFNGR4K49L77pMRJt8rkYidXL7F5mtcxH+F4Aq41xSkvFVku9MHEJqdmlrcLjM6aLEf/ACrV9y7C64zV4jLtehfAegnKh3mNURl6mczK7FnNlBnk0R5fwZ+QIwZ2CEjHzPfjPA0c5V21gio42I3nOtcTMoLpB/pXzeTbHiOcW7UkZNYv55lFPbCqrsDu308e9Q+wE8kPM9ahHfFGDPS6xE2Zv7sUauRP8G/CfTRvtKXqdlSnxrGFNkl014EaY8cZY2hPan7lF2rohaJb5bqnfqN9F3Gv0iuR1DbIRLdRZnFJXaGK/i74ifglwFap4NqfUqCOgHT1mywVohT6NJPoeYRUp68c3qEPcfXotJkqN2sU/A44Drkki2qfuAcYO9Yi0hV5XVzvGjlFJq/jGY3ea1Wox/6R2dWec1ipyISvVMlm5S00N9disWCIgdk96zTbdUPfklKsbTVnVof1F15AcLdsnUkV8yrFymI09SpVxxYvz1zlvLduskxpjKb4/B8cupbqGnrI2Y6NpbaL+9kyVRIyHzXtgckLXu2gbTM0MQQ+Mp3zw6TqXPUlmfgdmmznVovcm55chPjfy+W4gHX9I6nK5Y1ZWWTbBaDieWS7vceTSP3okEdd4MX0dcy3fjX/WxMyq2DeDF84gSsZmBmlVdIe5MX/lh2oTvWATWVX2p0RaKAmfTmKMH445HwJXeC+oJQRFKnw4nh9VxmAflJHxWJdzeR0WNnjsfA45WybzNIeuqGyKQ6p6uVbJy3l5o+nLZZ1xvpdlNX25rFAumZue6QcpsU4r5DLGQSSCiSw9rJCKuCdourb59Qr2hdxr+yYvEmPhSYVkwB6jVxWZqnxZaV/GLpgYn4jMmw/w4CX734hPQNf1YyGbXcubhdzVB/m0kB9c07OFZHBN/xaS7yskXe4N3pgmMq8qa0q4Diq5dD5V7QkudpP8+UVXW43XDrwoe7ZjOpNl9hvJCr077rIH1Fbqz7P7jlRys7KO++y+HfahhJqdJbVzdGNoRQT7K6wvZb4X58tV6mue3Op5s8c4zOuO+Yx+AimeQvLTQzqxP3nEL04WQz4FDsxOlj8PbjEDdAx/QS98iWesETX9iOe7Jq4vSrveqqhyzuqbfjmJuw3kHexX+9zXro8bm9PQVunMVNbxHBry38icLgtu35wBFiFdR+98ZJ11zsxZZFqTftrsgUufwK4qEIs8no+3PL7zkfWbkogngOIcdhPIk2fM1fCKnFn8RqDqPcXi03OMexdXnx09Nbz7zDqfnd1Fj9EnfP30tDfzrkLydxtVIbk7fHB/Sl8Zf/r0e87Tax8cBY8+xWSU1rRJxEXn6JgymXlaTHnmLE7W9dJJWeNN25ZO5FVg8HZNP3LNbrF2tr7ZnH77Njs43q43n9Qbrx5vPntq3szdUd+qB+oH+GlHPUNlHsCvtvpT/a3+Uf9uPNl4u9HZcDTr7VtG5htV+mwE/wPzuQuZ</latexit>

X ??G Y | Z (= X ??P Y | Z<latexit sha1_base64="TZPsAeWD3LFNlhH1rFZ4zmsmB1o=">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</latexit>

A

B

C

D

Brady Neal

Assumptions

6 / 45

Violation of Faithfulness
Faithfulness:<latexit sha1_base64="hcvMBKxaM6ynzjkNMhCc6VBJEEg=">AAAUs3icrVhLb9tGEN6kr8R9OalvvbBxCvQgK5KbwLkYDWAnLYoGcAC/kigwJJKyCPNVkrLqCPo5vfaP9Noe+m/6zbdLUdSLcmAJIlezM9/Mzmu57MS+l2aNxn+3bn/08Seffnbn7trnX3z51dfr9+4fp1E/sd0jO/Kj5LTTTl3fC92jzMt89zRO3HbQ8d2TzsWezJ9cuknqReFhdhW774L2eeh1PbudgXS2/tOp1fJCx41dXMLs7GfrtdUKPMd6A3oAC1ync2VNMbXiMdfZ+maj3uDHmh00zWBTmc9BdO/uX6qlHBUpW/VVoFwVqgxjX7VViu9b1VQNFYP2Tg1BSzDyOO+qkVqDbB9cLjjaoF7geo5/bw01xH/BTCltQ4uPXwJJS31veByMu6Tqu+i3JngX6RgSW2y8wr1jMANQM9UDtUou51xdroNvULHqDGt4ytV6WElMivjBLq25i7uP/xlWKNcrcLoYOZBKMLJB80HVFNGR4K49L77pMRJt8rkYidXL7F5mtcxH+F4Aq41xSkvFVku9MHEJqdmlrcLjM6aLEf/ACrV9y7C64zV4jLtehfAegnKh3mNURl6mczK7FnNlBnk0R5fwZ+QIwZ2CEjHzPfjPA0c5V21gio42I3nOtcTMoLpB/pXzeTbHiOcW7UkZNYv55lFPbCqrsDu308e9Q+wE8kPM9ahHfFGDPS6xE2Zv7sUauRP8G/CfTRvtKXqdlSnxrGFNkl014EaY8cZY2hPan7lF2rohaJb5bqnfqN9F3Gv0iuR1DbIRLdRZnFJXaGK/i74ifglwFap4NqfUqCOgHT1mywVohT6NJPoeYRUp68c3qEPcfXotJkqN2sU/A44Drkki2qfuAcYO9Yi0hV5XVzvGjlFJq/jGY3ea1Wox/6R2dWec1ipyISvVMlm5S00N9disWCIgdk96zTbdUPfklKsbTVnVof1F15AcLdsnUkV8yrFymI09SpVxxYvz1zlvLduskxpjKb4/B8cupbqGnrI2Y6NpbaL+9kyVRIyHzXtgckLXu2gbTM0MQQ+Mp3zw6TqXPUlmfgdmmznVovcm55chPjfy+W4gHX9I6nK5Y1ZWWTbBaDieWS7vceTSP3okEdd4MX0dcy3fjX/WxMyq2DeDF84gSsZmBmlVdIe5MX/lh2oTvWATWVX2p0RaKAmfTmKMH445HwJXeC+oJQRFKnw4nh9VxmAflJHxWJdzeR0WNnjsfA45WybzNIeuqGyKQ6p6uVbJy3l5o+nLZZ1xvpdlNX25rFAumZue6QcpsU4r5DLGQSSCiSw9rJCKuCdourb59Qr2hdxr+yYvEmPhSYVkwB6jVxWZqnxZaV/GLpgYn4jMmw/w4CX734hPQNf1YyGbXcubhdzVB/m0kB9c07OFZHBN/xaS7yskXe4N3pgmMq8qa0q4Diq5dD5V7QkudpP8+UVXW43XDrwoe7ZjOpNl9hvJCr077rIH1Fbqz7P7jlRys7KO++y+HfahhJqdJbVzdGNoRQT7K6wvZb4X58tV6mue3Op5s8c4zOuO+Yx+AimeQvLTQzqxP3nEL04WQz4FDsxOlj8PbjEDdAx/QS98iWesETX9iOe7Jq4vSrveqqhyzuqbfjmJuw3kHexX+9zXro8bm9PQVunMVNbxHBry38icLgtu35wBFiFdR+98ZJ11zsxZZFqTftrsgUufwK4qEIs8no+3PL7zkfWbkogngOIcdhPIk2fM1fCKnFn8RqDqPcXi03OMexdXnx09Nbz7zDqfnd1Fj9EnfP30tDfzrkLydxtVIbk7fHB/Sl8Zf/r0e87Tax8cBY8+xWSU1rRJxEXn6JgymXlaTHnmLE7W9dJJWeNN25ZO5FVg8HZNP3LNbrF2tr7ZnH77Njs43q43n9Qbrx5vPntq3szdUd+qB+oH+GlHPUNlHsCvtvpT/a3+Uf9uPNl4u9HZcDTr7VtG5htV+mwE/wPzuQuZ</latexit>

X ??G Y | Z (= X ??P Y | Z

A ?? D<latexit sha1_base64="28pEVlnQhwmLIAdGi6E7jp1a97o=">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</latexit><latexit sha1_base64="TZPsAeWD3LFNlhH1rFZ4zmsmB1o=">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</latexit>

A

B

C

D

Brady Neal

Assumptions

6 / 45

Violation of Faithfulness
Faithfulness:<latexit sha1_base64="hcvMBKxaM6ynzjkNMhCc6VBJEEg=">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</latexit>

X ??G Y | Z (= X ??P Y | Z

A ?? D
but A and D aren’t d-separated<latexit sha1_base64="28pEVlnQhwmLIAdGi6E7jp1a97o=">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</latexit><latexit sha1_base64="TZPsAeWD3LFNlhH1rFZ4zmsmB1o=">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</latexit>

A

B

C

D

Brady Neal

Assumptions

6 / 45

Violation of Faithfulness
Faithfulness:<latexit sha1_base64="hcvMBKxaM6ynzjkNMhCc6VBJEEg=">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</latexit>

X ??G Y | Z (= X ??P Y | Z

A ?? D
but A and D aren’t d-separated<latexit sha1_base64="28pEVlnQhwmLIAdGi6E7jp1a97o=">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</latexit><latexit sha1_base64="ihyvfKMQteuW8f8Iwqmc0pqYghM=">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</latexit>

A

↵
B

C

D

Brady Neal

Assumptions

6 / 45

Violation of Faithfulness
Faithfulness:<latexit sha1_base64="hcvMBKxaM6ynzjkNMhCc6VBJEEg=">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</latexit>

X ??G Y | Z (= X ??P Y | Z<latexit sha1_base64="GPZAItzzBzV43Hq4LUOy6ep/upA=">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</latexit>

A ?? D
but A and D aren’t d-separated<latexit sha1_base64="28pEVlnQhwmLIAdGi6E7jp1a97o=">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</latexit><latexit sha1_base64="ihyvfKMQteuW8f8Iwqmc0pqYghM=">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</latexit>

A

B := ↵A
C := A

D := B + C

↵
B

C

D

Brady Neal

Assumptions

6 / 45

Violation of Faithfulness
Faithfulness:<latexit sha1_base64="hcvMBKxaM6ynzjkNMhCc6VBJEEg=">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</latexit>

X ??G Y | Z (= X ??P Y | Z<latexit sha1_base64="GPZAItzzBzV43Hq4LUOy6ep/upA=">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</latexit>

A ?? D
but A and D aren’t d-separated<latexit sha1_base64="28pEVlnQhwmLIAdGi6E7jp1a97o=">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</latexit><latexit sha1_base64="ihyvfKMQteuW8f8Iwqmc0pqYghM=">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</latexit>

A

B := ↵A
C := A

D := B + C↵<latexit sha1_base64="IyMNvAxKZ/+ioFHKajhqN/Dpd94=">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</latexit>

BC

D = (↵ +)A

D

Brady Neal

Assumptions

6 / 45

Violation of Faithfulness
Faithfulness:<latexit sha1_base64="hcvMBKxaM6ynzjkNMhCc6VBJEEg=">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</latexit>

X ??G Y | Z (= X ??P Y | Z<latexit sha1_base64="GPZAItzzBzV43Hq4LUOy6ep/upA=">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</latexit>

A ?? D
but A and D aren’t d-separated<latexit sha1_base64="28pEVlnQhwmLIAdGi6E7jp1a97o=">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</latexit><latexit sha1_base64="ihyvfKMQteuW8f8Iwqmc0pqYghM=">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</latexit>

A

B := ↵A
C := A

D := B + C↵<latexit sha1_base64="IyMNvAxKZ/+ioFHKajhqN/Dpd94=">AAAUmXicrVjbbttGEN2kt8S9Oanf8sJGCZCisiq5KZwXAQnsBEHQAA7qWxsFBkWuJEK8laSsOoK+oP9SoK/tl/RvOnN2KZK6kFJgCSJXszNnZue2XHZD14mTZvO/Gzc/+viTTz+7dXvr8y++/Orr7Tt3T+NgFFnyxArcIDrvmrF0HV+eJE7iyvMwkqbXdeVZd3jA82eXMoqdwD9OrkL5zjP7vtNzLDMh0sX2w0OjbTzqmG44MI1OVyam8b3R6ZueR39t6Sbmd8Yz42K71mw08TEWBy09qAn9OQru3P5LdIQtAmGJkfCEFL5IaOwKU8T0fStaoilCor0TE6JFNHIwL8VUbJHsiLgkcZhEHdK1T//eaqpP/xkzhrRFWlz6RSRpiIeax6ZxD1R1Z/1GjneVjgmw2cYrunc1pkfURAyIWiWXcq4v16WvV7HqhNbwBKt1aCUhKOwHq7DmHt1d+p/QCvl6RZySRjZJRTSyiOYSVVFYR0R35Xn2zQCRMMEnacRWl9ldZjXPB/QdEpZJ4xiWsq2GeKHj4kOzhK3M4yKmqxH/oBUq+8qwerM1OIi7WgXzHhNlKN7TqIhcpjOfXau5Eo08XaKL+RNw+MQdEyVA5jvkP4c4irlqESbrMBHJPtYSIoMaGvkV5tNsDimeu7AnRtQM5JsDPaGurMzu1E6X7l1gRyQ/obkB9LAv6mSPBHaE7E29WAd3RP/G+GfBRmuO3kBlcjzrtCbOrjrhBjTjzLCUJ5Q/U4uUdROiGfq7K36Gfklxr8MrnNd1kg1gocriGLp8Hfs29RX2i0dXprJnU0odOjzYMUC2DImW6VNIrO8HWkWM+nE16oTuLrwWAqUO7eyfMcYe1sQRHUH3mMY29LC0Qb2uIfa1HdOCVvaNg+60qNVA/nHtqs44r5XlfFSqobOyDU1N8VivmCPAdue9ZuluqHpyjNVN56zqwv6sa3COFu1jqSw+xVjZyMYBpIq47MXl61y2lj3USR2xZN/3iaMNqZ6mx6jNUGvaytXfga6SAPGwcPd0Tqh6Z23juZkJ0T3tKZf4VJ3znsQzvxOmiZzqwHv5+TLE51o+3Q24409ALZc7RWUVZSMaTWYz5fIORhL+USOOuMIL4esQa/l29jNyM+tiXw+ev4DIGZtopHXRbeTG8pUfixr1ghplVdGfHGmmRHg6CWn8YMb5gHCZdwgtPlG4wiez+WllDA6JMtUe62EurcPMBgedzwZnR2ee4lAVlcxxcFWXa+W8XJY3il4ua8/yvSir6OWyTLlEbjq6H8TAOq+QSxAHlvByWXpcIRVgT1B0ZfOva9jnY68d6byItIVnFZIeeoxaVaCr8nWlfQm6YKR9wjK/fYAHL9H/pngC2tSPmWyykTczuasP8mkmP97Qs5mkt6F/M8n3FZISe4Mzo7HMm8qaYq6jSi6VT1V7gqTdJH1+UdVWx7VLXuQ929adydD7DWeF2h3b6AH1tfrz4r7DldyqrOMRum8XfSiCZrukdk6uDS2L4GiN9cXI9+x8uU59LZNbP28OEIdl3TGdUU8g2VNIenqIc/uTA/zsZDHBU+BY72Tp8+AuMkDF8CX1wtf0jDWFph/p+a5F1xeFXW9dVD5njXS/zOPuEfI+7VeH2Nc2xw31aWi3cGYq6nhOGtLfVJ8uM25XnwFWIW2idzmyyjp74Swyr0k9bQ6IS53ArioQszxejlce3+XI6k1JgBNAdg67DuT8GXM9vCxnVr8RqHpPsfr0HNK9R1cXHT3WvIfIOhedXVKPUSd89fR0sPCugvN3j6qCc3dy/+6cviL+/Om3j9PriDgyHnWKSSCtaHnEVefoEDKJflqMcebMTtaNwklZ4c3bFufyytN4bd2PpN4tti62a635t2+Lg9O9RuunRvPN49rTJ/rN3C1xT9wXj8hP++IpVeYR+dUSf4q/xT/i3517O892Xu68Uqw3b2iZb0Ths/PL/8W9/6E=</latexit>

BC

D = (↵ +)A

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Brady Neal

Assumptions

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Causal Sufficiency and Acyclicity

Brady Neal

Assumptions

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Causal Sufficiency and Acyclicity
Causal Sufficiency: there are no unobserved confounders of any of thevariables in the graph.

