-
-Factorizing the Joint Distribution
-
-
-
-Factorizing the Joint Distribution
-
-We have seen how these graphical models can be used to determine whether or not we can identify a treatment effect.
However, we can also use them as a computational tool, to help us find the simplest representation of the joint distribution of the data.
This is useful because it allows us to find the simplest model that is consistent with our assumptions about conditional independence.
-
-
-Factorizing the Joint Distribution
-
-To understand why this is useful, we’ll follow this example (from Mark Paskin).
Suppose we have a DGP with five variables:
-
-- \(E\in\{true, false\}\) - Has an earthquake happened? Earthquakes are unlikely
-- \(B\in\{true, false\}\) - Has a burglary happened? Burglaries are unlikely, but more likely than earthquakes
-- \(A\in\{true, false\}\) - Is the alarm going off? The alarm is triggered by both earthquakes and burglaries
-- \(J\in\{true, false\}\) - Is John calling? John calls when he hears the alarm, but he often misses it
-- \(M\in\{true, false\}\) - Is Mary calling? Mary calls when she hears the alarm, but she also calls to chat
-
-
-
-Factorizing the Joint Distribution
-
-- The goal is to compute \(P(B|J=true)\) from the joint distribution \(P(E,B,A,J,M)\). We’ll start by drawing the graphical model, to understand the conditional independence relationships.
-
-
-![]()
-
-Factorizing the Joint Distribution
-
-![]()
-- In order to represent the full joint distribution, we could use the chain rule of probabilities:
-
-\[
-P(E,B,A,J,M) = P(E)P(B|E)P(A|E,B)P(J|E,B,A)P(M|E,B,A,J)
-\]
-
-- Q: How many probabilities would we need to compute to represent the joint distribution this way?
-
-
-
-Factorizing the Joint Distribution
-
-![]()
-- In order to represent the full joint distribution, we could use the chain rule of probabilities:
-
-\[
-\underbrace{P(E,B,A,J,M)}_{31} = \underbrace{P(E)}_{1} \underbrace{P(B|E)}_{2} \underbrace{P(A|E,B)}_{4} \underbrace{P(J|E,B,A)}_{8} \underbrace{P(M|E,B,A,J)}_{16}
-\]
-
-- Q: How many probabilities would we need to store to represent the joint distribution this way?
-
-- A: There are \(2^5\) possible outcomes. We need 31 probabilities. (why not 32?)
-
-
-
-
-Factorizing the Joint Distribution
-
-![]()
-- However, we can use the conditional independence relationships to simplify this representation.
-
-- For example, we know that \(P(B|E) = P(B)\), because \(B\) and \(E\) are independent.
-- We also know that \(P(A|E,B) = P(A|B)\), because \(A\) is independent of \(E\), given \(B\).
-
-- This means that we can simplify the joint distribution to:
-
-\[
-\underbrace{P(E,B,A,J,M)}_{10} = \underbrace{P(E)}_1 \underbrace{P(B)}_1 \underbrace{P(A|E,B)}_4 \underbrace{P(J|A)}_2 \underbrace{P(M|A)}_2
-\]
-
-
-Factorizing the Joint Distribution
-
-Beyond causal inference, graphical models are a useful tool for computing all kinds of conditional probabilities.
This is useful in many inference problems, and in structural econometrics
-
-- The likelihood function is the conditional probability of the data, given the parameter values
-- In Bayesian inference the posterior is the conditional probability of the parameters, given the data (and our priors)
-
-
-
-Variable Elimination
-
-In order to simplify a conditional distribution, we can use variable elimination.
For example, let’s say we want to compute the probability of a burglary, given that John calls.
-\[
-\begin{aligned}
-&p_{B \mid J}(b, \text { true }) \propto \sum_e \sum_a \sum_m p_{E B A J M}(e, b, a, \text { true }, m) \\
-& =\sum_e \sum_a \sum_m p_E(e) \cdot p_B(b) \cdot p_{A \mid E B}(a, e, b) \cdot p_{J \mid A}(\text { true }, a) \cdot p_{M \mid A}(m, a)
-\end{aligned}
-\]
-
-- Then, we can reduce the computational complexity by eliminating variables one at a time, exploiting the distributive property of multiplication
-
-- \(xy + xz = x(y + z)\).
-
-
-
-
-Variable Elimination
-
-- Variable elimination works like this:
-
-- Repeat the following steps:
-
-
-- choose a variable to eliminate
-- push in its sum as far as possible
-- compute the sum, resulting in a new factor
-
-
-
-
-Variable Elimination
-\[
-\begin{aligned}
-&\sum_e \sum_a \sum_m p_E(e) \cdot p_B(b) \cdot p_{A \mid E B}(a, e, b) \cdot p_{J \mid A}(\text { true }, a) \cdot p_{M \mid A}(m, a) \\
-& =\sum_e \sum_a p_E(e) \cdot p_B(b) \cdot p_{A \mid E B}(a, e, b) \cdot p_{J \mid A}(\text { true }, a) \cdot \sum_m p_{M \mid A}(m, a) \\
-& =\sum_e \sum_a p_E(e) \cdot p_B(b) \cdot p_{A \mid E B}(a, e, b) \cdot p_{J \mid A}(\text { true }, a) \cdot \psi_A(a) \\
-& =\sum_e p_E(e) \cdot p_B(b) \cdot \sum_a p_{A \mid E B}(a, e, b) \cdot p_{J \mid A}(\text { true }, a) \cdot \psi_A(a) \\
-& =\sum_e p_E(e) \cdot p_B(b) \cdot \psi_{E B}(e, b) \\
-& =p_B(b) \cdot \sum_e p_E(e) \cdot \psi_{E B}(e, b) \\
-& =p_B(b) \cdot \psi_B(b)
-\end{aligned}
-\]
-
-
-Variable Elimination
-
-- That’s a lot of math! But let’s focus on the first step: \[
-\begin{aligned}
-& p_{B \mid J}(b, \text { true }) \propto \\
-&\underbrace{\underbrace{\sum_e \sum_a \sum_m}_{2^3 = 8\text{ iterations}} \underbrace{p_E(e) \cdot p_B(b) \cdot p_{A \mid E B}(a, e, b) \cdot p_{J \mid A}(\text { true }, a) \cdot p_{M \mid A}(m, a)}_{4\text{ multiplications}}}_{8*4=32\text{ multiplications }+7\text{ additions}=39\text{ total operations}}
-\end{aligned}
-\]
-
-
-
-Variable Elimination
-
-- That’s a lot of math! But let’s focus on the first step: \[
-\begin{aligned}
-& p_{B \mid J}(b, \text { true }) \propto \\
-&\underbrace{\underbrace{\sum_e \sum_a \sum_m}_{2^3 = 8\text{ iterations}} \underbrace{p_E(e) \cdot p_B(b) \cdot p_{A \mid E B}(a, e, b) \cdot p_{J \mid A}(\text { true }, a) \cdot p_{M \mid A}(m, a)}_{4\text{ multiplications}}}_{8*4=32\text{ multiplications }+7\text{ additions}=39\text{ total operations}} \\\\
-& =\underbrace{\underbrace{\sum_e \sum_a}_{2^2 = 4\text{ iterations}} \underbrace{p_E(e) \cdot p_B(b) \cdot p_{A \mid E B}(a, e, b) \cdot p_{J \mid A}(\text { true }, a) \cdot }_{4\text{ multiplications}} \underbrace{\sum_m p_{M \mid A}(m, a)}_{1\text{ addition}}}_{4*(4\text{ multiplications} + 1\text{ addition}) + 3\text{ additions} = 23\text{ total operations}} \\
-\end{aligned}
-\]
-
-
-
-Variable Elimination
-
-Variable elimination is an algorithm that exploits our conditional independence assumptions to reduce the computational complexity of conditional probabilities.
In this case, we were able to reduce the number of operations from 39 to 23 operations in just one step, each further step will continue to reduce the complexity.
This is a very simple example, because we only have 5 variables. In practice, we might have hundreds or thousands of variables, and the computational complexity can become very large.