Brady Neal

Assumptions

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Causal Sufficiency and Acyclicity
Causal Sufficiency: there are no unobserved confounders of any of thevariables in the graph.
Acyclicity: still assuming there are no cycles in the graph.

Brady Neal

Assumptions

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Causal Sufficiency and Acyclicity
Causal Sufficiency: there are no unobserved confounders of any of thevariables in the graph.
Acyclicity: still assuming there are no cycles in the graph.
All assumptions:• Markov assumption• Faithfulness• Causal sufficiency• Acyclicity

Brady Neal

Assumptions

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Causal Sufficiency and Acyclicity
Causal Sufficiency: there are no unobserved confounders of any of thevariables in the graph.
Acyclicity: still assuming there are no cycles in the graph.
All assumptions:• Markov assumption• Faithfulness• Causal sufficiency• Acyclicity

Brady Neal

Assumptions

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Question:
Why is the Markov assumption (plus causal
sufficiency and acyclicity) not enough forlearning causal graphs from data?

Independence-Based Causal Discovery
Assumptions
Markov Equivalence and Main Theorem
The PC Algorithm
Can We Do Better?

Semi-Parametric Causal Discovery
No Identifiability Without Parametric Assumptions
Linear Non-Gaussian Setting
Nonlinear Additive Noise Setting

Brady Neal

Markov Equivalence and Main Theorem

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Markov Equivalence and Main Theorem

X3

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Chains and Forks Encode Same Independencies<latexit sha1_base64="5wIKTKq+kukbfiQxkoSFOlUFIW0=">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</latexit>

X2<latexit sha1_base64="y4Hz1UpYTd2lIeNIvpwZN4oVH9k=">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</latexit><latexit sha1_base64="ma7LRbptppcWKpltbdWvSQM2GIg=">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</latexit>

X1

X2

X3

X1

X2

X3
X1

Markov:

Brady Neal

X3

X1 ?? X3 | X2
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Markov Equivalence and Main Theorem

10 / 45

Chains and Forks Encode Same Independencies<latexit sha1_base64="5wIKTKq+kukbfiQxkoSFOlUFIW0=">AAAVl3icrVjbbttGEF2lt8S9OSn8UPSFjR0gBmRVklu4QGA0gJMgKJoiAZzEqW0YFLmSCPFWkrLqEPqA/kz72n5K/6YzZ5ciqQspB5Egcjk7c2Z2bstVL3SdOGm3/2vc+ODDjz7+5OatjU8/+/yLLzdv33kVB+PIki+twA2ik54ZS9fx5cvESVx5EkbS9HqufN0bHfH860sZxU7gHydXoTz3zIHv9B3LTIh0cbtx96wnB46fJs7obehYyTiS0w1Df+4Zvwa2jGfPZz49nsaJmchz4/5Jd9cwE+N+u9neNdKdk4vuzvTBKt7OrnHak24wMVzZT4zDoG+cdM8h1qkQ25+JRc5gWJbbZ7mCrY/tQdHW0EyGykhJEzDhQcXs/q4GO5O+XXLHxeZ2u9XGx1gcdPRgW+jP8+D2rb/EmbBFICwxFp6QwhcJjV1hipi+p6Ij2iIk2rlIiRbRyMG8FFOxQbJj4pLEYRJ1RNcBPZ1qqk/PjBlD2iItLv0ikjTEPc1j07gPqrqzfqPAu0pHCmy28YruPY3pETURQ6LWyWWc68v16OvVrDqhNfyI1Tq0khAU9oNVWnOf7i49J7RCvl4Rp6SRTVIRjSyiuURVFNYR0V15nn0zRCRM8EkasdVVdldZzfMBfUeEZdI4hqVsqyGe6Lj40CxhK/O4iOlqxD9ohcq+Kqz+bA0O4q5WwbzHRBmJtzQqI1fpLGbXaq5EI0+X6GL+BBw+ccdECZD5DvnPIY5yrlqEyTpMRHKAtYTIoJZG/hnzWTaHFM892BMjagbyzYGeUFdWbndmp0v3HrAjkk9pbgg97Ism2SOBHSF7My82wR3R0wRPFmy05ugtVCbHs0lr4uxqEm5AM84MS3lC+TOzSFmXEs3Q3z3xC/RLinsTXuG8bpJsAAtVFsfQ5evYH1JfYb94dGUqezajNKHDgx1DZMuIaLk+hcT6vqNVxKgfV6OmdHfhtRAoTWhn/0ww9rAmjugYuic0tqGHpQ3qdS1xoO2YlrSybxx0p0WtBvKPa1d1xnmtLOejUg2dlYfQ1Bbf6xVzBNjuotcs3Q1VT46xuumcVT3Yn3cNztGyfSyVx6ccKxvZOIRUGZe9uHydy9bSRZ00EUv2/YA4DiHV1/QYtRlqTRuF+jvSVRIgHhbuns4JVe+sbTI3kxLd055yiU/VOe9JPPM7YZrIqTN4rzhfhfhYy2e7AXf8FNRquVeorLJsRKN0NlMt72Ak4R814ogrvBC+DrGWb2c/ozCzLvb7wfMXEDljE420LrqN3Fi+8mOxTb1gm7Kq7E+ONFMivJ2ENN6Zce4QLvOOoMUnCld4Opuf1sbgEVGm2mN9zGV1mNvgoPPZ4DzTmac4VEUlcxxc1dVaOS+X5Y2iV8vas3wvyyp6tSxTLpGbju4HMbBOauQSxIElvEKWHtdIBdgTFF3Z/GYN+3zstWOdF5G28HWNpIceo1YV6Kp8Vmtfgi4YaZ+wzG/v4MFL9L8p3oCu68dcNrmWN3O5q3fyaS4/uaZnc0nvmv7NJd/WSErsDc6MxjIvamuKuZ7Xcql8qtsTJO0m2fuLqrYmrj3yIu/Ztu5Mht5vOCvU7niIHtBcqz8v7jtcyZ3aOh6j+/bQhyJotitq5+V7Q8sjOF5jfTHyPT9frlNfy+TWz5sjxGFZd8xm1BtI/haSnR7iwv7kAD8/WaR4C5zonSx7H9xDBqgYPqVe+IzesabQtE/vdx26Pinteuui8jlrrPtlEbdLyAe0Xz3CvnZ93FCfhvZKZ6ayjsekIftN9eky53b1GWAV0nX0LkdWWWcvnEXmNam3zSFxqRPYVQ1insfL8arjuxxZ/VMS4ASQn8PeB3LxjLkeXp4zq/8RqPufYvXpOaR7n64uOnqseR8h61x0dkk9Rp3w1dvT0cJ/FZy/XaoKzt307p05fWX8+dPvAKfXMXHkPOoUk0Ba0YqIq87RIWQS/bYY48yZn6xbpZOywpu3LS7klafxDnU/knq32LjY3O7M//u2OHjVbXV+aLVfdLcfdvU/czfFN+KuuE9+OhAPqTKfk1+txp+Nvxv/NP7d+nrrp60nW08V642GlvlKlD5bL/4H2hxEcQ==</latexit>

X2<latexit sha1_base64="y4Hz1UpYTd2lIeNIvpwZN4oVH9k=">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</latexit><latexit sha1_base64="ma7LRbptppcWKpltbdWvSQM2GIg=">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</latexit>

X1

X2

X3

X1

X2

X3
X1

Markov:

X3

X1 ?? X3 | X2
<latexit sha1_base64="kDYAbMLEmlbdSowRTYEz+hGzx+M=">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</latexit>

Minimality: X1 6?? X2 and X2 6?? X3<latexit sha1_base64="FWaKkfiaXf4ywF9UByKO9yw4KFQ=">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</latexit>

Brady Neal<latexit sha1_base64="v5llvLyKb57cohMlUhqh5fZzhMM=">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</latexit>

Markov Equivalence and Main Theorem

10 / 45

Chains and Forks Encode Same Independencies<latexit sha1_base64="5wIKTKq+kukbfiQxkoSFOlUFIW0=">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</latexit>

X2<latexit sha1_base64="y4Hz1UpYTd2lIeNIvpwZN4oVH9k=">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</latexit><latexit sha1_base64="ma7LRbptppcWKpltbdWvSQM2GIg=">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</latexit>

X1

X2

X3

X1

X2

X3
X1

Markov:

X3

X1 ?? X3 | X2
<latexit sha1_base64="kDYAbMLEmlbdSowRTYEz+hGzx+M=">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</latexit>

Minimality: X1 6?? X2 and X2 6?? X3<latexit sha1_base64="FWaKkfiaXf4ywF9UByKO9yw4KFQ=">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</latexit><latexit sha1_base64="v5llvLyKb57cohMlUhqh5fZzhMM=">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</latexit>

Faithfulness: X1 6?? X3<latexit sha1_base64="+cPdfwzFN3rkMtWA0GecChlBa9o=">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</latexit>

Brady Neal

Markov Equivalence and Main Theorem

10 / 45

Chains and Forks Encode Same Independencies
Markov equivalent (all in the same Markov equivalence class)<latexit sha1_base64="5wIKTKq+kukbfiQxkoSFOlUFIW0=">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</latexit>

X2<latexit sha1_base64="y4Hz1UpYTd2lIeNIvpwZN4oVH9k=">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</latexit><latexit sha1_base64="ma7LRbptppcWKpltbdWvSQM2GIg=">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</latexit>

X1

X2

X3

X1

X2

X3
X1

Markov:

X3

X1 ?? X3 | X2
<latexit sha1_base64="kDYAbMLEmlbdSowRTYEz+hGzx+M=">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</latexit>

Minimality: X1 6?? X2 and X2 6?? X3<latexit sha1_base64="FWaKkfiaXf4ywF9UByKO9yw4KFQ=">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</latexit><latexit sha1_base64="v5llvLyKb57cohMlUhqh5fZzhMM=">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</latexit>

Faithfulness: X1 6?? X3<latexit sha1_base64="+cPdfwzFN3rkMtWA0GecChlBa9o=">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</latexit>

Brady Neal

Markov Equivalence and Main Theorem

10 / 45

Immoralities are Special
Markov equivalence class where
X1 ?? X3 | X2 and X1 6?? 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X2

X3

X1

X2

X3

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X1

Brady Neal

Markov Equivalence and Main Theorem

X3

11 / 45

Immoralities are Special
Markov equivalence class where
X1 ?? X3 | X2 and X1 6?? 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X1

X2

X3

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X2

X3

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X1

X3<latexit sha1_base64="5wIKTKq+kukbfiQxkoSFOlUFIW0=">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</latexit>

X2
X2
X1

Brady Neal

Markov Equivalence and Main Theorem

X3

11 / 45

Immoralities are Special
Markov equivalence class where
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X1 ?? X3
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X2

X3

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X1

X3<latexit sha1_base64="5wIKTKq+kukbfiQxkoSFOlUFIW0=">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</latexit>

X2
X2
X1

Brady Neal

Markov Equivalence and Main Theorem

X3

11 / 45

Immoralities are Special
Markov equivalence class where
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X1 ?? X3 and X1 6?? X3 | X2
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X1

X2

X3

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X2

X3

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X1

X3<latexit sha1_base64="5wIKTKq+kukbfiQxkoSFOlUFIW0=">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</latexit>

X2
X2
X1

Brady Neal

Markov Equivalence and Main Theorem

X3

11 / 45

Immoralities are Special
Markov equivalence class where
X1 ?? X3 | X2 and X1 6?? X3<latexit sha1_base64="kDYAbMLEmlbdSowRTYEz+hGzx+M=">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</latexit><latexit sha1_base64="+cPdfwzFN3rkMtWA0GecChlBa9o=">AAAU/XicrVjZbttGFJ2kW+JuTmqgD31hoxTog6xKTgrnxUAAO0FRNIADeGvjwNBCWYS5laTsOoLQjyn6lva139Iv6G/03DNDkdRCSoEliBzduffcO3cbDjuh68RJs/nvrdvvvf/Bhx/dubv28Seffvb5+r37R3EwjLr2YTdwg+ik045t1/Htw8RJXPskjOy213Ht487FrswfX9pR7AT+QXId2q+99rnv9J1uOwHpbP3Lk7OWdeoHieP37NDGxU+sk7NHZ+u1ZqPJjzU7aJlBTZnPfnDv7n/qVPVUoLpqqDxlK18lGLuqrWJ8X6mWaqoQtNdqBFqEkcN5W43VGmSH4LLB0Qb1Atdz/HtlqD7+C2ZM6S60uPhFkLTUN4anh3GfVH0X/VaOd5GOEbHFxmvcOwbTAzVRA1Cr5FLO5eU6+HoVq06whidcrYOVhKSIH7qFNfdxd/E/wQrleg1OG6MepCKMuqC5oGqK6Ihw154X3wwYiTb5bIzE6jK7y6yW+QDfC2C1MY5pqdhqqecmLj4127RVeFzGdDHib1ihtq8Mqz9Zg8O461UI7wEoF+oNRkXkMp357FrMlRjk8Rxdwp+Qwwd3DErAzHfgPwccxVztAlN0tBnJc64lZAY1DPKPnE+zOUQ8N2lPzKhZzDeHekJTWZndqZ0u7h1iR5AfYW5APeKLOuyxiR0xe1Mv1skd4d8V/3VpY3eK3mBlSjzrWJNkVx24AWacCZb2hPZnapG2bgSaZb6b6ifqtxH3Or0ieV2HbEALdRbH1OWb2O+gr4hfPFyFKp5NKXXq8GjHgNlyAVqmTyOJvu+wipj14xrUEe4uvRYSpU7t4p8rjj2uSSI6pO4rjHvUI9IWel1DbRs7xgWt4huH3WlWq8X8k9rVnXFaq8j5rFTLZOUONTXVY7NiiYDYnfda13RD3ZNjrm48ZVWH9mddQ3K0aJ9IZfEpxqrHbBxQqogrXpy/znlr2WKd1BlL8f05OHYo1Tf0mLUZGk1rufrbNVUSMB5d3j2TE7reRdvV1MwIdM94ygWfrnPZk2TmV2C2mVOn9F5+vgzxmZFPdwPp+CNSy+WOWFlF2Qij0WSmXN7hyKZ/9EgirvFC+jrkWr6e/KzczLLYN4PnzyBKxiYGaVn0HnNj/soPVA29oIasKvpTIi2UiE8nIcYPJ5wPgSu8F9TigyIVPprMjytjsAfK2Hisz7m0DjMbHHa+HjlPTeZpDl1RyRSHVHW5VsnLeXmj6eWyvUm+F2U1vVxWKJfMTcf0g5hYJxVyCeMgEl4uSw8qpALuCZqubf55Cft87rVDkxeRsfC4QtJjj9GrCkxVvqi0L2EXjIxPROaXd/DgJfvfmE9Aq/oxk01W8mYmd/1OPs3kr1b0bCbprejfTPJNhaTNvcGZ0ETmZWVNCdd+JZfOp6o9wcZukj6/6Gqr89qBF2XP7pnOZJn9RrJC74477AH1pfrz7L4jldyqrOMhu2+HfSii5l5J7RzeGFoWweES64uZ79n5cpn6mie3fN7sMg7zumM6U7XLBeYksGhf1hzWzP5WvvOt5Z570vNKnNsRHVqUnWVGfO68MgjpE+gmc05nzQ/ovi/wVDfm2h7hibKF6/PCPrssqpzshqZD53G3gLyNHXKPO+nquKE5f20WTmlFHc+gIf2NzXk245ZnycTsi/OQVtE7H1nneW/m9DOtST/fDsClz3zXFYhZ5czHK4/vfGT9bibgmSM7+d0Ecv5UuxxeljOL30FUvRlZfF4Pce/j6rLaYsO7x6xzuZfY6Gr6nYJ+XtudeTsi+buFqpDcHT24P6WviD993j7neXkIjoxHn5sSSmtaHnHRyT2kTGKeT2OecrOzfKNwNtd407bFubzyDN6O6WC22Z/WztZrren3fbODo61G6/tG8+Xj2tMn5l3gHfWVeqC+hZ+21VNU5j78KrH5Q71Vf238vvHnxtuNvzXr7VtG5gtV+Gz88z8+8hOI</latexit>

Markov equivalence class where
X1 ?? X3 and X1 6?? X3 | X2
<latexit 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X1

X2

X3

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X2

X3

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Brady Neal

Markov Equivalence and Main Theorem

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Skeletons
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X2

X1

Brady Neal

Markov Equivalence and Main Theorem

X3

12 / 45

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X1 ?? X3 | X2
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Brady Neal

Markov Equivalence and Main Theorem

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X1 ?? X3 | X2
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X1

X2

X3

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X2

X1

Brady Neal

Markov Equivalence and Main Theorem

X3

12 / 45

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X1 ?? X3 | X2
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X1 6?? X3 | X2
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Brady Neal

Markov Equivalence and Main Theorem

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X1 ?? X3 | X2
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X1 6?? X3 | X2
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Brady Neal

Markov Equivalence and Main Theorem

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Markov Equivalence via Immoral Skeletons

Brady Neal

Markov Equivalence and Main Theorem

13 / 45

Markov Equivalence via Immoral Skeletons
Two important graph qualities that we can use to distinguish graphs:

Brady Neal

Markov Equivalence and Main Theorem

13 / 45

Markov Equivalence via Immoral Skeletons
Two important graph qualities that we can use to distinguish graphs:
1. Immoralities

Brady Neal

Markov Equivalence and Main Theorem

13 / 45

Markov Equivalence via Immoral Skeletons
Two important graph qualities that we can use to distinguish graphs:
1. Immoralities
2. Skeleton

Brady Neal

Markov Equivalence and Main Theorem

13 / 45

Markov Equivalence via Immoral Skeletons
Two important graph qualities that we can use to distinguish graphs:
1. Immoralities
2. Skeleton
Theorem: Two graphs are Markov equivalent if and only if they have the same
skeleton and same immoralities (Verma & Pearl, 1990; Frydenburg, 1990).

Brady Neal

Markov Equivalence and Main Theorem

13 / 45

Markov Equivalence via Immoral Skeletons
Two important graph qualities that we can use to distinguish graphs:
1. Immoralities
2. Skeleton
Theorem: Two graphs are Markov equivalent if and only if they have the same
skeleton and same immoralities (Verma & Pearl, 1990; Frydenburg, 1990).
Essential graph (aka CPDAG): skeleton + immoralities

Brady Neal

Markov Equivalence and Main Theorem

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Question:
What graphs are Markov equivalent to thebasic fork graph?
<latexit sha1_base64="5wIKTKq+kukbfiQxkoSFOlUFIW0=">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</latexit>

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Question:
What graphs are Markov equivalent to thebasic immorality?
<latexit sha1_base64="vcMVYs41M8xH0M+3wQpL73YVA0M=">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</latexit>

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Question:
What graphs is the following graphMarkov equivalent to?
<latexit sha1_base64="9uhgdMllqMmTAPQwzT8Li+1oJ2U=">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</latexit>

A

B

C

D

Question:
What graphs is the following graphMarkov equivalent to?
<latexit sha1_base64="9uhgdMllqMmTAPQwzT8Li+1oJ2U=">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</latexit>

A

B

C

D

Question:
What graphs is the following graphMarkov equivalent to?
<latexit sha1_base64="9uhgdMllqMmTAPQwzT8Li+1oJ2U=">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</latexit><latexit sha1_base64="TxnHGw9Y/fS7PZVHqd3DmS5QMSg=">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</latexit>

C

A

A

B

B

D

C

D

Question:
What graphs is the following graphMarkov equivalent to?
<latexit sha1_base64="9uhgdMllqMmTAPQwzT8Li+1oJ2U=">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</latexit><latexit sha1_base64="TxnHGw9Y/fS7PZVHqd3DmS5QMSg=">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</latexit>

C<latexit sha1_base64="elxMQqUeTYkfY2zB/m6tkkrFvsQ=">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</latexit>

A

A

A

B

B

B

D

C

D

C

D

Question:
What graphs is the following graphMarkov equivalent to?
<latexit sha1_base64="9uhgdMllqMmTAPQwzT8Li+1oJ2U=">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</latexit><latexit sha1_base64="TxnHGw9Y/fS7PZVHqd3DmS5QMSg=">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</latexit>

C<latexit sha1_base64="elxMQqUeTYkfY2zB/m6tkkrFvsQ=">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</latexit><latexit sha1_base64="MasacBT/b8ps4xn3LfGY+Nx78sM=">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</latexit>

A

A

A

A

B

B

B

B

D

C

D

C

D

C

D

Question:
What graphs is the following graphMarkov equivalent to?
<latexit sha1_base64="5iw4e3tdeXSoIX/I0DdOufJlW/M=">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</latexit>

A

B

C

D

Question:
Give a few graphs that thefollowing graph is Markov
equivalent to:<latexit sha1_base64="3DQ7r19kWAZMzfs23R4DTL741SA=">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</latexit>

A

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H

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G

Independence-Based Causal Discovery
Assumptions
Markov Equivalence and Main Theorem
The PC Algorithm
Can We Do Better?

Semi-Parametric Causal Discovery
No Identifiability Without Parametric Assumptions
Linear Non-Gaussian Setting
Nonlinear Additive Noise Setting

Brady Neal

The PC Algorithm

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The PC Algorithm: Overview

Brady Neal

The PC Algorithm

20 / 45

The PC Algorithm: Overview

True graph<latexit sha1_base64="IynnaVgXE4s3SYQQJW2SymTdHKk=">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</latexit>

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Brady Neal

The PC Algorithm

E20 / 45

The PC Algorithm: Overview

True graph<latexit sha1_base64="IynnaVgXE4s3SYQQJW2SymTdHKk=">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</latexit>

A

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Start with complete undirected graph
CD

Brady Neal

The PC Algorithm

E20 / 45

True graph

The PC Algorithm: Overview<latexit sha1_base64="IynnaVgXE4s3SYQQJW2SymTdHKk=">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</latexit>

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Start with complete undirected graphCD

E<latexit sha1_base64="zQnUGvERJzPDl1TcxcZOBr5WXzU=">AAAXbXicrVjbbttGEN3YvSTuzUnQ+qFpwcYOmgCyKrspUiAQkFhyUBRNkQBO4tY2AopcSYR4K0lZdQS99h/7BX0q+gudObsUSV1IOYgEicvZmTP3XS47oevESaPx95W19ffe/+DDq9c2Pvr4k08/27x+42UcDCNLvrACN4iOO2YsXceXLxInceVxGEnT67jyVWfQ4vlX5zKKncA/Si5CeeaZPd/pOpaZEOn19bW/Tjuy5/jjxBm8CR0rGUZycrJ7tmHozx3j18CW8fT+1KfbkzgxE3lm3G3dM8zEuNuoNe4Z453WzuThEsbH94wTsxOcS8OV3cQwmkHXaJ2RzOPlMm2S6Ug3GM3KtJfLHEz1RE6vn2QyB8tlDqd6lEwmdMhCuVAc2r18KEIz6cM1SXQOxsPZuYOSuVY61y6ZO8zNLVV8MA/wuARcz5HPvq1cbt5vnBV1zdq/fE7BcIYUygKF7TwKJk9JplByrze3G/UGPsb8YE8PtoX+PAuuX/tHnApbBMISQ+EJKXyR0NgVpojpeyL2REOERDsTY6JFNHIwL8VEbJDskLgkcZhEHdB/j+5ONNWne8aMIW2RFpd+EUka4o7msWncBVVdWb+R412mYwxstvGCrh2N6RE1EX2iVsmlnKvLdejrVXidkA8/wluHPAlB4ThYBZ+7dHXpPiEP+f+COCWNbJKKaGQRzSWqorCOiK4q8hybPjJhgk/SiK0us7vMap4P6DsgLJPGMSxlWw3xROfFh2YJW5nHRU6XI/5JHir7yrC6Ux8c5F15wbxHRBmINzQqIpfpzFfXcq5EI08W6GL+BBw+ccdECVD5DsXPIY5irVqEyTpMZLIHX0JUUF0j/4z5tJpDyucu7ImRNQP15kBPqDsrszu106VrB9gRyY9prg89HIsa2SOBHaF60yjWwB3R3Qh3Fmy0Zuh1dCbns0Y+cXXVCDegGWeKpSKh4plapKwbE83Q313xC/RLynsNUeG6rpFsAAtVFcfQ5evcN2ld4bh49M9UjmxKqUGHBzv6qJYB0TJ9Con1fUdexOgfV6OO6eoiaiFQatDO8Rlh7MEnzugQukc0tqGHpQ1a6+rigbZjUtDKsXGwOs1rNVB/3LtqZZzVynI+OtXQVdmEpoa4rz3mDLDd+ahZejVUa3IM7yYzVnVgf7ZqcI0W7WOpLD/FXNmoxj6kirgcxcV+LvJlH31SQy459j3iaEKqq+kxejPUmjZy/dfSXRIgHxaunq4J1e+sbTQzMya6pyPlEp/qc96TeOYPwjRRU6eIXn6+DPFQy6e7Aa/4Y1DL5V6is4qyEY3G05lyeQcjifioEWdc4YWIdQhfvpn+jNzMqtjvBs+fQ+SKTTTSqug2amOx50dim9aCbaqqYjw500yJ8HQS0nhnyrlDuMw7gBafKNzh4+n8pDIHbaJMdMS6mEv7MLPBwcpng/NUV57iUB2VzHBwV5dr5bpcVDeKXi5rT+u9KKvo5bJMOUdtOno9iIF1XCGXIA8s4eWq9KhCKsCeoOjK5t9WsM/HXjvUdRFpC19VSHpYY5RXge7Kp5X2JVgFIx0Tlvn9LSJ4jvVvgiegy8Yxk00uFc1M7uKtYprJjy4Z2UzSu2R8M8k3FZISe4MzpbHM88qeYq5nlVyqnqr2BEm7Sfr8orqthv8ORZH3bFuvTIbeb7gq1O7YxBpQW2l9nt93uJP3Kvt4iNW3g3Uogma7pHdevDO0LIPDFfyLUe/Z+XKV/lokt3rdtJCHRatjOlO1ywX6JLBsX1Ycxtz+Vr7zbeSee9LzSpzbER1YlJ1lxnjuHGmE9Al0FzWnquYnWn2f0lPdBL59T0+Ue/T/pLDProrKJ7uhXqHzuPuE/IB2yDZ20svjhvr8tVs4pRV1HJKG9DfR59mMm58lE70vLkK6jN7FyKrO7bnTz6wm9XzbJy515ruoQMw6ZzFeeX4XI6t3MwHOHNnJ710g50+1q+FlNbP8HUTVm5Hl5/WQrl36d9FtseZto+pc7CWSVjX1TkE9r7Xm3o5w/e5TV3Dtjm/fmNFXxJ89b/dwXh4SR8ajzk0JpBUtj7js5B5CJtHPpzFOudlZvl44myu8WdviXF15Gq+pVzCp96eN15vbe7Pv++YHL/frez/UG8/3tx/t63eBV8WX4ra4S3F6IB5RZz6juFpr/67fXL+1/tUX/219vnVr62vFunZFy9wUhc/Wt/8D7tSx7A==</latexit>