Systematic patterns of conditional independence can be exploited to reduce the complexity of inference problems in some large models.
Credits
diff --git a/lectures/lectures/causal_graphical_models.ipynb b/lectures/lectures/causal_graphical_models.ipynb
index a9bd2b7..901d0e6 100644
--- a/lectures/lectures/causal_graphical_models.ipynb
+++ b/lectures/lectures/causal_graphical_models.ipynb
@@ -6,7 +6,7 @@
"source": [
"# ECON526: Quantitative Economics with Data Science Applications\n",
"\n",
- "Directed Graphical Models and Causality\n",
+ "Introduction to Directed Acyclic Graphs\n",
"\n",
"Phil Solimine (University of British Columbia)\n",
"\n",
@@ -84,7 +84,7 @@
"\n",
"- A directed graphical model might look something like this:"
],
- "id": "275de8ce-9151-4e06-b894-00611be3713a"
+ "id": "85cc77ac-a74c-44e4-afdc-29fa59e143ba"
},
{
"cell_type": "code",
@@ -112,7 +112,7 @@
"g.edge('Z', 'Y')\n",
"g"
],
- "id": "2a5ee5d2"
+ "id": "9dd3b9a6"
},
{
"cell_type": "markdown",
@@ -124,7 +124,7 @@
"\n",
"## Directed Graphical Models"
],
- "id": "fd5e110b-5897-4d09-99e3-ef0a2a966f60"
+ "id": "34a34acc-8cef-416c-a6d8-76a69815f7a3"
},
{
"cell_type": "code",
@@ -148,7 +148,7 @@
"g.edge(\"severeness\", \"medicine\")\n",
"g"
],
- "id": "57be38c5"
+ "id": "a75db644"
},
{
"cell_type": "markdown",
@@ -173,7 +173,7 @@
"\n",
"## Directed Graphical Models"
],
- "id": "4889b381-ee31-481a-9405-d9493ff037da"
+ "id": "496d6749-371a-4218-b834-5efbc07d1bd3"
},
{
"cell_type": "code",
@@ -205,7 +205,7 @@
"\n",
"g"
],
- "id": "f5d47833"
+ "id": "63fd93a3"
},
{
"cell_type": "markdown",
@@ -253,7 +253,7 @@
"\n",
"Now let’s look at another common structure:"
],
- "id": "ae7c3355-83ee-4510-9e17-04f732e5ae0d"
+ "id": "171c8f12-d930-430c-a318-0f92b1067554"
},
{
"cell_type": "code",
@@ -284,7 +284,7 @@
"\n",
"g"
],
- "id": "0c97f505"
+ "id": "3adadb58"
},
{
"cell_type": "markdown",
@@ -295,7 +295,7 @@
"\n",
"## Directed Graphical Models"
],
- "id": "ebe6faec-5ace-44bc-a37e-994b33165750"
+ "id": "00c19592-fd0d-4afd-b137-37b8a33de17f"
},
{
"cell_type": "code",
@@ -313,7 +313,7 @@
"source": [
"g"
],
- "id": "3c903f53"
+ "id": "797fc24c"
},
{
"cell_type": "markdown",
@@ -330,7 +330,7 @@
"\n",
"## Directed Graphical Models"
],
- "id": "aec8e61c-ed8f-4150-99a2-bd6e6b10aad8"
+ "id": "465dd26a-b281-42d9-a870-d1c9340b3322"
},
{
"cell_type": "code",
@@ -348,7 +348,7 @@
"source": [
"g"
],
- "id": "cb0623af"
+ "id": "c2c392cd"
},
{
"cell_type": "markdown",
@@ -363,7 +363,7 @@
"\n",
"Finally, let’s look at a third common structure, called a **collider**:"
],
- "id": "28fbe80e-235c-4dfa-b273-d5f3a818bd1d"
+ "id": "75cc42a0-c68a-412c-8ba2-e205d94d8c93"
},
{
"cell_type": "code",
@@ -394,7 +394,7 @@
"\n",
"g"
],
- "id": "023b3fbf"
+ "id": "eadccd31"
},
{
"cell_type": "markdown",
@@ -435,7 +435,7 @@
"- Let’s say we were interested in the effect of online classes on\n",
" college dropout rates. The treatment effect we want to identify is:"
],
- "id": "17a0d5bd-9cfe-4501-b726-d7809748cc4d"
+ "id": "dc86fee0-90fd-4b90-ad05-90d18732cf07"
},
{
"cell_type": "code",
@@ -455,7 +455,7 @@
"g.edge(\"OnlineClass\", \"Dropout\", style=\"dashed\")\n",
"g"
],
- "id": "a9a51bf5"
+ "id": "09fbd1eb"
},
{
"cell_type": "markdown",
@@ -466,7 +466,7 @@
"- In addition to `OnlineClass`, there are **many** variables that\n",
" might cause a student to drop out of college."
],
- "id": "684a804a-2e8d-4409-ad4f-a6f6ae2ca032"
+ "id": "cbc4651e-856f-47f1-8d8d-38d6ed344434"
},
{
"cell_type": "code",
@@ -490,7 +490,7 @@
"\n",
"g"
],
- "id": "caabe963"
+ "id": "779be6e4"
},
{
"cell_type": "markdown",
@@ -501,7 +501,7 @@
"- Some of these variables might **also** determine whether or not a\n",
" student takes an online class."
],
- "id": "e977a379-da4e-4d51-bca9-e057a0430c01"
+ "id": "b09b0cf1-6a64-4f4a-8d09-67c15aa79731"
},
{
"cell_type": "code",
@@ -521,7 +521,7 @@