A

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C

D

Brady Neal

The PC Algorithm

E20 / 45

True graph

The PC Algorithm: Overview<latexit sha1_base64="IynnaVgXE4s3SYQQJW2SymTdHKk=">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</latexit>

A

B

Start with complete undirected graphThree steps:

C

D

E

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A

B

C

D

Brady Neal

The PC Algorithm

E20 / 45

True graph

The PC Algorithm: Overview<latexit sha1_base64="IynnaVgXE4s3SYQQJW2SymTdHKk=">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</latexit>

A

B

Start with complete undirected graphThree steps:
1. Identify the skeleton

C

D

E

<latexit sha1_base64="manHgzpfs/psOrYrf/dCaA10eUE=">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</latexit>

A

B

C

D

Brady Neal

The PC Algorithm

E20 / 45

True graph

The PC Algorithm: Overview<latexit sha1_base64="IynnaVgXE4s3SYQQJW2SymTdHKk=">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</latexit>

A

Start with complete undirected graphThree steps:
1. Identify the skeleton
2. Identify immoralities and orient them

B

C

D

E

<latexit sha1_base64="NmgZoBtMfPRIlhPMF2nLd8BXymQ=">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</latexit>

A

B

C

D

Brady Neal

The PC Algorithm

E20 / 45

True graph

The PC Algorithm: Overview
Start with complete undirected graphThree steps:
1. Identify the skeleton
2. Identify immoralities and orient them
3. Orient qualifying edges that are incident
on colliders<latexit sha1_base64="IynnaVgXE4s3SYQQJW2SymTdHKk=">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</latexit>

A

B

C

D

E

<latexit sha1_base64="IynnaVgXE4s3SYQQJW2SymTdHKk=">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</latexit>

A

B

C

D

Brady Neal

The PC Algorithm

E20 / 45

Identifying the Skeleton

Brady Neal

The PC Algorithm

21 / 45

Identifying the Skeleton
Start with complete undirected graph and remove
edges X – Y where X ?? Y | Z for some (potentiallyempty) conditioning set Z, starting with the empty
conditioning set and increasing the size<latexit sha1_base64="q0ihpu/VVZobt5/GyIm9ZV3HzT4=">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</latexit>

Brady Neal

The PC Algorithm

21 / 45

Identifying the Skeleton
Start with complete undirected graph and remove
edges X – Y where X ?? Y | Z for some (potentiallyempty) conditioning set Z, starting with the empty
conditioning set and increasing the size

True graph<latexit sha1_base64="IynnaVgXE4s3SYQQJW2SymTdHKk=">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</latexit>

A

B<latexit sha1_base64="q0ihpu/VVZobt5/GyIm9ZV3HzT4=">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</latexit>

Brady Neal

The PC Algorithm

C

D

E

21 / 45

True graph

Identifying the Skeleton<latexit sha1_base64="IynnaVgXE4s3SYQQJW2SymTdHKk=">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</latexit>

A

Start with complete undirected graph and remove
edges X – Y where X ?? Y | Z for some (potentiallyempty) conditioning set Z, starting with the empty
conditioning set and increasing the sizeB<latexit sha1_base64="q0ihpu/VVZobt5/GyIm9ZV3HzT4=">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</latexit>

C

D

E

<latexit sha1_base64="zQnUGvERJzPDl1TcxcZOBr5WXzU=">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</latexit>

A

B

C

D

Brady Neal

The PC Algorithm

E21 / 45

True graph

Identifying the Skeleton<latexit sha1_base64="IynnaVgXE4s3SYQQJW2SymTdHKk=">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</latexit>

A

Start with complete undirected graph and remove
edges X – Y where X ?? Y | Z for some (potentiallyempty) conditioning set Z, starting with the empty
conditioning set and increasing the sizeB<latexit sha1_base64="q0ihpu/VVZobt5/GyIm9ZV3HzT4=">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</latexit>

C

D

E

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A ?? B<latexit sha1_base64="zFYufAa/HS++C9mbuXuhk/qMoAc=">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</latexit>

A

B

C

D

Brady Neal

The PC Algorithm

E21 / 45

True graph

Identifying the Skeleton<latexit sha1_base64="IynnaVgXE4s3SYQQJW2SymTdHKk=">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</latexit>

A

Start with complete undirected graph and remove
edges X – Y where X ?? Y | Z for some (potentiallyempty) conditioning set Z, starting with the empty
conditioning set and increasing the sizeB<latexit sha1_base64="q0ihpu/VVZobt5/GyIm9ZV3HzT4=">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</latexit>

C

D

E

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A ?? B | {}<latexit sha1_base64="P6rRl21ORtHYorYMuGk8jrmEFjo=">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</latexit>

A

B

C

D

Brady Neal

The PC Algorithm

E21 / 45

True graph

Identifying the Skeleton<latexit sha1_base64="IynnaVgXE4s3SYQQJW2SymTdHKk=">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</latexit>

A

Start with complete undirected graph and remove
edges X – Y where X ?? Y | Z for some (potentiallyempty) conditioning set Z, starting with the empty
conditioning set and increasing the sizeB<latexit sha1_base64="q0ihpu/VVZobt5/GyIm9ZV3HzT4=">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</latexit>

C

D

E

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A ?? B | {}<latexit sha1_base64="P6rRl21ORtHYorYMuGk8jrmEFjo=">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</latexit>

A

B

C

D

Brady Neal

The PC Algorithm

E21 / 45

True graph

Identifying the Skeleton<latexit sha1_base64="IynnaVgXE4s3SYQQJW2SymTdHKk=">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</latexit>

A

Start with complete undirected graph and remove
edges X – Y where X ?? Y | Z for some (potentiallyempty) conditioning set Z, starting with the empty
conditioning set and increasing the sizeB<latexit sha1_base64="q0ihpu/VVZobt5/GyIm9ZV3HzT4=">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</latexit>

C

D

E

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A ?? B | {}<latexit sha1_base64="P6rRl21ORtHYorYMuGk8jrmEFjo=">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</latexit><latexit sha1_base64="wvLSO2j4pP+2woUwkrnLhYy4rHk=">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</latexit>

8 other pairs (X, Y ) ,A

X ?? Y | {C}C

DBrady NealB

The PC Algorithm

E21 / 45

True graph

Identifying the Skeleton<latexit sha1_base64="IynnaVgXE4s3SYQQJW2SymTdHKk=">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</latexit>

A

Start with complete undirected graph and remove
edges X – Y where X ?? Y | Z for some (potentiallyempty) conditioning set Z, starting with the empty
conditioning set and increasing the sizeB<latexit sha1_base64="q0ihpu/VVZobt5/GyIm9ZV3HzT4=">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</latexit>

C

D

E

<latexit sha1_base64="manHgzpfs/psOrYrf/dCaA10eUE=">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</latexit>

A ?? B | {}<latexit sha1_base64="P6rRl21ORtHYorYMuGk8jrmEFjo=">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</latexit><latexit sha1_base64="wvLSO2j4pP+2woUwkrnLhYy4rHk=">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</latexit>

8 other pairs (X, Y ) ,A

X ?? Y | {C}C

DBrady NealB

The PC Algorithm

E21 / 45

Identifying the Immoralities

True graph<latexit sha1_base64="IynnaVgXE4s3SYQQJW2SymTdHKk=">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</latexit>

A

B

C

D

E

A

B

<latexit sha1_base64="manHgzpfs/psOrYrf/dCaA10eUE=">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</latexit>

C

D

Brady Neal

The PC Algorithm

E

22 / 45

Identifying the Immoralities

True graph<latexit sha1_base64="IynnaVgXE4s3SYQQJW2SymTdHKk=">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</latexit>A

Now for any paths X – Z – Y in our working
graph where the following are true:

B

C

D

E

A

B

<latexit sha1_base64="manHgzpfs/psOrYrf/dCaA10eUE=">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</latexit>

C

D

Brady Neal

The PC Algorithm

E

22 / 45

Identifying the Immoralities

True graph<latexit sha1_base64="IynnaVgXE4s3SYQQJW2SymTdHKk=">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</latexit>A

Now for any paths X – Z – Y in our working
graph where the following are true:
1. We discovered that there is no edge between
X and Y in our previous step.

B

C

D

E

A

B

<latexit sha1_base64="manHgzpfs/psOrYrf/dCaA10eUE=">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</latexit>

C

D

Brady Neal

The PC Algorithm

E

22 / 45

Identifying the Immoralities
Now for any paths X – Z – Y in our working
graph where the following are true:
1. We discovered that there is no edge between
X and Y in our previous step.
2. Z was not in the conditioning set that makes
X and Y conditionally independent.

True graph<latexit sha1_base64="IynnaVgXE4s3SYQQJW2SymTdHKk=">AAAWnHicrVjpbttGEN64V+JeSQr4T4uCjR0gBmRVdlukQGAgseSgaJIiARwnqW0EFLmSCPEqubLrCHqsPkyfoK/RmW+XIqmDlINIkLicnW9mdq7lshv7XqparX+vrX308Seffnb9xvrnX3z51dc3b90+TqNR4siXTuRHyeuunUrfC+VL5Slfvo4TaQddX77qDts8/+pcJqkXhUfqMpZngd0PvZ7n2IpIb29d++e0K/teOFbe8F3sOWqUyMm6ZT53rT8iV6bT+9OQbk9SZSt5Zt1rb1u2su61Gq1ta7zV3po8WML4aNs6sbvRubR82VOWtR/1rPYZYR4tx3QI05V+dDGL6SzHHEz1JF5/oHLMwXLM4VSPxuSgQwYVXHHo9ouuiG01wNIk0dkZD2bnDirm2tlcp2LucNsYcCpDtxShtzc3W80WPtb8YNcMNoX5PI9u3fhPnApXRMIRIxEIKUKhaOwLW6T0PRG7oiViop2JMdESGnmYl2Ii1gk7Ii5JHDZRh/Tfp7sTQw3pnmWmQDukxadfQkhL3DU8Lo17oOor67cKvMt0jCGbbbyka9fIDIiqxICodbiMc3Vcl75BzaoVreFXrNajlcSgsB+c0pp7dPXpXtEK+f+SOCWNXEIlNHKI5hNVU1hHQlftefbNAJGwwSdpxFZX2V1lNc9H9B2SLJvGKSxlWy3x2MQlhGYJW5nHR0yXS/ybVqjtq5LVm67BQ9z1Kpj3iChD8Y5GZclVOovZtZxLGcmTBbqYX4EjJO6UKBEy3yP/ecRRzlWHZLIOG5HsYy0xMqhpJP+O+SybY4rnDuxJETUL+eZBT2wqK7c7s9OnaxeyE8KPaW4APeyLBtkjITtB9mZebIA7obsL3Dmw0ZmhN1GZHM8GrYmzq0FyI5rxprK0J7Q/M4u0dWOiWea7I55Cv6S4N+AVzusGYSNYqLM4ha7QxH6f+gr7JaB/prJnM0oDOgLYMUC2DImW69OSWN+PtIoU9eMbqWO6+vBaDCkNaGf/XGAcYE0c0RF0X9DYhR5GW9TrmuK+sWNS0sq+8dCd5rVayD+uXd0ZZ7UyLkSlWiYr96GpJX42K+YIsN1FrzmmG+qenGJ1kxmrurA/7xqco2X7GJXHpxwrF9k4AKosl724eJ2L1rKHOmkgluz7PnHsA9Uz9BS1GRtN64X6a5sqiRAPB9fA5ISud9Z2MTMzJnpgPOUTn65z3pN45i+SaSOnTuG94nyVxEODz3YD7vhjUKtxx6isMjah0Xg6U433MJLwjx5xxLW8GL6OsZYfpj+rMLOq7A8jL5yTyBmrjKRVpbvIjcUrPxKb1As2KavK/uRIMyXB00lM460p5xbJZd4htIRE4QofT+cntTHoEGViPNbDXFaHuQ0eOp8LzlOTeZpDV5Sa4eCqrtbKebkobzS9GutO872M1fRqLFPOkZue6QcpZL2uwSnEgRFBIUuPalAR9gRN1za/WcG+EHvtyORFYix8VYMM0GP0qiJTlc9q7VPogonxCWP+fA8PnqP/TfAEdFU/5lh1JW/muMv38mmOv7iiZ3NkcEX/5sh3NUiJvcGb0hjzorammOt5LZfOp7o9QdJukj2/6Gpr4L9LXuQ92zWdyTL7DWeF3h330QMaK/Xn+X2HK3m3to5H6L5d9KEEmt2K2nn5waTlERytsL4U+Z6fL1epr0W41fOmjTgs6o7ZTN0uF5mTwLJ9WXNYc/tb9c63Xnjuyc4raWFH9GBRfpYZ47nzwkjInkB3kHM6a36j7vuMnuomWNtP9ES5S/+PS/vsqlL5ZDcyHbood48k36cdsoOd9OpyY3P+2imd0so6DklD9puY82zOzc+SyuyLiyRdRe9iyTrP3bnTz6wm/Xw7IC595ruskZhXzmJ51fFdLFm/m4lw5shPfh9CcvFUu5q8PGeWv4OoezOy/Lwe07VH/z6qLTW8HWSdj71EUlfT7xT081p77u0I5+8eVQXn7vjO7Rl9Zfmz5+0+zssj4sh59LlJAa1pRYnLTu4xMMo8n6Y45eZn+WbpbK7lzdqWFvIqMPL2TQeTZn9af3tzc3f2fd/84HivuftLs/Vib/PhnnkXeF18K+6Ie+Sn++IhVeZz8quz9t3awdqTtacb3290Np5sPNOsa9cM5htR+mwc/w9+SH9c</latexit>

A

B

C

D

E

A

B

<latexit sha1_base64="manHgzpfs/psOrYrf/dCaA10eUE=">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</latexit>

C

D

Brady Neal

The PC Algorithm

E

22 / 45

Identifying the Immoralities
Now for any paths X – Z – Y in our working
graph where the following are true:
1. We discovered that there is no edge between
X and Y in our previous step.
2. Z was not in the conditioning set that makes
X and Y conditionally independent.
Then, we know X – Z – Y forms an immortality.

True graph<latexit sha1_base64="IynnaVgXE4s3SYQQJW2SymTdHKk=">AAAWnHicrVjpbttGEN64V+JeSQr4T4uCjR0gBmRVdlukQGAgseSgaJIiARwnqW0EFLmSCPEqubLrCHqsPkyfoK/RmW+XIqmDlINIkLicnW9mdq7lshv7XqparX+vrX308Seffnb9xvrnX3z51dc3b90+TqNR4siXTuRHyeuunUrfC+VL5Slfvo4TaQddX77qDts8/+pcJqkXhUfqMpZngd0PvZ7n2IpIb29d++e0K/teOFbe8F3sOWqUyMm6ZT53rT8iV6bT+9OQbk9SZSt5Zt1rb1u2su61Gq1ta7zV3po8WML4aNs6sbvRubR82VOWtR/1rPYZYR4tx3QI05V+dDGL6SzHHEz1JF5/oHLMwXLM4VSPxuSgQwYVXHHo9ouuiG01wNIk0dkZD2bnDirm2tlcp2LucNsYcCpDtxShtzc3W80WPtb8YNcMNoX5PI9u3fhPnApXRMIRIxEIKUKhaOwLW6T0PRG7oiViop2JMdESGnmYl2Ii1gk7Ii5JHDZRh/Tfp7sTQw3pnmWmQDukxadfQkhL3DU8Lo17oOor67cKvMt0jCGbbbyka9fIDIiqxICodbiMc3Vcl75BzaoVreFXrNajlcSgsB+c0pp7dPXpXtEK+f+SOCWNXEIlNHKI5hNVU1hHQlftefbNAJGwwSdpxFZX2V1lNc9H9B2SLJvGKSxlWy3x2MQlhGYJW5nHR0yXS/ybVqjtq5LVm67BQ9z1Kpj3iChD8Y5GZclVOovZtZxLGcmTBbqYX4EjJO6UKBEy3yP/ecRRzlWHZLIOG5HsYy0xMqhpJP+O+SybY4rnDuxJETUL+eZBT2wqK7c7s9OnaxeyE8KPaW4APeyLBtkjITtB9mZebIA7obsL3Dmw0ZmhN1GZHM8GrYmzq0FyI5rxprK0J7Q/M4u0dWOiWea7I55Cv6S4N+AVzusGYSNYqLM4ha7QxH6f+gr7JaB/prJnM0oDOgLYMUC2DImW69OSWN+PtIoU9eMbqWO6+vBaDCkNaGf/XGAcYE0c0RF0X9DYhR5GW9TrmuK+sWNS0sq+8dCd5rVayD+uXd0ZZ7UyLkSlWiYr96GpJX42K+YIsN1FrzmmG+qenGJ1kxmrurA/7xqco2X7GJXHpxwrF9k4AKosl724eJ2L1rKHOmkgluz7PnHsA9Uz9BS1GRtN64X6a5sqiRAPB9fA5ISud9Z2MTMzJnpgPOUTn65z3pN45i+SaSOnTuG94nyVxEODz3YD7vhjUKtxx6isMjah0Xg6U433MJLwjx5xxLW8GL6OsZYfpj+rMLOq7A8jL5yTyBmrjKRVpbvIjcUrPxKb1As2KavK/uRIMyXB00lM460p5xbJZd4htIRE4QofT+cntTHoEGViPNbDXFaHuQ0eOp8LzlOTeZpDV5Sa4eCqrtbKebkobzS9GutO872M1fRqLFPOkZue6QcpZL2uwSnEgRFBIUuPalAR9gRN1za/WcG+EHvtyORFYix8VYMM0GP0qiJTlc9q7VPogonxCWP+fA8PnqP/TfAEdFU/5lh1JW/muMv38mmOv7iiZ3NkcEX/5sh3NUiJvcGb0hjzorammOt5LZfOp7o9QdJukj2/6Gpr4L9LXuQ92zWdyTL7DWeF3h330QMaK/Xn+X2HK3m3to5H6L5d9KEEmt2K2nn5waTlERytsL4U+Z6fL1epr0W41fOmjTgs6o7ZTN0uF5mTwLJ9WXNYc/tb9c63Xnjuyc4raWFH9GBRfpYZ47nzwkjInkB3kHM6a36j7vuMnuomWNtP9ES5S/+PS/vsqlL5ZDcyHbood48k36cdsoOd9OpyY3P+2imd0so6DklD9puY82zOzc+SyuyLiyRdRe9iyTrP3bnTz6wm/Xw7IC595ruskZhXzmJ51fFdLFm/m4lw5shPfh9CcvFUu5q8PGeWv4OoezOy/Lwe07VH/z6qLTW8HWSdj71EUlfT7xT081p77u0I5+8eVQXn7vjO7Rl9Zfmz5+0+zssj4sh59LlJAa1pRYnLTu4xMMo8n6Y45eZn+WbpbK7lzdqWFvIqMPL2TQeTZn9af3tzc3f2fd/84HivuftLs/Vib/PhnnkXeF18K+6Ie+Sn++IhVeZz8quz9t3awdqTtacb3290Np5sPNOsa9cM5htR+mwc/w9+SH9c</latexit>

A

B

C

D

E

A

B

<latexit sha1_base64="manHgzpfs/psOrYrf/dCaA10eUE=">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</latexit>

C

D

Brady Neal

The PC Algorithm

E

22 / 45

Identifying the Immoralities
Now for any paths X – Z – Y in our working
graph where the following are true:
1. We discovered that there is no edge between
X and Y in our previous step.
2. Z was not in the conditioning set that makes
X and Y conditionally independent.
Then, we know X – Z – Y forms an immortality.

True graph<latexit sha1_base64="IynnaVgXE4s3SYQQJW2SymTdHKk=">AAAWnHicrVjpbttGEN64V+JeSQr4T4uCjR0gBmRVdlukQGAgseSgaJIiARwnqW0EFLmSCPEqubLrCHqsPkyfoK/RmW+XIqmDlINIkLicnW9mdq7lshv7XqparX+vrX308Seffnb9xvrnX3z51dc3b90+TqNR4siXTuRHyeuunUrfC+VL5Slfvo4TaQddX77qDts8/+pcJqkXhUfqMpZngd0PvZ7n2IpIb29d++e0K/teOFbe8F3sOWqUyMm6ZT53rT8iV6bT+9OQbk9SZSt5Zt1rb1u2su61Gq1ta7zV3po8WML4aNs6sbvRubR82VOWtR/1rPYZYR4tx3QI05V+dDGL6SzHHEz1JF5/oHLMwXLM4VSPxuSgQwYVXHHo9ouuiG01wNIk0dkZD2bnDirm2tlcp2LucNsYcCpDtxShtzc3W80WPtb8YNcMNoX5PI9u3fhPnApXRMIRIxEIKUKhaOwLW6T0PRG7oiViop2JMdESGnmYl2Ii1gk7Ii5JHDZRh/Tfp7sTQw3pnmWmQDukxadfQkhL3DU8Lo17oOor67cKvMt0jCGbbbyka9fIDIiqxICodbiMc3Vcl75BzaoVreFXrNajlcSgsB+c0pp7dPXpXtEK+f+SOCWNXEIlNHKI5hNVU1hHQlftefbNAJGwwSdpxFZX2V1lNc9H9B2SLJvGKSxlWy3x2MQlhGYJW5nHR0yXS/ybVqjtq5LVm67BQ9z1Kpj3iChD8Y5GZclVOovZtZxLGcmTBbqYX4EjJO6UKBEy3yP/ecRRzlWHZLIOG5HsYy0xMqhpJP+O+SybY4rnDuxJETUL+eZBT2wqK7c7s9OnaxeyE8KPaW4APeyLBtkjITtB9mZebIA7obsL3Dmw0ZmhN1GZHM8GrYmzq0FyI5rxprK0J7Q/M4u0dWOiWea7I55Cv6S4N+AVzusGYSNYqLM4ha7QxH6f+gr7JaB/prJnM0oDOgLYMUC2DImW69OSWN+PtIoU9eMbqWO6+vBaDCkNaGf/XGAcYE0c0RF0X9DYhR5GW9TrmuK+sWNS0sq+8dCd5rVayD+uXd0ZZ7UyLkSlWiYr96GpJX42K+YIsN1FrzmmG+qenGJ1kxmrurA/7xqco2X7GJXHpxwrF9k4AKosl724eJ2L1rKHOmkgluz7PnHsA9Uz9BS1GRtN64X6a5sqiRAPB9fA5ISud9Z2MTMzJnpgPOUTn65z3pN45i+SaSOnTuG94nyVxEODz3YD7vhjUKtxx6isMjah0Xg6U433MJLwjx5xxLW8GL6OsZYfpj+rMLOq7A8jL5yTyBmrjKRVpbvIjcUrPxKb1As2KavK/uRIMyXB00lM460p5xbJZd4htIRE4QofT+cntTHoEGViPNbDXFaHuQ0eOp8LzlOTeZpDV5Sa4eCqrtbKebkobzS9GutO872M1fRqLFPOkZue6QcpZL2uwSnEgRFBIUuPalAR9gRN1za/WcG+EHvtyORFYix8VYMM0GP0qiJTlc9q7VPogonxCWP+fA8PnqP/TfAEdFU/5lh1JW/muMv38mmOv7iiZ3NkcEX/5sh3NUiJvcGb0hjzorammOt5LZfOp7o9QdJukj2/6Gpr4L9LXuQ92zWdyTL7DWeF3h330QMaK/Xn+X2HK3m3to5H6L5d9KEEmt2K2nn5waTlERytsL4U+Z6fL1epr0W41fOmjTgs6o7ZTN0uF5mTwLJ9WXNYc/tb9c63Xnjuyc4raWFH9GBRfpYZ47nzwkjInkB3kHM6a36j7vuMnuomWNtP9ES5S/+PS/vsqlL5ZDcyHbood48k36cdsoOd9OpyY3P+2imd0so6DklD9puY82zOzc+SyuyLiyRdRe9iyTrP3bnTz6wm/Xw7IC595ruskZhXzmJ51fFdLFm/m4lw5shPfh9CcvFUu5q8PGeWv4OoezOy/Lwe07VH/z6qLTW8HWSdj71EUlfT7xT081p77u0I5+8eVQXn7vjO7Rl9Zfmz5+0+zssj4sh59LlJAa1pRYnLTu4xMMo8n6Y45eZn+WbpbK7lzdqWFvIqMPL2TQeTZn9af3tzc3f2fd/84HivuftLs/Vib/PhnnkXeF18K+6Ie+Sn++IhVeZz8quz9t3awdqTtacb3290Np5sPNOsa9cM5htR+mwc/w9+SH9c</latexit>

A

B

C

D

E

A

B

<latexit sha1_base64="manHgzpfs/psOrYrf/dCaA10eUE=">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</latexit>

C

D

Brady Neal

The PC Algorithm

A ?? B | {}<latexit sha1_base64="P6rRl21ORtHYorYMuGk8jrmEFjo=">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</latexit>E

22 / 45

Identifying the Immoralities
Now for any paths X – Z – Y in our working
graph where the following are true:
1. We discovered that there is no edge between
X and Y in our previous step.
2. Z was not in the conditioning set that makes
X and Y conditionally independent.
Then, we know X – Z – Y forms an immortality.

True graph<latexit sha1_base64="IynnaVgXE4s3SYQQJW2SymTdHKk=">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</latexit>

A

B

C

D

E

A

B

<latexit sha1_base64="qfjNYC8ZtInOScZrliQHryStpSY=">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</latexit>

C

D

Brady Neal

The PC Algorithm

A ?? B | {}<latexit sha1_base64="P6rRl21ORtHYorYMuGk8jrmEFjo=">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</latexit>E

22 / 45

Orienting Qualifying Edges Incident on Colliders
True graph<latexit sha1_base64="IynnaVgXE4s3SYQQJW2SymTdHKk=">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</latexit>

A

B

C<latexit sha1_base64="qfjNYC8ZtInOScZrliQHryStpSY=">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</latexit><latexit sha1_base64="manHgzpfs/psOrYrf/dCaA10eUE=">AAAXW3icrVhZb9tGEN7YPRIlbZMUrR/6wsYOkACyKrspUiAwkFhyUBRNkQDOVdsIKHElEeJVcmXVEfRDi6LP/Rud+XYpkjpIOYgEicvZmW/uXS47kecmqtn8+8rG5ieffvb51Wu16ze++PKrm7duv0rCUdyVL7uhF8ZvOnYiPTeQL5WrPPkmiqXtdzz5ujNs8fzrcxknbhgcq4tInvl2P3B7btdWRHp3a0OddmTfDSbKHb6P3K4axXJ6sntWs8znrvV76Mhkdn8a0O1Jomwlz6x7rfuWrax7zXrzvjXZae1MH61gfHLfOrE74bm0PNlTlnUQ9qzWGck8WS3TJpmO9MLxvEx7tczhTE/s9gcqkzlcLXM006NlMqEjFsqF4sjp50MR2WoA1yTRORiP5ucOS+Za6Vy7ZO4oN5czZF71YY5tcbZdMkueB452/OBB86yo8e6CH2WzGopzpZFYLZhPiV4osHc3t5uNJj7W4mDPDLaF+TwPb137T5wKR4SiK0bCF1IEQtHYE7ZI6Hsi9kRTREQ7ExOixTRyMS/FVNRIdkRckjhsog7pv093J4Ya0D1jJpDukhaPfjFJWuKu4XFo3ANVX1m/leNdpWMCbLbxgq4dg+kTVYkBUavkUs715Tr09Su8VuTDz/DWJU8iUDgO3YLPPbp6dK/IQ/6/IE5JI4ekYhp1ieYRVVNYR0xXHXmOzQCZsMEnacRWl9ldZjXPh/QdEpZN4wSWsq2WeGryEkCzhK3M4yGnqxH/Ig+1fWVYvZkPLvKuvWDeY6IMxXsaFZHLdOarazWXMsjTJbqYX4EjIO6EKCEq36X4ucRRrNUuYbIOG5nsw5cIFdQwyL9iPq3miPK5C3sSZM1CvbnQE5nOyuxO7fTo2gF2TPITmhtAD8eiTvZIYMeo3jSKdXDHdDfGXRc2dufoDXQm57NOPnF11Qk3pBl3hqUjoeOZWqStmxDNMt9d8Rv0S8p7HVHhuq6TbAgLdRUn0BWY3B/QusJx8emfqRzZlFKHDh92DFAtQ6Jl+jQS6/uBvEjQP55BndDVQ9QioNShneMzxtiHT5zREXSPaexAD0tbtNY1xENjx7SglWPjYnVa1Gqh/rh39co4r5XlAnSqZaryAJqa4oHxmDPAduej1jWroV6TE3g3nbOqA/uzVYNrtGgfS2X5KebKQTUOIFXE5Sgu93OZL/vokzpyybHvE8cBpHqGnqA3I6Opluu/lumSEPno4uqbmtD9ztrGczMTovsmUh7x6T7nPYln/iRMGzV1iujl58sQj4x8uhvwij8BtVzuFTqrKBvTaDKbKZd3MZKIjx5xxjVehFhH8OX72c/KzayL/XHwggVErlhlkNZFd1Abyz0/Ftu0FmxTVRXjyZlmSoynk4jGOzPOHcJl3iG0BEThDp/M5qeVOWgTZWoi1sNc2oeZDS5WPgecp6byNIfuKDXHwV1drpXrclndaHq5rDOr96KsppfLMuUctema9SAB1psKOYU8sISfq9LjCqkQe4Kma5vfrmFfgL12ZOoiNha+rpD0scZor0LTlc8q7VNYBWMTE5b54wMieI71b4onoMvGMZNVl4pmJnfxQTHN5MeXjGwm6V8yvpnk+wpJib3BndFY5kVlTzHX80ouXU9Ve4Kk3SR9ftHdVsd/h6LIe7ZjVibL7DdcFXp3PMAaUF9rfV7cd7iT9yr7eITVt4N1KIZmp6R3Xn40tCyDozX8S1Dv2flynf5aJrd+3bSQh2WrYzpTtcuF5iSwal/WHNbC/la+89Vyzz3peSXJ7YguLMrOMhM8d44NQvoEuoua01XzC62+z+ipbgrffqQnyj36f1rYZ9dF5ZPdyKzQedx9Qn5IO2QbO+nlcSNz/totnNKKOo5IQ/qbmvNsxs3Pksrsi8uQLqN3ObKuc2fh9DOvST/fDohLn/kuKhCzzlmOV57f5cj63UyIM0d28vsYyPlT7Xp4Wc2sfgdR9WZk9Xk9omuP/j10W2J426g6D3uJpFVNv1PQz2uthbcjXL/71BVcu5M7t+f0FfHnz9t9nJdHxJHx6HOTgrSm5RFXndwjyCjzfJrglJud5RuFs7nGm7ctydWVb/AOzAomzf5Ue3dze2/+fd/i4NV+Y++nRvPF/vbjffMu8Kr4TtwR9yhOD8Vj6sznFNfuxj+bVzZrm9e//Xdrc6u2dUOzblwxMl+Lwmfrm/8BbRmsIw==</latexit>