"\n",
"g"
],
- "id": "4dc4eafc"
+ "id": "b6b15dfa"
},
{
"cell_type": "markdown",
@@ -532,7 +532,7 @@
"- Then there are some other variables that affect just `OnlineClass`\n",
" and not `Dropout`"
],
- "id": "aff1b5ba-9ee6-4e43-a7b0-2bbbc59c409d"
+ "id": "bf47df39-8d40-447f-9d15-99dbba5a7f40"
},
{
"cell_type": "code",
@@ -553,7 +553,7 @@
"\n",
"g"
],
- "id": "5a55117a"
+ "id": "5ed83819"
},
{
"cell_type": "markdown",
@@ -563,7 +563,7 @@
"\n",
"- There are relationships between some of these variables as well"
],
- "id": "43d48b68-8696-4fd7-957a-8cf788454485"
+ "id": "cc3c2f99-e1f1-4f66-a558-d5fba395d6d1"
},
{
"cell_type": "code",
@@ -586,7 +586,7 @@
"\n",
"g"
],
- "id": "62897f63"
+ "id": "d8e0b476"
},
{
"cell_type": "markdown",
@@ -598,7 +598,7 @@
" variables, there might be an indirect relationship through a third\n",
" variable"
],
- "id": "7ecffc84-d215-4160-b653-a0d11449daf8"
+ "id": "c53a26d8-170c-422e-8381-c8e406f55614"
},
{
"cell_type": "code",
@@ -627,7 +627,7 @@
"\n",
"g"
],
- "id": "8eed9cd8"
+ "id": "d06bfd0e"
},
{
"cell_type": "markdown",
@@ -638,7 +638,7 @@
"- And then there are some variables that we know are correlated, but\n",
" due to some other combination of unknown factors"
],
- "id": "7ecbde3b-faab-48d6-a88d-332e0b3d978b"
+ "id": "7dc322d9-32e5-4983-9d0f-ca2c05590f40"
},
{
"cell_type": "code",
@@ -676,7 +676,7 @@
"\n",
"g"
],
- "id": "6d4e03d5"
+ "id": "aef21e18"
},
{
"cell_type": "markdown",
@@ -798,7 +798,7 @@
"image/svg+xml": 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nZSI+Cjx0aXRsZT5BZ2UmIzQ1OyZndDtQcmVmZXJlbmNlczwvdGl0bGU+CjxwYXRoIGZp\nbGw9Im5vbmUiIHN0cm9rZT0iYmxhY2siIGQ9Ik0zMjQuOTMsLTI4OC4xNUMzMzEuNDEsLTI2My45\nNCAzNDMuNDMsLTIxOS4wNSAzNTEuMTcsLTE5MC4xMyIvPgo8cG9seWdvbiBmaWxsPSJibGFjayIg\nc3Ryb2tlPSJibGFjayIgcG9pbnRzPSIzNTQuNiwtMTkwLjgzIDM1My44MSwtMTgwLjI3IDM0Ny44\nNCwtMTg5LjAyIDM1NC42LC0xOTAuODMiLz4KPC9nPgo8IS0tIFNFUyAtLT4KPGcgaWQ9Im5vZGUx\nMSIgY2xhc3M9Im5vZGUiPgo8dGl0bGU+U0VTPC90aXRsZT4KPGVsbGlwc2UgZmlsbD0ibm9uZSIg\nc3Ryb2tlPSJibGFjayIgY3g9IjUyNy40MyIgY3k9Ii0yMzQiIHJ4PSIyNyIgcnk9IjE4Ii8+Cjx0\nZXh0IHRleHQtYW5jaG9yPSJtaWRkbGUiIHg9IjUyNy40MyIgeT0iLTIzMC4zIiBmb250LWZhbWls\neT0iVGltZXMgTmV3IFJvbWFuLHNlcmlmIiBmb250LXNpemU9IjE0LjAwIj5TRVM8L3RleHQ+Cjwv\nZz4KPCEtLSBBZ2UmIzQ1OyZndDtTRVMgLS0+CjxnIGlkPSJlZGdlMTUiIGNsYXNzPSJlZGdlIj4K\nPHRpdGxlPkFnZSYjNDU7Jmd0O1NFUzwvdGl0bGU+CjxwYXRoIGZpbGw9Im5vbmUiIHN0cm9rZT0i\nYmxhY2siIGQ9Ik0zNDMuODMsLTI5Ni42MUMzNTEuNjQsLTI5My44NCAzNjAuNCwtMjkwLjc2IDM2\nOC40MywtMjg4IDQxMS42MywtMjczLjE1IDQ2MS42NSwtMjU2LjU4IDQ5My44NywtMjQ1Ljk4Ii8+\nCjxwb2x5Z29uIGZpbGw9ImJsYWNrIiBzdHJva2U9ImJsYWNrIiBwb2ludHM9IjQ5NS4yNywtMjQ5\nLjIxIDUwMy42NywtMjQyLjc2IDQ5My4wOCwtMjQyLjU2IDQ5NS4yNywtMjQ5LjIxIi8+CjwvZz4K\nPCEtLSBXb3JrSG91cnMgLS0+CjxnIGlkPSJub2RlMTIiIGNsYXNzPSJub2RlIj4KPHRpdGxlPldv\ncmtIb3VyczwvdGl0bGU+CjxlbGxpcHNlIGZpbGw9Im5vbmUiIHN0cm9rZT0iYmxhY2siIGN4PSIy\nMDIuNDMiIGN5PSItMjM0IiByeD0iNTEuOTkiIHJ5PSIxOCIvPgo8dGV4dCB0ZXh0LWFuY2hvcj0i\nbWlkZGxlIiB4PSIyMDIuNDMiIHk9Ii0yMzAuMyIgZm9udC1mYW1pbHk9IlRpbWVzIE5ldyBSb21h\nbixzZXJpZiIgZm9udC1zaXplPSIxNC4wMCI+V29ya0hvdXJzPC90ZXh0Pgo8L2c+CjwhLS0gQWdl\nJiM0NTsmZ3Q7V29ya0hvdXJzIC0tPgo8ZyBpZD0iZWRnZTE2IiBjbGFzcz0iZWRnZSI+Cjx0aXRs\nZT5BZ2UmIzQ1OyZndDtXb3JrSG91cnM8L3RpdGxlPgo8cGF0aCBmaWxsPSJub25lIiBzdHJva2U9\nImJsYWNrIiBkPSJNMzAwLjk0LC0yOTMuNDRDMjgzLjQxLC0yODMuMDQgMjU3LjI4LC0yNjcuNTQg\nMjM2LjMzLC0yNTUuMTEiLz4KPHBvbHlnb24gZmlsbD0iYmxhY2siIHN0cm9rZT0iYmxhY2siIHBv\naW50cz0iMjM4LjA5LC0yNTIuMDkgMjI3LjcxLC0yNTAgMjM0LjUyLC0yNTguMTEgMjM4LjA5LC0y\nNTIuMDkiLz4KPC9nPgo8IS0tIEdlbmRlciAtLT4KPGcgaWQ9Im5vZGU1IiBjbGFzcz0ibm9kZSI+\nCjx0aXRsZT5HZW5kZXI8L3RpdGxlPgo8ZWxsaXBzZSBmaWxsPSJub25lIiBzdHJva2U9ImJsYWNr\nIiBjeD0iNDEzLjQzIiBjeT0iLTMwNiIgcng9IjM2LjI5IiByeT0iMTgiLz4KPHRleHQgdGV4dC1h\nbmNob3I9Im1pZGRsZSIgeD0iNDEzLjQzIiB5PSItMzAyLjMiIGZvbnQtZmFtaWx5PSJUaW1lcyBO\nZXcgUm9tYW4sc2VyaWYiIGZvbnQtc2l6ZT0iMTQuMDAiPkdlbmRlcjwvdGV4dD4KPC9nPgo8IS0t\nIEdlbmRlciYjNDU7Jmd0O0Ryb3BvdXQgLS0+CjxnIGlkPSJlZGdlNCIgY2xhc3M9ImVkZ2UiPgo8\ndGl0bGU+R2VuZGVyJiM0NTsmZ3Q7RHJvcG91dDwvdGl0bGU+CjxwYXRoIGZpbGw9Im5vbmUiIHN0\ncm9rZT0iYmxhY2siIGQ9Ik00MTkuNTEsLTI4OC4wN0M0MjIuODgsLTI3Ny43OSA0MjYuNzIsLTI2\nNC4zMSA0MjguNDMsLTI1MiA0MzkuODEsLTE2OS45OSA0NDUuNzEsLTEyMi45MyAzODAuNDMsLTcy\nIDMzNS4yNSwtMzYuNzUgMjY5LjIyLC0yNC43NSAyMjMuOTYsLTIwLjc2Ii8+Cjxwb2x5Z29uIGZp\nbGw9ImJsYWNrIiBzdHJva2U9ImJsYWNrIiBwb2ludHM9IjIyNC4wMiwtMTcuMjYgMjEzLjc4LC0x\nOS45NyAyMjMuNDgsLTI0LjI0IDIyNC4wMiwtMTcuMjYiLz4KPC9nPgo8IS0tIEdlbmRlciYjNDU7\nJmd0O0F2YWlsYWJsZVRpbWUgLS0+CjxnIGlkPSJlZGdlMjMiIGNsYXNzPSJlZGdlIj4KPHRpdGxl\nPkdlbmRlciYjNDU7Jmd0O0F2YWlsYWJsZVRpbWU8L3RpdGxlPgo8cGF0aCBmaWxsPSJub25lIiBz\ndHJva2U9ImJsYWNrIiBkPSJNMzkzLjc2LC0yOTAuNkMzNTkuNjcsLTI2NS42NSAyODkuNDQsLTIx\nNC4yNSAyNDkuMywtMTg0Ljg3Ii8+Cjxwb2x5Z29uIGZpbGw9ImJsYWNrIiBzdHJva2U9ImJsYWNr\nIiBwb2ludHM9IjI1MS4zNSwtMTgyLjAzIDI0MS4yMSwtMTc4Ljk1IDI0Ny4yMiwtMTg3LjY4IDI1\nMS4zNSwtMTgyLjAzIi8+CjwvZz4KPCEtLSBHZW5kZXImIzQ1OyZndDtQcmVmZXJlbmNlcyAtLT4K\nPGcgaWQ9ImVkZ2UxMiIgY2xhc3M9ImVkZ2UiPgo8dGl0bGU+R2VuZGVyJiM0NTsmZ3Q7UHJlZmVy\nZW5jZXM8L3RpdGxlPgo8cGF0aCBmaWxsPSJub25lIiBzdHJva2U9ImJsYWNrIiBkPSJNNDE2LjQx\nLC0yODcuODFDNDE4LjgzLC0yNjkuMjggNDIwLjQxLC0yMzkuMzEgNDEwLjQzLC0yMTYgNDA1LjQz\nLC0yMDQuMzQgMzk2LjU5LC0xOTMuODEgMzg3LjU5LC0xODUuMjYiLz4KPHBvbHlnb24gZmlsbD0i\nYmxhY2siIHN0cm9rZT0iYmxhY2siIHBvaW50cz0iMzg5Ljg4LC0xODIuNjEgMzgwLjA5LC0xNzgu\nNTcgMzg1LjIyLC0xODcuODQgMzg5Ljg4LC0xODIuNjEiLz4KPC9nPgo8IS0tIEdlbmRlciYjNDU7\nJmd0O1dvcmtIb3VycyAtLT4KPGcgaWQ9ImVkZ2UyMSIgY2xhc3M9ImVkZ2UiPgo8dGl0bGU+R2Vu\nZGVyJiM0NTsmZ3Q7V29ya0hvdXJzPC90aXRsZT4KPHBhdGggZmlsbD0ibm9uZSIgc3Ryb2tlPSJi\nbGFjayIgZD0iTTM4NC4wOSwtMjk1LjI3QzM0OC44NiwtMjgzLjU4IDI4OS4zOSwtMjYzLjg1IDI0\nOC4xMywtMjUwLjE2Ii8+Cjxwb2x5Z29uIGZpbGw9ImJsYWNrIiBzdHJva2U9ImJsYWNrIiBwb2lu\ndHM9IjI0OS4xNywtMjQ2LjgyIDIzOC41OCwtMjQ2Ljk5IDI0Ni45NywtMjUzLjQ2IDI0OS4xNywt\nMjQ2LjgyIi8+CjwvZz4KPCEtLSBSYWNlIC0tPgo8ZyBpZD0ibm9kZTYiIGNsYXNzPSJub2RlIj4K\nPHRpdGxlPlJhY2U8L3RpdGxlPgo8ZWxsaXBzZSBmaWxsPSJub25lIiBzdHJva2U9ImJsYWNrIiBj\neD0iMjI0LjQzIiBjeT0iLTMwNiIgcng9IjI4LjciIHJ5PSIxOCIvPgo8dGV4dCB0ZXh0LWFuY2hv\ncj0ibWlkZGxlIiB4PSIyMjQuNDMiIHk9Ii0zMDIuMyIgZm9udC1mYW1pbHk9IlRpbWVzIE5ldyBS\nb21hbixzZXJpZiIgZm9udC1zaXplPSIxNC4wMCI+UmFjZTwvdGV4dD4KPC9nPgo8IS0tIFJhY2Um\nIzQ1OyZndDtEcm9wb3V0IC0tPgo8ZyBpZD0iZWRnZTUiIGNsYXNzPSJlZGdlIj4KPHRpdGxlPlJh\nY2UmIzQ1OyZndDtEcm9wb3V0PC90aXRsZT4KPHBhdGggZmlsbD0ibm9uZSIgc3Ryb2tlPSJibGFj\nayIgZD0iTTIwMi4zMywtMjk0LjU2QzE3OC4zNSwtMjgyLjg0IDE0Mi4xNiwtMjYzLjkyIDEzMi40\nMywtMjUyIDExMS4wOSwtMjI1Ljg4IDExMy44MywtMjEzLjMgMTA4LjQzLC0xODAgMTAwLjc0LC0x\nMzIuNjIgODcuODMsLTExNS4zNiAxMDguNDMsLTcyIDExNS4wNywtNTggMTI3LjQzLC00Ni40OCAx\nMzkuNiwtMzcuOCIvPgo8cG9seWdvbiBmaWxsPSJibGFjayIgc3Ryb2tlPSJibGFjayIgcG9pbnRz\nPSIxNDEuNTUsLTQwLjcxIDE0Ny45MiwtMzIuMjQgMTM3LjY2LC0zNC44OCAxNDEuNTUsLTQwLjcx\nIi8+CjwvZz4KPCEtLSBSYWNlJiM0NTsmZ3Q7QXZhaWxhYmxlVGltZSAtLT4KPGcgaWQ9ImVkZ2Uy\nMiIgY2xhc3M9ImVkZ2UiPgo8dGl0bGU+UmFjZSYjNDU7Jmd0O0F2YWlsYWJsZVRpbWU8L3RpdGxl\nPgo8cGF0aCBmaWxsPSJub25lIiBzdHJva2U9ImJsYWNrIiBkPSJNMjAyLjU4LC0yOTQuMTVDMTgw\nLjMzLC0yODIuNTkgMTQ4LjE2LC0yNjQuMjkgMTQxLjQzLC0yNTIgMTMzLjc0LC0yMzcuOTcgMTMz\nLjk0LC0yMzAuMTQgMTQxLjQzLC0yMTYgMTQ4Ljk3LC0yMDEuNzYgMTYyLjI0LC0xOTAuNjMgMTc1\nLjczLC0xODIuMzMiLz4KPHBvbHlnb24gZmlsbD0iYmxhY2siIHN0cm9rZT0iYmxhY2siIHBvaW50\ncz0iMTc3LjYyLC0xODUuMjggMTg0LjU2LC0xNzcuMjggMTc0LjE0LC0xNzkuMjEgMTc3LjYyLC0x\nODUuMjgiLz4KPC9nPgo8IS0tIFJhY2UmIzQ1OyZndDtQcmVmZXJlbmNlcyAtLT4KPGcgaWQ9ImVk\nZ2UxMyIgY2xhc3M9ImVkZ2UiPgo8dGl0bGU+UmFjZSYjNDU7Jmd0O1ByZWZlcmVuY2VzPC90aXRs\nZT4KPHBhdGggZmlsbD0ibm9uZSIgc3Ryb2tlPSJibGFjayIgZD0iTTIzOC41MSwtMjkwLjA3QzI2\nMS43MSwtMjY1LjQ5IDMwOC4xMiwtMjE2LjMxIDMzNS45MSwtMTg2Ljg2Ii8+Cjxwb2x5Z29uIGZp\nbGw9ImJsYWNrIiBzdHJva2U9ImJsYWNrIiBwb2ludHM9IjMzOC41NywtMTg5LjE0IDM0Mi44OSwt\nMTc5LjQ3IDMzMy40OCwtMTg0LjM0IDMzOC41NywtMTg5LjE0Ii8+CjwvZz4KPCEtLSBSYWNlJiM0\nNTsmZ3Q7V29ya0hvdXJzIC0tPgo8ZyBpZD0iZWRnZTIwIiBjbGFzcz0iZWRnZSI+Cjx0aXRsZT5S\nYWNlJiM0NTsmZ3Q7V29ya0hvdXJzPC90aXRsZT4KPHBhdGggZmlsbD0ibm9uZSIgc3Ryb2tlPSJi\nbGFjayIgZD0iTTIxOS4xLC0yODguMDVDMjE2LjY1LC0yODAuMjYgMjEzLjY4LC0yNzAuODIgMjEw\nLjk0LC0yNjIuMDgiLz4KPHBvbHlnb24gZmlsbD0iYmxhY2siIHN0cm9rZT0iYmxhY2siIHBvaW50\ncz0iMjE0LjE5LC0yNjAuNzcgMjA3Ljg2LC0yNTIuMjggMjA3LjUyLC0yNjIuODYgMjE0LjE5LC0y\nNjAuNzciLz4KPC9nPgo8IS0tIEFjYWRlbWljcyAtLT4KPGcgaWQ9Im5vZGU3IiBjbGFzcz0ibm9k\nZSI+Cjx0aXRsZT5BY2FkZW1pY3M8L3RpdGxlPgo8ZWxsaXBzZSBmaWxsPSJub25lIiBzdHJva2U9\nImJsYWNrIiBjeD0iMTI3LjQzIiBjeT0iLTMwNiIgcng9IjUwLjA5IiByeT0iMTgiLz4KPHRleHQg\ndGV4dC1hbmNob3I9Im1pZGRsZSIgeD0iMTI3LjQzIiB5PSItMzAyLjMiIGZvbnQtZmFtaWx5PSJU\naW1lcyBOZXcgUm9tYW4sc2VyaWYiIGZvbnQtc2l6ZT0iMTQuMDAiPkFjYWRlbWljczwvdGV4dD4K\nPC9nPgo8IS0tIEFjYWRlbWljcyYjNDU7Jmd0O0Ryb3BvdXQgLS0+CjxnIGlkPSJlZGdlNiIgY2xh\nc3M9ImVkZ2UiPgo8dGl0bGU+QWNhZGVtaWNzJiM0NTsmZ3Q7RHJvcG91dDwvdGl0bGU+CjxwYXRo\nIGZpbGw9Im5vbmUiIHN0cm9rZT0iYmxhY2siIGQ9Ik04OS4xMSwtMjk0LjEzQzY4LjQyLC0yODYu\nMSA0NC42NywtMjcyLjg1IDMyLjQzLC0yNTIgLTguMDgsLTE4My4wMSAtMTMuMTIsLTEzNy43NyAz\nMi40MywtNzIgNTIuOTEsLTQyLjQxIDkyLjExLC0yOS4zMiAxMjMuOTksLTIzLjU0Ii8+Cjxwb2x5\nZ29uIGZpbGw9ImJsYWNrIiBzdHJva2U9ImJsYWNrIiBwb2ludHM9IjEyNC43NCwtMjYuOTYgMTM0\nLjA1LC0yMS45MSAxMjMuNjIsLTIwLjA1IDEyNC43NCwtMjYuOTYiLz4KPC9nPgo8IS0tIEFjYWRl\nbWljcyYjNDU7Jmd0O1dvcmtIb3VycyAtLT4KPGcgaWQ9ImVkZ2UxOSIgY2xhc3M9ImVkZ2UiPgo8\ndGl0bGU+QWNhZGVtaWNzJiM0NTsmZ3Q7V29ya0hvdXJzPC90aXRsZT4KPHBhdGggZmlsbD0ibm9u\nZSIgc3Ryb2tlPSJibGFjayIgZD0iTTE0NC44MiwtMjg4Ljc2QzE1NC41NywtMjc5LjY2IDE2Ni45\nLC0yNjguMTYgMTc3LjY0LC0yNTguMTMiLz4KPHBvbHlnb24gZmlsbD0iYmxhY2siIHN0cm9rZT0i\nYmxhY2siIHBvaW50cz0iMTgwLjIsLTI2MC41MyAxODUuMTMsLTI1MS4xNSAxNzUuNDMsLTI1NS40\nMSAxODAuMiwtMjYwLjUzIi8+CjwvZz4KPCEtLSBJbnRlcm5ldEFjY2VzcyAtLT4KPGcgaWQ9Im5v\nZGU4IiBjbGFzcz0ibm9kZSI+Cjx0aXRsZT5JbnRlcm5ldEFjY2VzczwvdGl0bGU+CjxlbGxpcHNl\nIGZpbGw9Im5vbmUiIHN0cm9rZT0iYmxhY2siIGN4PSI1ODguNDMiIGN5PSItMTYyIiByeD0iNjMu\nMDkiIHJ5PSIxOCIvPgo8dGV4dCB0ZXh0LWFuY2hvcj0ibWlkZGxlIiB4PSI1ODguNDMiIHk9Ii0x\nNTguMyIgZm9udC1mYW1pbHk9IlRpbWVzIE5ldyBSb21hbixzZXJpZiIgZm9udC1zaXplPSIxNC4w\nMCI+SW50ZXJuZXRBY2Nlc3M8L3RleHQ+CjwvZz4KPCEtLSBJbnRlcm5ldEFjY2VzcyYjNDU7Jmd0\nO09ubGluZUNsYXNzIC0tPgo8ZyBpZD0iZWRnZTgiIGNsYXNzPSJlZGdlIj4KPHRpdGxlPkludGVy\nbmV0QWNjZXNzJiM0NTsmZ3Q7T25saW5lQ2xhc3M8L3RpdGxlPgo8cGF0aCBmaWxsPSJub25lIiBz\ndHJva2U9ImJsYWNrIiBkPSJNNTQzLjM3LC0xNDkuMzZDNDk1LjI2LC0xMzYuOTMgNDE5LjE0LC0x\nMTcuMjcgMzY4Ljc1LC0xMDQuMjYiLz4KPHBvbHlnb24gZmlsbD0iYmxhY2siIHN0cm9rZT0iYmxh\nY2siIHBvaW50cz0iMzY5LjU4LC0xMDAuODYgMzU5LjAzLC0xMDEuNzUgMzY3LjgzLC0xMDcuNjQg\nMzY5LjU4LC0xMDAuODYiLz4KPC9nPgo8IS0tIFByZWZlcmVuY2VzJiM0NTsmZ3Q7T25saW5lQ2xh\nc3MgLS0+CjxnIGlkPSJlZGdlOSIgY2xhc3M9ImVkZ2UiPgo8dGl0bGU+UHJlZmVyZW5jZXMmIzQ1\nOyZndDtPbmxpbmVDbGFzczwvdGl0bGU+CjxwYXRoIGZpbGw9Im5vbmUiIHN0cm9rZT0iYmxhY2si\nIGQ9Ik0zNDguNSwtMTQ0LjA1QzM0My42NywtMTM1LjggMzM3Ljc1LC0xMjUuNyAzMzIuMzgsLTEx\nNi41NCIvPgo8cG9seWdvbiBmaWxsPSJibGFjayIgc3Ryb2tlPSJibGFjayIgcG9pbnRzPSIzMzUu\nMzMsLTExNC42NSAzMjcuMjYsLTEwNy43OSAzMjkuMjksLTExOC4xOSAzMzUuMzMsLTExNC42NSIv\nPgo8L2c+CjwhLS0gTG9jYXRpb24gLS0+CjxnIGlkPSJub2RlMTAiIGNsYXNzPSJub2RlIj4KPHRp\ndGxlPkxvY2F0aW9uPC90aXRsZT4KPGVsbGlwc2UgZmlsbD0ibm9uZSIgc3Ryb2tlPSJibGFjayIg\nY3g9IjYxNC40MyIgY3k9Ii0yMzQiIHJ4PSI0Mi40OSIgcnk9IjE4Ii8+Cjx0ZXh0IHRleHQtYW5j\naG9yPSJtaWRkbGUiIHg9IjYxNC40MyIgeT0iLTIzMC4zIiBmb250LWZhbWlseT0iVGltZXMgTmV3\nIFJvbWFuLHNlcmlmIiBmb250LXNpemU9IjE0LjAwIj5Mb2NhdGlvbjwvdGV4dD4KPC9nPgo8IS0t\nIExvY2F0aW9uJiM0NTsmZ3Q7SW50ZXJuZXRBY2Nlc3MgLS0+CjxnIGlkPSJlZGdlMTQiIGNsYXNz\nPSJlZGdlIj4KPHRpdGxlPkxvY2F0aW9uJiM0NTsmZ3Q7SW50ZXJuZXRBY2Nlc3M8L3RpdGxlPgo8\ncGF0aCBmaWxsPSJub25lIiBzdHJva2U9ImJsYWNrIiBkPSJNNjA4LjEzLC0yMTYuMDVDNjA1LjIx\nLC0yMDguMTggNjAxLjY2LC0xOTguNjIgNTk4LjM4LC0xODkuNzkiLz4KPHBvbHlnb24gZmlsbD0i\nYmxhY2siIHN0cm9rZT0iYmxhY2siIHBvaW50cz0iNjAxLjYxLC0xODguNDMgNTk0Ljg0LC0xODAu\nMjggNTk1LjA0LC0xOTAuODcgNjAxLjYxLC0xODguNDMiLz4KPC9nPgo8IS0tIFNFUyYjNDU7Jmd0\nO1ByZWZlcmVuY2VzIC0tPgo8ZyBpZD0iZWRnZTE3IiBjbGFzcz0iZWRnZSI+Cjx0aXRsZT5TRVMm\nIzQ1OyZndDtQcmVmZXJlbmNlczwvdGl0bGU+CjxwYXRoIGZpbGw9Im5vbmUiIHN0cm9rZT0iYmxh\nY2siIGQ9Ik01MDQuOTgsLTIyMy43QzQ3OC4yNiwtMjEyLjY0IDQzMy4wOCwtMTkzLjkyIDM5OS45\nNCwtMTgwLjE5Ii8+Cjxwb2x5Z29uIGZpbGw9ImJsYWNrIiBzdHJva2U9ImJsYWNrIiBwb2ludHM9\nIjQwMS4wOCwtMTc2Ljg4IDM5MC41LC0xNzYuMjkgMzk4LjQsLTE4My4zNSA0MDEuMDgsLTE3Ni44\nOCIvPgo8L2c+CjwhLS0gV29ya0hvdXJzJiM0NTsmZ3Q7QXZhaWxhYmxlVGltZSAtLT4KPGcgaWQ9\nImVkZ2UxOCIgY2xhc3M9ImVkZ2UiPgo8dGl0bGU+V29ya0hvdXJzJiM0NTsmZ3Q7QXZhaWxhYmxl\nVGltZTwvdGl0bGU+CjxwYXRoIGZpbGw9Im5vbmUiIHN0cm9rZT0iYmxhY2siIGQ9Ik0yMDYuNTQs\nLTIxNi4wNUMyMDguNDMsLTIwOC4yNiAyMTAuNzMsLTE5OC44MiAyMTIuODUsLTE5MC4wOCIvPgo8\ncG9seWdvbiBmaWxsPSJibGFjayIgc3Ryb2tlPSJibGFjayIgcG9pbnRzPSIyMTYuMjcsLTE5MC44\nMiAyMTUuMjMsLTE4MC4yOCAyMDkuNDcsLTE4OS4xNyAyMTYuMjcsLTE5MC44MiIvPgo8L2c+Cjwh\nLS0gVTEgLS0+CjxnIGlkPSJub2RlMTMiIGNsYXNzPSJub2RlIj4KPHRpdGxlPlUxPC90aXRsZT4K\nPGVsbGlwc2UgZmlsbD0ibm9uZSIgc3Ryb2tlPSJibGFjayIgc3Ryb2tlLWRhc2hhcnJheT0iNSwy\nIiBjeD0iMTk1LjQzIiBjeT0iLTM3OCIgcng9IjI3IiByeT0iMTgiLz4KPHRleHQgdGV4dC1hbmNo\nb3I9Im1pZGRsZSIgeD0iMTk1LjQzIiB5PSItMzc0LjMiIGZvbnQtZmFtaWx5PSJUaW1lcyBOZXcg\nUm9tYW4sc2VyaWYiIGZvbnQtc2l6ZT0iMTQuMDAiPlUxPC90ZXh0Pgo8L2c+CjwhLS0gVTEmIzQ1\nOyZndDtSYWNlIC0tPgo8ZyBpZD0iZWRnZTI0IiBjbGFzcz0iZWRnZSI+Cjx0aXRsZT5VMSYjNDU7\nJmd0O1JhY2U8L3RpdGxlPgo8cGF0aCBmaWxsPSJub25lIiBzdHJva2U9ImJsYWNrIiBkPSJNMjAy\nLjMsLTM2MC40MUMyMDUuNjQsLTM1Mi4zNCAyMDkuNzUsLTM0Mi40MyAyMTMuNTEsLTMzMy4zNSIv\nPgo8cG9seWdvbiBmaWxsPSJibGFjayIgc3Ryb2tlPSJibGFjayIgcG9pbnRzPSIyMTYuODEsLTMz\nNC41MyAyMTcuNCwtMzIzLjk2IDIxMC4zNCwtMzMxLjg2IDIxNi44MSwtMzM0LjUzIi8+CjwvZz4K\nPCEtLSBVMSYjNDU7Jmd0O0FjYWRlbWljcyAtLT4KPGcgaWQ9ImVkZ2UyNSIgY2xhc3M9ImVkZ2Ui\nPgo8dGl0bGU+VTEmIzQ1OyZndDtBY2FkZW1pY3M8L3RpdGxlPgo8cGF0aCBmaWxsPSJub25lIiBz\ndHJva2U9ImJsYWNrIiBkPSJNMTgxLjM0LC0zNjIuNUMxNzIuMzgsLTM1My4yOCAxNjAuNTksLTM0\nMS4xNCAxNTAuMzEsLTMzMC41NiIvPgo8cG9seWdvbiBmaWxsPSJibGFjayIgc3Ryb2tlPSJibGFj\nayIgcG9pbnRzPSIxNTIuNjQsLTMyNy45MyAxNDMuMTYsLTMyMy4yIDE0Ny42MiwtMzMyLjgxIDE1\nMi42NCwtMzI3LjkzIi8+CjwvZz4KPCEtLSBVMiAtLT4KPGcgaWQ9Im5vZGUxNCIgY2xhc3M9Im5v\nZGUiPgo8dGl0bGU+VTI8L3RpdGxlPgo8ZWxsaXBzZSBmaWxsPSJub25lIiBzdHJva2U9ImJsYWNr\nIiBzdHJva2UtZGFzaGFycmF5PSI1LDIiIGN4PSIzNjYuNDMiIGN5PSItMzc4IiByeD0iMjciIHJ5\nPSIxOCIvPgo8dGV4dCB0ZXh0LWFuY2hvcj0ibWlkZGxlIiB4PSIzNjYuNDMiIHk9Ii0zNzQuMyIg\nZm9udC1mYW1pbHk9IlRpbWVzIE5ldyBSb21hbixzZXJpZiIgZm9udC1zaXplPSIxNC4wMCI+VTI8\nL3RleHQ+CjwvZz4KPCEtLSBVMiYjNDU7Jmd0O1JhY2UgLS0+CjxnIGlkPSJlZGdlMjYiIGNsYXNz\nPSJlZGdlIj4KPHRpdGxlPlUyJiM0NTsmZ3Q7UmFjZTwvdGl0bGU+CjxwYXRoIGZpbGw9Im5vbmUi\nIHN0cm9rZT0iYmxhY2siIGQ9Ik0zNDUuMTcsLTM2Ni41MkMzMjEuMzYsLTM1NC43OCAyODIuNTYs\nLTMzNS42NiAyNTUuMzgsLTMyMi4yNiIvPgo8cG9seWdvbiBmaWxsPSJibGFjayIgc3Ryb2tlPSJi\nbGFjayIgcG9pbnRzPSIyNTYuODEsLTMxOS4wNiAyNDYuMywtMzE3Ljc4IDI1My43MiwtMzI1LjM0\nIDI1Ni44MSwtMzE5LjA2Ii8+CjwvZz4KPCEtLSBVMiYjNDU7Jmd0O1NFUyAtLT4KPGcgaWQ9ImVk\nZ2UyNyIgY2xhc3M9ImVkZ2UiPgo8dGl0bGU+VTImIzQ1OyZndDtTRVM8L3RpdGxlPgo8cGF0aCBm\naWxsPSJub25lIiBzdHJva2U9ImJsYWNrIiBkPSJNMzg4LjQxLC0zNjcuMzVDNDA4LjA4LC0zNTgu\nMDQgNDM3LjAzLC0zNDIuNjUgNDU4LjQzLC0zMjQgNDgwLjA1LC0zMDUuMTUgNDk5LjU2LC0yNzgu\nNjYgNTEyLjI3LC0yNTkuNDkiLz4KPHBvbHlnb24gZmlsbD0iYmxhY2siIHN0cm9rZT0iYmxhY2si\nIHBvaW50cz0iNTE1LjI4LC0yNjEuMjggNTE3Ljc4LC0yNTAuOTkgNTA5LjQsLTI1Ny40OCA1MTUu\nMjgsLTI2MS4yOCIvPgo8L2c+CjwhLS0gVTMgLS0+CjxnIGlkPSJub2RlMTUiIGNsYXNzPSJub2Rl\nIj4KPHRpdGxlPlUzPC90aXRsZT4KPGVsbGlwc2UgZmlsbD0ibm9uZSIgc3Ryb2tlPSJibGFjayIg\nc3Ryb2tlLWRhc2hhcnJheT0iNSwyIiBjeD0iMjg1LjQzIiBjeT0iLTM3OCIgcng9IjI3IiByeT0i\nMTgiLz4KPHRleHQgdGV4dC1hbmNob3I9Im1pZGRsZSIgeD0iMjg1LjQzIiB5PSItMzc0LjMiIGZv\nbnQtZmFtaWx5PSJUaW1lcyBOZXcgUm9tYW4sc2VyaWYiIGZvbnQtc2l6ZT0iMTQuMDAiPlUzPC90\nZXh0Pgo8L2c+CjwhLS0gVTMmIzQ1OyZndDtHZW5kZXIgLS0+CjxnIGlkPSJlZGdlMjgiIGNsYXNz\nPSJlZGdlIj4KPHRpdGxlPlUzJiM0NTsmZ3Q7R2VuZGVyPC90aXRsZT4KPHBhdGggZmlsbD0ibm9u\nZSIgc3Ryb2tlPSJibGFjayIgZD0iTTMwNS43MSwtMzY1LjkxQzMyNS44NywtMzU0Ljg4IDM1Ny4x\nMiwtMzM3Ljc5IDM4MC43OCwtMzI0Ljg1Ii8+Cjxwb2x5Z29uIGZpbGw9ImJsYWNrIiBzdHJva2U9\nImJsYWNrIiBwb2ludHM9IjM4Mi42NywtMzI3LjgxIDM4OS43NiwtMzE5Ljk0IDM3OS4zMSwtMzIx\nLjY3IDM4Mi42NywtMzI3LjgxIi8+CjwvZz4KPCEtLSBVMyYjNDU7Jmd0O0FjYWRlbWljcyAtLT4K\nPGcgaWQ9ImVkZ2UyOSIgY2xhc3M9ImVkZ2UiPgo8dGl0bGU+VTMmIzQ1OyZndDtBY2FkZW1pY3M8\nL3RpdGxlPgo8cGF0aCBmaWxsPSJub25lIiBzdHJva2U9ImJsYWNrIiBkPSJNMjYzLjQ2LC0zNjcu\nMjdDMjM4LjY3LC0zNTYuMjggMTk3LjgzLC0zMzguMTkgMTY3LjM2LC0zMjQuNjkiLz4KPHBvbHln\nb24gZmlsbD0iYmxhY2siIHN0cm9rZT0iYmxhY2siIHBvaW50cz0iMTY4LjQzLC0zMjEuMzQgMTU3\nLjg3LC0zMjAuNDkgMTY1LjU5LC0zMjcuNzQgMTY4LjQzLC0zMjEuMzQiLz4KPC9nPgo8IS0tIFU0\nIC0tPgo8ZyBpZD0ibm9kZTE2IiBjbGFzcz0ibm9kZSI+Cjx0aXRsZT5VNDwvdGl0bGU+CjxlbGxp\ncHNlIGZpbGw9Im5vbmUiIHN0cm9rZT0iYmxhY2siIHN0cm9rZS1kYXNoYXJyYXk9IjUsMiIgY3g9\nIjQ0Mi40MyIgY3k9Ii0zNzgiIHJ4PSIyNyIgcnk9IjE4Ii8+Cjx0ZXh0IHRleHQtYW5jaG9yPSJt\naWRkbGUiIHg9IjQ0Mi40MyIgeT0iLTM3NC4zIiBmb250LWZhbWlseT0iVGltZXMgTmV3IFJvbWFu\nLHNlcmlmIiBmb250LXNpemU9IjE0LjAwIj5VNDwvdGV4dD4KPC9nPgo8IS0tIFU0JiM0NTsmZ3Q7\nR2VuZGVyIC0tPgo8ZyBpZD0iZWRnZTMwIiBjbGFzcz0iZWRnZSI+Cjx0aXRsZT5VNCYjNDU7Jmd0\nO0dlbmRlcjwvdGl0bGU+CjxwYXRoIGZpbGw9Im5vbmUiIHN0cm9rZT0iYmxhY2siIGQ9Ik00MzUu\nNTUsLTM2MC40MUM0MzIuMjEsLTM1Mi4zNCA0MjguMSwtMzQyLjQzIDQyNC4zNCwtMzMzLjM1Ii8+\nCjxwb2x5Z29uIGZpbGw9ImJsYWNrIiBzdHJva2U9ImJsYWNrIiBwb2ludHM9IjQyNy41MSwtMzMx\nLjg2IDQyMC40NSwtMzIzLjk2IDQyMS4wNCwtMzM0LjUzIDQyNy41MSwtMzMxLjg2Ii8+CjwvZz4K\nPCEtLSBVNCYjNDU7Jmd0O1NFUyAtLT4KPGcgaWQ9ImVkZ2UzMSIgY2xhc3M9ImVkZ2UiPgo8dGl0\nbGU+VTQmIzQ1OyZndDtTRVM8L3RpdGxlPgo8cGF0aCBmaWxsPSJub25lIiBzdHJva2U9ImJsYWNr\nIiBkPSJNNDU5LjcsLTM2NC4wOUM0NzEuNzUsLTM1NC4yNyA0ODcuNDIsLTMzOS44IDQ5Ny40Mywt\nMzI0IDUwOS41MSwtMzA0LjkxIDUxNy4zMSwtMjgwLjQxIDUyMS45MSwtMjYxLjk1Ii8+Cjxwb2x5\nZ29uIGZpbGw9ImJsYWNrIiBzdHJva2U9ImJsYWNrIiBwb2ludHM9IjUyNS4zNCwtMjYyLjY2IDUy\nNC4yMSwtMjUyLjEzIDUxOC41MiwtMjYxLjA3IDUyNS4zNCwtMjYyLjY2Ii8+CjwvZz4KPCEtLSBV\nNSAtLT4KPGcgaWQ9Im5vZGUxNyIgY2xhc3M9Im5vZGUiPgo8dGl0bGU+VTU8L3RpdGxlPgo8ZWxs\naXBzZSBmaWxsPSJub25lIiBzdHJva2U9ImJsYWNrIiBzdHJva2UtZGFzaGFycmF5PSI1LDIiIGN4\nPSI1OTYuNDMiIGN5PSItMzA2IiByeD0iMjciIHJ5PSIxOCIvPgo8dGV4dCB0ZXh0LWFuY2hvcj0i\nbWlkZGxlIiB4PSI1OTYuNDMiIHk9Ii0zMDIuMyIgZm9udC1mYW1pbHk9IlRpbWVzIE5ldyBSb21h\nbixzZXJpZiIgZm9udC1zaXplPSIxNC4wMCI+VTU8L3RleHQ+CjwvZz4KPCEtLSBVNSYjNDU7Jmd0\nO0xvY2F0aW9uIC0tPgo8ZyBpZD0iZWRnZTMyIiBjbGFzcz0iZWRnZSI+Cjx0aXRsZT5VNSYjNDU7\nJmd0O0xvY2F0aW9uPC90aXRsZT4KPHBhdGggZmlsbD0ibm9uZSIgc3Ryb2tlPSJibGFjayIgZD0i\nTTYwMC43OCwtMjg4LjA1QzYwMi43OSwtMjgwLjI2IDYwNS4yMSwtMjcwLjgyIDYwNy40NiwtMjYy\nLjA4Ii8+Cjxwb2x5Z29uIGZpbGw9ImJsYWNrIiBzdHJva2U9ImJsYWNrIiBwb2ludHM9IjYxMC44\nOCwtMjYyLjgzIDYwOS45OCwtMjUyLjI4IDYwNC4xLC0yNjEuMDkgNjEwLjg4LC0yNjIuODMiLz4K\nPC9nPgo8IS0tIFU1JiM0NTsmZ3Q7U0VTIC0tPgo8ZyBpZD0iZWRnZTMzIiBjbGFzcz0iZWRnZSI+\nCjx0aXRsZT5VNSYjNDU7Jmd0O1NFUzwvdGl0bGU+CjxwYXRoIGZpbGw9Im5vbmUiIHN0cm9rZT0i\nYmxhY2siIGQ9Ik01ODIuMTMsLTI5MC41QzU3Mi41MSwtMjgwLjczIDU1OS42NSwtMjY3LjY5IDU0\nOC44MiwtMjU2LjciLz4KPHBvbHlnb24gZmlsbD0iYmxhY2siIHN0cm9rZT0iYmxhY2siIHBvaW50\ncz0iNTUxLjIyLC0yNTQuMTUgNTQxLjcsLTI0OS40OSA1NDYuMjMsLTI1OS4wNiA1NTEuMjIsLTI1\nNC4xNSIvPgo8L2c+CjwvZz4KPC9zdmc+Cg==\n"
}
},
- "id": "7d5f84ed-e13d-480d-a69f-97c544f4e1af"
+ "id": "6877493a-8806-4f59-a66b-20709be19c9d"
},
{
"cell_type": "code",
@@ -843,7 +843,7 @@
"\n",
"g"
],
- "id": "f5aa195a"
+ "id": "c2e63ae0"
},
{
"cell_type": "markdown",
@@ -862,7 +862,7 @@
"\n",
"- The simplest cycles look like this:"
],
- "id": "a0d0626d-a1dc-43f3-86e9-3e68565d4d4a"
+ "id": "706d6fa2-da17-4c18-a00b-9f8a5ad88afd"
},
{
"cell_type": "code",
@@ -888,7 +888,7 @@
"\n",
"g"
],
- "id": "0b7d6e2b"
+ "id": "aa4e719f"
},
{
"cell_type": "markdown",
@@ -899,7 +899,7 @@
"- Why can’t we have cycles? Surely there are feedback loops in the\n",
" real world."