A

B

D

E

C

D

Brady Neal

The PC Algorithm

E

23 / 45

Orienting Qualifying Edges Incident on Colliders
Idea: use fact that we discovered all immoralities to orient
more edges

True graph<latexit sha1_base64="IynnaVgXE4s3SYQQJW2SymTdHKk=">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</latexit>

A

B

C<latexit sha1_base64="qfjNYC8ZtInOScZrliQHryStpSY=">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</latexit><latexit sha1_base64="manHgzpfs/psOrYrf/dCaA10eUE=">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</latexit>

A

B

D

E

C

D

Brady Neal

The PC Algorithm

E

23 / 45

Orienting Qualifying Edges Incident on Colliders
Idea: use fact that we discovered all immoralities to orient
more edges
Any edge Z Y part of a partially directed path of theform X ! Z Y , where there is no edge connecting X
and Y can be oriented as Z ! Y

True graph<latexit sha1_base64="IynnaVgXE4s3SYQQJW2SymTdHKk=">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</latexit>

A

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A

B

D

E

C

D

Brady Neal

The PC Algorithm

E

23 / 45

Orienting Qualifying Edges Incident on Colliders
Idea: use fact that we discovered all immoralities to orient
more edges
Any edge Z Y part of a partially directed path of theform X ! Z Y , where there is no edge connecting X
and Y can be oriented as Z ! Y

True graph<latexit sha1_base64="IynnaVgXE4s3SYQQJW2SymTdHKk=">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</latexit>

A

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A

B

D

E

C

D

Brady Neal

The PC Algorithm

E

23 / 45

Removing Assumptions

Brady Neal

The PC Algorithm

24 / 45

Removing Assumptions
No assumed causal sufficiency: FCI algorithm (Spirtes et al., 2001)

Brady Neal

The PC Algorithm

24 / 45

Removing Assumptions
No assumed causal sufficiency: FCI algorithm (Spirtes et al., 2001)
No assumed acyclicity: CCD algorithm (Richardson, 1996)

Brady Neal

The PC Algorithm

24 / 45

Removing Assumptions
No assumed causal sufficiency: FCI algorithm (Spirtes et al., 2001)
No assumed acyclicity: CCD algorithm (Richardson, 1996)
Neither causal sufficiency nor acyclicity: SAT-based causal discovery(Hyttinen et al., 2013; 2014)

Brady Neal

The PC Algorithm

24 / 45

Hardness of Conditional Independence Testing

Brady Neal

The PC Algorithm

25 / 45

Hardness of Conditional Independence Testing
Independence-based causal discovery algorithms rely on accurateconditional independence testing.

Brady Neal

The PC Algorithm

25 / 45

Hardness of Conditional Independence Testing
Independence-based causal discovery algorithms rely on accurateconditional independence testing.
Conditional independence testing is simple if we have infinite data.

Brady Neal

The PC Algorithm

25 / 45

Hardness of Conditional Independence Testing
Independence-based causal discovery algorithms rely on accurateconditional independence testing.
Conditional independence testing is simple if we have infinite data.
However, it is a quite hard problem with finite data, and it can sometimes
require a lot of data to get accurate test results (Shah & Peters, 2020).

Brady Neal

The PC Algorithm

25 / 45

Questions:
1. What are the essential graphs of the following graphs?

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A

B<latexit sha1_base64="vcMVYs41M8xH0M+3wQpL73YVA0M=">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</latexit>

X2

X1

X3
C

B
X1

X3

X2
C

D

D

Questions:
1. What are the essential graphs of the following graphs?

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A

B<latexit sha1_base64="vcMVYs41M8xH0M+3wQpL73YVA0M=">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</latexit>

X2

X1

X3
C

B
X1

X3

X2
C

D

D

Questions:
1. What are the essential graphs of the following graphs?

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A

B<latexit sha1_base64="vcMVYs41M8xH0M+3wQpL73YVA0M=">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</latexit>

X2

X1

X3
C

B
X1

X3

X2
C

D

D

Questions:
1. What are the essential graphs of the following graphs?

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A

B<latexit sha1_base64="vcMVYs41M8xH0M+3wQpL73YVA0M=">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</latexit>

X2

X1

X3
C

B
X1

X3

X2
C

D

D

Questions:
1. What are the essential graphs of the following graphs?

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A

B<latexit sha1_base64="vcMVYs41M8xH0M+3wQpL73YVA0M=">AAAVlXicrVhZb9tGEF6lV+JeTgqjD31hYwdIAFmV5BYpEAgIYCcoiqZNACdxaxsGj5VEiFdJyqpD6Lm/pn1sf0v/TWe+XYmkDlIOIkHkcnbmm9m5lisr8twkbbf/a9x47/0PPvzo5q2tjz/59LPPt2/feZWE49iWL+3QC+MTy0yk5wbyZeqmnjyJYmn6lidfW6NDnn99KePEDYPj9CqS5745CNy+a5spkS5uN4wzSw7cIEvd0ZvItdNxLKdbhv7cM34OHZnMn88CejxNUjOV58b9k84Dw0yN++1m+4GR7Z1cdPamj9bxdh8Yp5b0wokRu4Nhahi9sG+cdM4h2K0QPCBB0wovpRaEXFfJHbBcwdonzqBobWSmQ2WmpAkY8Whp9qA0u3UmA6fkjIvt3XarjY+xPOjowa7Qn+fh7Vt/izPhiFDYYix8IUUgUhp7whQJfU9FR7RFRLRzkREtppGLeSmmYotkx8QlicMk6oiuA3o61dSAnhkzgbRNWjz6xSRpiHuax6FxH1R1Z/1GgXedjgzYbOMV3S2N6RM1FUOi1snNODeXs+jr16w6pTV8j9W6tJIIFPaDXVpzn+4ePae0Qr5eEaekkUNSMY1sonlEVRTWEdNdeZ59M0QkTPBJGrHVVXZXWc3zIX1HhGXSOIGlbKshnuq4BNAsYSvzeIjpesQ/aIXKviqs/nwNLuKuVsG8x0QZiTc0KiNX6Sxm13quVCNPV+hi/hQcAXEnRAmR+S75zyWOcq7ahMk6TERygLVEyKCWRv4R87Nsjiie+7AnQdQM5JsLPZGurNzumZ0e3S1gxySf0dwQetgXTbJHAjtG9s682AR3TE8TPNmw0V6gt1CZHM8mrYmzq0m4Ic24cyzlCeXPmUXKuoxohv7ui5+gX1Lcm/AK53WTZENYqLI4ga5Ax75HfYX94tOVqezZGaUJHT7sGCJbRkTL9Skk1vcNrSJB/XgaNaO7B69FQGlCO/tngrGPNXFEx9A9obEDPSxtUK9riYfajmlJK/vGRXda1mog/7h2VWdc1MpyASrV0FnZg6a2+FavmCPAdhe9ZutuqHpygtVNF6yyYH/eNThHy/axVB6fcqwcZOMQUmVc9uLqda5aSxd10kQs2fcD4uhBqq/pCWoz0pq2CvV3qKskRDxs3H2dE6reWdtkYSYjuq895RGfqnPek3jmd8I0kVNn8F5xvgrxiZaf7Qbc8TNQq+VeobLKsjGNsvlMtbyLkYR/1IgjrvAi+DrCWr6e/4zCzKbY7wYvWELkjE010qboDnJj9cqPxS71gl3KqrI/OdJMifF2EtF4b865R7jMO4KWgChc4dl8flobgyOiTLXH+pib1WFug4vO54DzTGee4lAVlS5wcFVXa+W8XJU3il4t68zzvSyr6NWyTLlEbrq6HyTAOqmRSxEHlvALWXpcIxViT1B0ZfOvG9gXYK8d67yItYWvayR99Bi1qlBX5bNa+1J0wVj7hGV+ewsPXqL/TfEGdF0/5rLptbyZy129lU9z+ck1PZtL+tf0by75pkZSYm9w5zSWeVFbU8z1vJZL5VPdniBpN5m9v6hqa+JqkRd5z3Z0ZzL0fsNZoXbHHnpAc6P+vLzvcCV3aut4jO5roQ/F0OxU1M7Ld4aWR3C8wfoS5Ht+vtykvlbJbZ43h4jDqu44m1FvIPlbyOz0kBT2Jxf4+ckiw1vgRO9ks/fBfWSAiuEP1Auf0TvWFJoO6P2uQ9enpV1vU1Q+Z411vyzidgn5Ie1XR9jXro8b6dPQfunMVNbxhDTMflN9usy5PX0GWId0Hb2rkVXWOUtnkUVN6m1zSFzqBHZVg5jn8Wq86viuRlb/lIQ4AeTnsHeBXDxjboaX58z6fwTq/qdYf3qO6N6nq4eOnmjeI2Sdh84uqceoE756ezpc+q+C87dLVcG5m929s6CvjL94+h3g9DomjpxHnWJSSCtaEXHdOTqCTKrfFhOcOfOTdat0UlZ4i7YlhbzyNV5P9yOpd4uti+3dzuK/b8uDV91W57tW+0V393FX/zN3U3wl7or75KeH4jFV5nPyq934s/FX45/Gvztf7vR2jnaeKtYbDS3zhSh9dn75H1TfRGM=</latexit>

X2

X1

X3
C

B
X1

X3

X2
C

D

D

Questions:
1. What are the essential graphs of the following graphs?
2. Walk through the steps of PC to get them.
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A

B<latexit sha1_base64="vcMVYs41M8xH0M+3wQpL73YVA0M=">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</latexit>

X2

X1

X3
C

B
X1

X3

X2
C

D

D

Question:
What is the essential
graph for this graph?
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F

A

E

B

C

D

H

G

Independence-Based Causal Discovery
Assumptions
Markov Equivalence and Main Theorem
The PC Algorithm
Can We Do Better?

Semi-Parametric Causal Discovery
No Identifiability Without Parametric Assumptions
Linear Non-Gaussian Setting
Nonlinear Additive Noise Setting

Brady Neal

Can We Do Better?

28 / 45

Can We Do Better?
With faithfulness, we saw we can identify the essential graph (Markov
equivalence class).

Brady Neal

Can We Do Better?

29 / 45

Can We Do Better?
With faithfulness, we saw we can identify the essential graph (Markov
equivalence class).
If we have multinomial distributions (Meek, 1995) or linear Gaussian
structural equations (Geiger & Pearl, 1988), we can only identify a graphup to its Markov equivalence class.

Brady Neal

Can We Do Better?

29 / 45

Can We Do Better?
With faithfulness, we saw we can identify the essential graph (Markov
equivalence class).
If we have multinomial distributions (Meek, 1995) or linear Gaussian
structural equations (Geiger & Pearl, 1988), we can only identify a graphup to its Markov equivalence class.
What about non-Gaussian structural equations?
Or nonlinear structural equations?
Brady Neal

Can We Do Better?

29 / 45

Independence-Based Causal Discovery
Assumptions
Markov Equivalence and Main Theorem
The PC Algorithm
Can We Do Better?

Semi-Parametric Causal Discovery
No Identifiability Without Parametric Assumptions
Linear Non-Gaussian Setting
Nonlinear Additive Noise Setting

Brady Neal

Semi-Parametric Causal Discovery

30 / 45Issues with Independence-Based Causal Discovery

Brady Neal

Semi-Parametric Causal Discovery

31 / 45Issues with Independence-Based Causal Discovery• Requires faithfulness assumption

Brady Neal

Semi-Parametric Causal Discovery

31 / 45Issues with Independence-Based Causal Discovery• Requires faithfulness assumption• Large samples can be necessary for conditional independence tests

Brady Neal

Semi-Parametric Causal Discovery

31 / 45Issues with Independence-Based Causal Discovery• Requires faithfulness assumption• Large samples can be necessary for conditional independence tests• Only identifies the Markov equivalence class

Brady Neal

Semi-Parametric Causal Discovery

31 / 45

Independence-Based Causal Discovery
Assumptions
Markov Equivalence and Main Theorem
The PC Algorithm
Can We Do Better?

Semi-Parametric Causal Discovery
No Identifiability Without Parametric Assumptions
Linear Non-Gaussian Setting
Nonlinear Additive Noise Setting

Brady NealNo Identifiability Without Parametric Assumptions

32 / 45

Two Variable Case: Markov Equivalence
Infinite data: P(x, y)<latexit sha1_base64="jBwbuIlti6jnLY2cn36E4bkUnGo=">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</latexit>

Brady NealNo Identifiability Without Parametric Assumptions

33 / 45

Two Variable Case: Markov Equivalence
Infinite data: P(x, y)<latexit sha1_base64="jBwbuIlti6jnLY2cn36E4bkUnGo=">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</latexit><latexit sha1_base64="lp8YKtng9vRFXp19nB6OK75Nx04=">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</latexit><latexit sha1_base64="EKn0aHdHoE8Fw38KTNqEHpxLHj0=">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</latexit>

X

Brady Neal

Y

vs.

X

Y

No Identifiability Without Parametric Assumptions

33 / 45

Two Variable Case: Markov Equivalence
Infinite data: P(x, y)<latexit 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X

vs.

Y

X

Y

<latexit sha1_base64="AX4vTCaI+N27kDWf8bJzbQ4xxKc=">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</latexit>

Essential graph:

Brady NealXY

No Identifiability Without Parametric Assumptions

33 / 45

Two Variable Case: SCMs Perspective

Brady NealNo Identifiability Without Parametric Assumptions

34 / 45

Two Variable Case: SCMs Perspective
Proposition: For every joint distribution P(x, y) on two real-valued
random variables, there is an SCM in either direction that generates dataconsistent with P(x, y) .
<latexit sha1_base64="jBwbuIlti6jnLY2cn36E4bkUnGo=">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</latexit><latexit sha1_base64="jBwbuIlti6jnLY2cn36E4bkUnGo=">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</latexit>

Brady NealNo Identifiability Without Parametric Assumptions

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Two Variable Case: SCMs Perspective
Proposition: For every joint distribution P(x, y) on two real-valued
random variables, there is an SCM in either direction that generates dataconsistent with P(x, y) .
<latexit sha1_base64="jBwbuIlti6jnLY2cn36E4bkUnGo=">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</latexit><latexit sha1_base64="jBwbuIlti6jnLY2cn36E4bkUnGo=">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</latexit>

Mathematically, there exists a function fY such that<latexit sha1_base64="sSBhXFcX+Mj8SOR6YDH4xFNvImA=">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</latexit><latexit sha1_base64="59DpBQw9pcKZJU71Dq6zk+FP84A=">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</latexit>

Y = fY (X, UY ) ,X ?? UY

and there exists a function fX such that<latexit sha1_base64="zxHR8hBubTO35mehygN57CdqPzc=">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</latexit><latexit sha1_base64="10IbNR118wFWiqavt5L3KPpn2AA=">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</latexit>

X = fX (Y, UX ) ,Y ?? UX

where UY and UX are real-valued random variables.
<latexit sha1_base64="LgV6oxEH5xLkx/oz6C2Zo20n45Q=">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</latexit>

Brady Neal<latexit sha1_base64="mJ2hoStsYJkCWCnBcIvqCGTLQPI=">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</latexit>

No Identifiability Without Parametric Assumptions

34 / 45

Two Variable Case: SCMs Perspective
Proposition: For every joint distribution P(x, y) on two real-valued
random variables, there is an SCM in either direction that generates dataconsistent with P(x, y) .
<latexit sha1_base64="jBwbuIlti6jnLY2cn36E4bkUnGo=">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</latexit><latexit sha1_base64="jBwbuIlti6jnLY2cn36E4bkUnGo=">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</latexit>

Mathematically, there exists a function fY such that<latexit sha1_base64="sSBhXFcX+Mj8SOR6YDH4xFNvImA=">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</latexit><latexit sha1_base64="iShsRbC2sEDwqtF690PAwnJyQu0=">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</latexit><latexit sha1_base64="59DpBQw9pcKZJU71Dq6zk+FP84A=">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</latexit>

Y = fY (X, UY ) ,X ?? UY

X

Y

and there exists a function fX such that<latexit sha1_base64="zxHR8hBubTO35mehygN57CdqPzc=">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</latexit><latexit sha1_base64="10IbNR118wFWiqavt5L3KPpn2AA=">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</latexit>

X = fX (Y, UX ) ,Y ?? UX

where UY and UX are real-valued random variables.
<latexit sha1_base64="LgV6oxEH5xLkx/oz6C2Zo20n45Q=">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</latexit>

Brady Neal<latexit sha1_base64="mJ2hoStsYJkCWCnBcIvqCGTLQPI=">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</latexit>

No Identifiability Without Parametric Assumptions

34 / 45

Two Variable Case: SCMs Perspective
Proposition: For every joint distribution P(x, y) on two real-valued
random variables, there is an SCM in either direction that generates dataconsistent with P(x, y) .
<latexit sha1_base64="jBwbuIlti6jnLY2cn36E4bkUnGo=">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</latexit><latexit sha1_base64="jBwbuIlti6jnLY2cn36E4bkUnGo=">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</latexit>

Mathematically, there exists a function fY such that<latexit sha1_base64="sSBhXFcX+Mj8SOR6YDH4xFNvImA=">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</latexit><latexit 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Y = fY (X, UY ) ,X ?? UY

X

Y

X

Y

and there exists a function fX such that<latexit 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X = fX (Y, UX ) ,Y ?? UX

where UY and UX are real-valued random variables.
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Brady Neal<latexit sha1_base64="mJ2hoStsYJkCWCnBcIvqCGTLQPI=">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</latexit>

No Identifiability Without Parametric Assumptions

34 / 45

We must make assumptions
about the parametric form.

Independence-Based Causal Discovery
Assumptions
Markov Equivalence and Main Theorem
The PC Algorithm
Can We Do Better?

Semi-Parametric Causal Discovery
No Identifiability Without Parametric Assumptions
Linear Non-Gaussian Setting
Nonlinear Additive Noise Setting

Brady Neal

Linear Non-Gaussian Setting

36 / 45

Linear Non-Gaussian Assumption

Brady Neal

Linear Non-Gaussian Setting

37 / 45

Linear Non-Gaussian Assumption
Recall: We cannot hope to identify the graph more precisely than the Markov
equivalence class in the linear Gaussian noise setting (Geiger & Pearl, 1988).

Brady Neal

Linear Non-Gaussian Setting

37 / 45

Linear Non-Gaussian Assumption
Recall: We cannot hope to identify the graph more precisely than the Markov
equivalence class in the linear Gaussian noise setting (Geiger & Pearl, 1988).
What if the noise is non-Gaussian?

Brady Neal

Linear Non-Gaussian Setting

37 / 45

Linear Non-Gaussian Assumption
Recall: We cannot hope to identify the graph more precisely than the Markov
equivalence class in the linear Gaussian noise setting (Geiger & Pearl, 1988).
What if the noise is non-Gaussian?
Linear Non-Gaussian Assumption:
All structural equations (causal mechanisms that generate the data) are of the following
form:<latexit sha1_base64="jNKO1AkF2p/JegXbZoEjgSF7DTo=">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</latexit>

Y := f(X) + U

where f is a linear function, X ?? U, and U is distributed as some non-Gaussian.
<latexit sha1_base64="ofzHQ1hQyr3M9/ZqEqr4z6AWDjg=">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</latexit>

Brady Neal<latexit sha1_base64="Gxaror+G7Gl7xgavL5BAswEeLs8=">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</latexit>

Linear Non-Gaussian Setting

37 / 45

Identifiability in Linear Non-Gaussian Setting
Theorem (Shimizu et al., 2006):
In the linear non-Gaussian setting, if the true SCM is<latexit sha1_base64="Apa+6rzKR8SL6bFZG+j0GQ9gS5U=">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</latexit>

Y := f (X) + U ,X ?? U ,then there does not exist an SCM in the reverse direction,<latexit sha1_base64="XjNLuZUUlBEpumFRcHxuDhMsxns=">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</latexit>

X := g(Y ) + Ũ ,Y ?? Ũ ,that can generate data consistent with P (x, y).
<latexit sha1_base64="qn1sOHxgspJ1uLaxcD4kNOQkPVE=">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</latexit>

Brady Neal

Linear Non-Gaussian Setting

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Identifiability in Linear Non-Gaussian Setting
Theorem (Shimizu et al., 2006):
In the linear non-Gaussian setting, if the true SCM is<latexit sha1_base64="Apa+6rzKR8SL6bFZG+j0GQ9gS5U=">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</latexit>

Y := f (X) + U ,X ?? U ,then there does not exist an SCM in the reverse direction,<latexit sha1_base64="XjNLuZUUlBEpumFRcHxuDhMsxns=">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</latexit>

X := g(Y ) + Ũ ,Y ?? Ũ ,that can generate data consistent with P (x, y).
<latexit sha1_base64="qn1sOHxgspJ1uLaxcD4kNOQkPVE=">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</latexit>

See proof in the course book
Brady Neal

Linear Non-Gaussian Setting

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Identifiability in Linear Non-Gaussian Setting:Linear Fit

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Linear Non-Gaussian Setting

39 / 45

Identifiability in Linear Non-Gaussian Setting:Linear Fit<latexit sha1_base64="Apa+6rzKR8SL6bFZG+j0GQ9gS5U=">AAAUo3icrVjpbttGEN4kPRL3clL/6x82SoAUpRXJTeCggIAAdoKibQCn8JVGhiGRK4sQr5CUVUfQY/Q1+gL92z5E36Yz3y5FUgcpBZYgcjU7883sXMtlN3SdOGk0/rtx89ZHH3/y6e07G599/sWXX23evXccB8PIkkdW4AbRabcTS9fx5VHiJK48DSPZ8bquPOkO9nj+5FJGsRP4h8lVKM+8zoXv9ByrkxDpfPOx8cb4sWX0Hp1+Z3xvHBlt0zCN9rthxzZOjbbj2zKUdPETPXe+WWvUG/gY84OmHtSE/hwEd+/8JdrCFoGwxFB4QgpfJDR2RUfE9H0rmqIhQqKdiTHRIho5mJdiIjZIdkhckjg6RB3Q9YL+vdVUn/4zZgxpi7S49ItI0hAPNY9N4x6o6s76jRzvMh1jYLONV3TvakyPqInoE7VKLuVcXa5LX69i1Qmt4RlW69BKQlDYD1ZhzT26u/Q/oRXy9Yo4JY1skopoZBHNJaqisI6I7srz7Js+ItEBn6QRW11md5nVPB/Qd0BYHRrHsJRtNcRLHRcfmiVsZR4XMV2O+AetUNlXhtWbrsFB3NUqmPeQKAPxnkZF5DKd+exazpVo5MkCXcyfgMMn7pgoATLfIf85xFHMVYswWUcHkbzAWkJkUF0j/4z5NJtDiuc27IkRNQP55kBPqCsrszu106V7F9gRyY9prg897AuT7JHAjpC9qRdNcEf0b4R/Fmy0Zuh1VCbH06Q1cXaZhBvQjDPFUp5Q/kwtUtaNiWbo77b4Ffolxd2EVzivTZINYKHK4hi6fB37FvUV9otHV6ayZ1OKCR0e7OgjWwZEy/QpJNb3mFYRo35cjTqmuwuvhUAxoZ39M8LYw5o4okPoHtHYhh6WNqjX1cWutmNS0Mq+cdCd5rUayD+uXdUZZ7WynI9KNXRWtqCpIZ7oFXME2O681yzdDVVPjrG6yYxVXdifdQ3O0aJ9LJXFpxgrG9nYh1QRl724eJ2L1rKDOjERS/b9BXG0INXT9Bi1GWpNG7n629NVEiAeFu6ezglV76xtNDMzJrqnPeUSn6pz3pN45h1hdpBTbXgvP1+G+ELLp7sBd/wxqOVyx6isomxEo/F0plzewUjCP2rEEVd4IXwdYi3fTn9GbmZV7OvB8+cQOWMTjbQquo3cWLzyQ1GjXlCjrCr6kyPNlAhPJyGNH0w5HxAu8w6gxScKV/h4Oj+pjME+USbaYz3MpXWY2eCg89ngbOvMUxyqopIZDq7qcq2cl4vyRtHLZe1pvhdlFb1climXyE1H94MYWKcVcgniwBJeLksPK6QC7AmKrmx+s4J9Pvbaoc6LSFt4UiHpoceoVQW6Kl9V2pegC0baJyzz+wd48BL9b4InoHX9mMkma3kzk7v6IJ9m8qM1PZtJemv6N5N8XyEpsTc4UxrLvK6sKeY6qORS+VS1J0jaTdLnF1VtJq5d8iLv2bbuTIbebzgr1O7YQg8wV+rP8/sOV3Kzso6H6L5d9KEImu2S2jm6NrQsgsMV1hcj37Pz5Sr1tUhu9bzZQxwWdcd0Rj2BZE8h6ekhzu1PDvCzk8UYT4EjvZOlz4PbyAAVw5+oF76iZ6wJNP1Az3dNur4s7HqrovI5a6j7ZR53h5B3ab/ax762Pm6oT0PbhTNTUccL0pD+Jvp0mXG7+gywDGkdvYuRVdbZc2eRWU3qabNPXOoEdlWBmOXxYrzy+C5GVm9KApwAsnPYdSDnz5ir4WU5s/yNQNV7iuWn55DuPbq66Oix5t1H1rno7JJ6jDrhq6envbl3FZy/O1QVnLvj+/dm9BXxZ0+/Fzi9Dokj41GnmATSipZHXHaODiGT6KfFGGfO7GRdL5yUFd6sbXEurzyN19L9SOrdYuN8s9acffs2PzjeqTef1huvn9SeP9Nv5m6Lb8R98Yj8tCueU2UekF8t8af4W/wj/t16uPXL1m9bh4r15g0t87UofLbO/gd5ZAG8</latexit>

Y := f (X) + U ,Brady Neal

Linear Non-Gaussian Setting

X ?? U ,39 / 45

Identifiability in Linear Non-Gaussian Setting:Linear Fit<latexit sha1_base64="Apa+6rzKR8SL6bFZG+j0GQ9gS5U=">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</latexit>

Y := f (X) + U ,Brady Neal

Linear Non-Gaussian Setting

X ?? U ,39 / 45

Identifiability in Linear Non-Gaussian Setting:Linear Fit<latexit sha1_base64="XjNLuZUUlBEpumFRcHxuDhMsxns=">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</latexit>

X := g(Y ) + Ũ ,Brady Neal

Linear Non-Gaussian Setting

Y ?? Ũ ,39 / 45

Identifiability in Linear Non-Gaussian Setting:Linear Fit<latexit sha1_base64="XjNLuZUUlBEpumFRcHxuDhMsxns=">AAAUs3icrVhZb9tGEN6kV+JeTuq3vrBRCqQorUpuAgcFhAawExRFAziAz1qBIZEriRCvkJRVh9DP6Wv/SF/bh/6bzny7FEldlIJIELmanflmdq7lshu6Tpw0Gv/duv3Bhx99/Mmdu1uffvb5F19u37t/GgejyJInVuAG0Xm3E0vX8eVJ4iSuPA8j2fG6rjzrDg94/uxaRrET+MfJTShfe52+7/Qcq5MQ6Wr753Pjp5bRf3TxnfG90U4c15bpycRom4ZptN+MOrZxYbQd35ahpIufGO3jjAdMV9u1Rr2BjzE/aOpBTejPUXDv7l+iLWwRCEuMhCek8EVCY1d0REzfS9EUDRES7bVIiRbRyMG8FBOxRbIj4pLE0SHqkK59+nepqT79Z8wY0hZpcekXkaQhvtU8No17oKo76zcKvMt0pMBmG2/o3tWYHlETMSBqlVzGub5cl75exaoTWsNTrNahlYSgsB+s0pp7dHfpf0Ir5OsNcUoa2SQV0cgimktURWEdEd2V59k3A0SiAz5JI7Z6ld2rrOb5gL5DwurQOIalbKshXui4+NAsYSvzuIjpcsQ/aIXKvlVYvekaHMRdrYJ5j4kyFG9pVEZepbOYXcu5Eo08WaCL+RNw+MQdEyVA5jvkP4c4yrlqESbr6CCSfawlRAbVNfKvmM+yOaR47sKeGFEzkG8O9IS6snK7MztduneBHZF8SnMD6GFfmGSPBHaE7M28aII7on9j/LNgozVDr6MyOZ4mrYmzyyTcgGacKZbyhPJnZpGyLiWaob+74jfolxR3E17hvDZJNoCFKotj6PJ17FvUV9gvHl2Zyp7NKCZ0eLBjgGwZEi3Xp5BY3w+0ihj142rUlO4uvBYCxYR29s8YYw9r4oiOoHtMYxt6WNqgXlcX+9qOSUkr+8ZBd5rXaiD/uHZVZ5zVynI+KtXQWdmCpoZ4rFfMEWC7i16zdDdUPTnG6iYzVnVhf941OEfL9rFUHp9yrGxk4wBSZVz24uJ1LlrLHurERCzZ933iaEGqp+kxajPUmrYK9XegqyRAPCzcPZ0Tqt5Z23hmJiW6pz3lEp+qc96TeOYNYXaQU214rzi/CvG5ls92A+74Kair5U5RWWXZiEbpdGa1vIORhH/UiCOu8EL4OsRavpn+jMLMutjvB8+fQ+SMTTTSuug2cmPxyo9FjXpBjbKq7E+ONFMiPJ2ENH445XxIuMw7hBafKFzh6XR+UhmDQ6JMtMd6mMvqMLfBQeezwdnWmac4VEUlMxxc1au1cl4uyhtFXy1rT/O9LKvoq2WZco3cdHQ/iIF1XiGXIA4s4RWy9LhCKsCeoOjK5os17POx1450XkTawrMKSQ89Rq0q0FX5stK+BF0w0j5hmd/fwYPX6H8TPAFt6sdcNtnIm7nczTv5NJcfb+jZXNLb0L+55NsKSYm9wZnSWOZVZU0x11Ell8qnqj1B0m6SPb+oajNx7ZIXec+2dWcy9H7DWaF2xxZ6gLlWf57fd7iSm5V1PEL37aIPRdBsr6idk/eGlkdwtMb6YuR7fr5cp74Wya2fNweIw6LumM2oJ5D8KSQ7PcSF/ckBfn6ySPEUONY7WfY8uIsMUDH8hXrhS3rGmkDTj/R816Tri9Kuty4qn7NGul8WcfcIeZ/2q0Psa5vjhvo0tFs6M5V1PCcN2W+iT5c5t6vPAMuQNtG7GFllnT13FpnVpJ42B8SlTmA3FYh5Hi/GWx3fxcjqTUmAE0B+DnsfyMUz5np4ec4sfyNQ9Z5i+ek5pHuPri46eqx5D5F1Ljq7pB6jTvjq6elg7l0F5+8eVQXnbvrg/oy+Mv7s6beP0+uIOHIedYpJIK1oRcRl5+gQMol+Woxx5sxP1vXSSVnhzdoWF/LK03gt3Y+k3i22rrZrzdm3b/OD071680m98epx7dlT/WbujvhaPBCPyE/74hlV5hH51RJ/ir/FP+LfnSc7lzvdHVux3r6lZb4Spc+O9z80awkK</latexit>

X := g(Y ) + Ũ ,Brady Neal

Linear Non-Gaussian Setting

Y ?? Ũ ,39 / 45

Identifiability in Linear Non-Gaussian Setting:Residuals<latexit sha1_base64="tOJPvOyJO4WVMXsigr8mI8+AKHk=">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</latexit>

Y := f (X) + U ,

id
s
Re

X ?? Ulsau<latexit sha1_base64="3TqMi5Hh4Whfqm01+zRMl/7BL6k=">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</latexit>

Brady NealX
Linear Non-Gaussian Setting

40 / 45

Identifiability in Linear Non-Gaussian Setting:Residuals<latexit sha1_base64="4B51l08uVYleGrk+rmAuaJd1H68=">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</latexit><latexit sha1_base64="tOJPvOyJO4WVMXsigr8mI8+AKHk=">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</latexit>

Y := f (X) + U ,

id
s
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X ?? U

X := g(Y ) + Ũ ,Y 6?? Ũ ,lsau<latexit sha1_base64="3TqMi5Hh4Whfqm01+zRMl/7BL6k=">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</latexit>

Brady NealX

Y<latexit sha1_base64="yi7fMoOEqm77T8L2JFokYABt1t0=">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</latexit>

Linear Non-Gaussian Setting

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Linear Non-Gaussian Identifiability Extensions

Brady Neal

Linear Non-Gaussian Setting

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Linear Non-Gaussian Identifiability Extensions• Multivariate: Shimizu et al., (2006)

Brady Neal

Linear Non-Gaussian Setting

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Linear Non-Gaussian Identifiability Extensions• Multivariate: Shimizu et al., (2006)• Drop causal sufficiency assumption: Hoyer et al. (2008)

Brady Neal

Linear Non-Gaussian Setting

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Linear Non-Gaussian Identifiability Extensions• Multivariate: Shimizu et al., (2006)• Drop causal sufficiency assumption: Hoyer et al. (2008)• Drop acyclicity assumption: Lacerda et al. (2008)

Brady Neal

Linear Non-Gaussian Setting

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Independence-Based Causal Discovery
Assumptions
Markov Equivalence and Main Theorem
The PC Algorithm
Can We Do Better?

Semi-Parametric Causal Discovery
No Identifiability Without Parametric Assumptions
Linear Non-Gaussian Setting
Nonlinear Additive Noise Setting

Brady Neal

Nonlinear Additive Noise Setting

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Identifiability in Nonlinear Additive Noise Setting

Brady Neal

Nonlinear Additive Noise Setting

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Identifiability in Nonlinear Additive Noise Setting
Recall: We cannot hope to identify the graph more precisely than the Markov
equivalence class in the linear Gaussian noise setting (Geiger & Pearl, 1988).

Brady Neal

Nonlinear Additive Noise Setting

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Identifiability in Nonlinear Additive Noise Setting
Recall: We cannot hope to identify the graph more precisely than the Markov
equivalence class in the linear Gaussian noise setting (Geiger & Pearl, 1988).
What if the structural equations are nonlinear?

Brady Neal

Nonlinear Additive Noise Setting

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Identifiability in Nonlinear Additive Noise Setting
Recall: We cannot hope to identify the graph more precisely than the Markov
equivalence class in the linear Gaussian noise setting (Geiger & Pearl, 1988).
What if the structural equations are nonlinear?
Nonlinear additive noise assumption:

Brady Neal<latexit sha1_base64="ILCc1R+ywcGUyAu5W574IqMhXLQ=">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</latexit>

8i ,where fi is nonlinear<latexit sha1_base64="aDLgw+EFlsId0PRMEp8K/TxDiGA=">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</latexit>

Xi := fi (pai ) + Ui

Nonlinear Additive Noise Setting

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Identifiability in Nonlinear Additive Noise Setting
Recall: We cannot hope to identify the graph more precisely than the Markov
equivalence class in the linear Gaussian noise setting (Geiger & Pearl, 1988).
What if the structural equations are nonlinear?
Nonlinear additive noise assumption:<latexit sha1_base64="ILCc1R+ywcGUyAu5W574IqMhXLQ=">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</latexit>

8i ,where fi is nonlinear<latexit sha1_base64="aDLgw+EFlsId0PRMEp8K/TxDiGA=">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</latexit>

Xi := fi (pai ) + Ui

Theorem (Hoyer et al. 2008): Under the Markov assumption, causal sufficiency,
acyclicity, the nonlinear additive noise assumption, and a technical condition from
Hoyer et al. (2008), we can identify the causal graph.
Brady Neal

Nonlinear Additive Noise Setting

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Post-Nonlinear Setting
Nonlinear additive noise setting: Y := f (X) + U , X ?? U<latexit sha1_base64="tOJPvOyJO4WVMXsigr8mI8+AKHk=">AAAUnnicrVjbbttGEN2kt8S9Oanf+sJGKZC2tCq5KRwUEBDAjlEUNeoAtqM0CgyJXFmEeAtJWXUE/UO/pI99bX+jf9OZs0uR1IWUAksQuZqdOTM7t+WyF7pOnDQa/926/d77H3z40Z27Wx9/8ulnn2/fu38eB6PIkmdW4AZRu9eNpev48ixxEle2w0h2vZ4rX/SGBzz/4kpGsRP4p8l1KF973Uvf6TtWNyHSxfa3xkvjp5bRf9T+xvjOODM6pmEanTejrm20jY7j2zKUdPET4+xiu9aoN/AxFgdNPagJ/TkJ7t39S3SELQJhiZHwhBS+SGjsiq6I6ftKNEVDhER7LSZEi2jkYF6Kqdgi2RFxSeLoEnVI10v690pTffrPmDGkLdLi0i8iSUN8rXlsGvdBVXfWb+R4V+mYAJttvKZ7T2N6RE3EgKhVcinn+nI9+noVq05oDU+wWodWEoLCfrAKa+7T3aX/Ca2Qr9fEKWlkk1REI4toLlEVhXVEdFeeZ98MEIku+CSN2Ooyu8us5vmAvkPC6tI4hqVsqyGOdFx8aJawlXlcxHQ14h+0QmVfGVZ/tgYHcVerYN5TogzFWxoVkct05rNrNVeikadLdDF/Ag6fuGOiBMh8h/znEEcxVy3CZB1dRPISawmRQXWN/Avm02wOKZ67sCdG1AzkmwM9oa6szO7UTpfuPWBHJD+huQH0sC9MskcCO0L2pl40wR3RvzH+WbDRmqPXUZkcT5PWxNllEm5AM84MS3lC+TO1SFk3IZqhv7viV+iXFHcTXuG8Nkk2gIUqi2Po8nXsW9RX2C8eXZnKnk0pJnR4sGOAbBkSLdOnkFjf97SKGPXjatQJ3V14LQSKCe3snzHGHtbEER1B95jGNvSwtEG9ri72tR3Tglb2jYPutKjVQP5x7arOOK+V5XxUqqGzsgVNDfFYr5gjwHbnvWbpbqh6cozVTees6sH+rGtwjhbtY6ksPsVY2cjGAaSKuOzF5etctpY91ImJWLLvL4mjBam+pseozVBr2srV34GukgDxsHD3dE6oemdt47mZCdE97SmX+FSd857EM28Is4uc6sB7+fkyxGdaPt0NuONPQC2XO0dlFWUjGk1mM+XyDkYS/lEjjrjCC+HrEGv5avYzcjPrYt8Mnr+AyBmbaKR10W3kxvKVn4oa9YIaZVXRnxxppkR4Oglp/HDG+ZBwmXcILT5RuMIns/lpZQwOiTLVHutjLq3DzAYHnc8GZ0dnnuJQFZXMcXBVl2vlvFyWN4peLmvP8r0oq+jlsky5Qm46uh/EwGpXyCWIA0t4uSw9rZAKsCcourL55Rr2+dhrRzovIm3hiwpJDz1GrSrQVXlcaV+CLhhpn7DM7+/gwSv0vymegDb1YyabbOTNTO76nXyayY839Gwm6W3o30zybYWkxN7gzGgs87yyppjrpJJL5VPVniBpN0mfX1S1mbj2yIu8Z9u6Mxl6v+GsULtjCz3AXKs/L+47XMnNyjoeofv20IciaLZLaufsxtCyCI7WWF+MfM/Ol+vU1zK59fPmAHFY1h3TGfUEkj2FpKeHOLc/OcDPThYTPAWO9U6WPg/uIgNUDH+mXnhMz1hTaPqBnu+adD0q7HrrovI5a6T7ZR53j5D3ab86xL62OW6oT0O7hTNTUccz0pD+pvp0mXG7+gywCmkTvcuRVdbZC2eReU3qaXNAXOoEdl2BmOXxcrzy+C5HVm9KApwAsnPYTSDnz5jr4WU5s/qNQNV7itWn55Dufbq66Oix5j1E1rno7JJ6jDrhq6eng4V3FZy/e1QVnLuTB/fn9BXx50+/lzi9jogj41GnmATSipZHXHWODiGT6KfFGGfO7GRdL5yUFd68bXEurzyN19L9SOrdYutiu9acf/u2ODjfqzd/rDeeP649faLfzN0RX4oH4hH5aV88pco8Ib9a4k/xt/hH/Ltj7BztHO/8plhv39IyX4jCZ6f9PwQXAJY=</latexit>

Brady Neal

Nonlinear Additive Noise Setting

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Post-Nonlinear Setting
Nonlinear additive noise setting: Y := f (X) + U , X ?? U<latexit sha1_base64="tOJPvOyJO4WVMXsigr8mI8+AKHk=">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</latexit><latexit sha1_base64="tOJPvOyJO4WVMXsigr8mI8+AKHk=">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</latexit>

Y := f (X) + U ,Brady Neal

X ?? U

Nonlinear Additive Noise Setting

44 / 45

Post-Nonlinear Setting
Nonlinear additive noise setting: Y := f (X) + U , X ?? U<latexit sha1_base64="tOJPvOyJO4WVMXsigr8mI8+AKHk=">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</latexit><latexit sha1_base64="nvFPVVHQUiWtoDiaJH3T44Q1QYQ=">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</latexit>

Y := g(f (X) + U ) ,Brady Neal

X ?? U

Nonlinear Additive Noise Setting

44 / 45

Post-Nonlinear Setting
Nonlinear additive noise setting: Y := f (X) + U , X ?? U<latexit sha1_base64="tOJPvOyJO4WVMXsigr8mI8+AKHk=">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</latexit>

Post-nonlinear (Zhang & Hyvärinen, 2009):<latexit sha1_base64="nvFPVVHQUiWtoDiaJH3T44Q1QYQ=">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</latexit>

Y := g(f (X) + U ) ,Brady Neal

X ?? U

Nonlinear Additive Noise Setting

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Review Articles and Book
• Introduction to the Foundations of Causal Discovery (Eberhardt, 2017)• Review of Causal Discovery Methods Based on Graphical Models(Glymour, Zhang, & Spirtes, 2019)• Elements of Causal Inference (Peters, Janzing, & Schölkopf, 2017)

Brady Neal

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