],
- "id": "f71ceb6d-e796-411e-afec-2c510263ba8e"
+ "id": "3ba790df-f6f1-4b99-89e6-30ef83783c36"
},
{
"cell_type": "code",
@@ -921,7 +921,7 @@
"\n",
"g"
],
- "id": "e9ba0317"
+ "id": "06685f87"
},
{
"cell_type": "markdown",
@@ -952,875 +952,6 @@
" causality, where the arrow of time is fundamental.\n",
" - Similar to the time-series concept of **Granger Causality**.\n",
"\n",
- "# Dependence Flows\n",
- "\n",
- "## Dependence Flows\n",
- "\n",
- "- In order to determine whether or not we can identify a treatment\n",
- " effect, we need to understand how dependence **flows** through a\n",
- " graphical model.\n",
- "\n",
- "- We have seen that conditioning on a node can either make or break\n",
- " the dependence relationship between two other nodes\n",
- "\n",
- "- To identify the treatment effect, we want the *link between the\n",
- " treatment and the outcome to be unblocked*.\n",
- "\n",
- "- However, we also need to make sure that there is *no other\n",
- " dependence path* between the treatment and the outcome that is\n",
- " unblocked.\n",
- "\n",
- "## The Rules of Bayes-Ball\n",
- "\n",
- "- We can think about the flow of dependence as a game of “Bayes-ball”\n",
- "- The rules of Bayes-ball are reasonably simple. A dependence path is\n",
- " blocked if and only if:\n",
- " 1. It contains a *non-collider* that is conditioned on\n",
- " 2. It contains a *collider* that is not conditioned on, and neither\n",
- " are *any of its descendants*\n",
- "\n",
- "\n",
- "\n",
- "## Directed Graphical Models\n",
- "\n",
- "Turning back to our **collider** example:"
- ],
- "attachments": {
- "./data/fig/graph-flow.png": {
- "image/png": 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ll1SybSBGmOekGXGE6S7pSrQb8xJ1nYNdxTrpVyGAFA3tCAN5l7UmwJ5gTNNm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\nQmCC\n"
- },
- "causal_graphical_models_files/figure-ipynb/cell-24-output-1.svg": {
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- }
- },
- "id": "2f139a07-6688-4d32-8f57-b9db9b0cdcc2"
- },
- {
- "cell_type": "code",
- "execution_count": 24,
- "metadata": {},
- "outputs": [],
- "source": [
- "g = gr.Digraph()\n",
- "g.edge(\"B\", \"C\")\n",
- "g.edge(\"A\", \"C\")\n",
- "\n",
- "g.edge(\"Y\", \"X\")\n",
- "g.edge(\"Z\", \"X\")\n",
- "g.node(\"X\", \"X\", style=\"filled\")"
- ],
- "id": "35e0dc40"
- },
- {
- "cell_type": "markdown",
- "metadata": {},
- "source": [
- "- Notice that when we do not condition on $C$, $A$ and $B$ are\n",
- " independent.\n",
- "- However, somewhat unintuitively, when we condition on $X$, $Y$ and\n",
- " $Z$ become dependent.\n",
- " - This is because conditioning on $X$ opens the flow of dependence\n",
- " from $Y$ to $Z$.\n",
- " - In any case that is *not* a collider, conditioning on a node\n",
- " *blocks* the flow of dependence.\n",
- "\n",
- "## Two Games of Bayes Ball\n",
- "\n",
- "\n",
- "\n",
- "# Viualizing Bias\n",
- "\n",
- "## Bias and Causality\n",
- "\n",
- "- In a causal inference framework, we can use graphical models to\n",
- " determine whether or not we can identify a treatment effect, and\n",
- " which covariates we need to condition on.\n",
- "\n",
- "- Typically, drawing out a graphical model is not necessary, but it\n",
- " can be a useful exercise to help you think through the problem.\n",
- "\n",
- " - The links you draw represent the assumptions you are making\n",
- " about the data generating process\n",
- "\n",
- "## Bias and Causality\n",
- "\n",
- "- There are two major types of bias that we need to worry about in\n",
- " causal inference:\n",
- " - **Confounding**: When there is an unobserved variable that is a\n",
- " common cause of both the treatment and the outcome\n",
- " - **Selection**: When there is an unobserved variable that is a\n",
- " common cause of both the treatment and the selection into the\n",
- " sample\n",
- "- Both of these types of bias can be represented using a graphical\n",
- " model\n",
- "\n",
- "# Confounding\n",
- "\n",
- "## Confounding\n",
- "\n",
- "- Let’s look at an example of confounding:"
- ],
- "attachments": {
- "./data/fig/two-games.png": {
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k0DkANEegBkKy7lXW8+gSNpV9Pqjoi0mNIPVAZ4WGa35UtrI+XkAZEaiBkDwq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\nAAAAQAoEagAAAKBtZv8/aFuleGxIskAAAAAASUVORK5CYII=\n"
- }
- },
- "id": "b7641d2b-b754-410b-9055-ae180bdd07a0"
- },
- {
- "cell_type": "code",
- "execution_count": 25,
- "metadata": {
- "output-location": "column"
- },
- "outputs": [
- {
- "output_type": "display_data",
- "metadata": {},
- "data": {
- "image/svg+xml": 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- }
- }
- ],
- "source": [
- "g = gr.Digraph()\n",
- "g.edge(\"X\", \"T\")\n",
- "g.edge(\"X\", \"Y\")\n",
- "g.edge(\"T\", \"Y\")\n",
- "\n",
- "g.edge(\"rain\", \"umbrella\")\n",
- "g.edge(\"rain\", \"wet\")\n",
- "g.edge(\"umbrella\", \"wet\")\n",
- "\n",
- "g.edge(\"severeness\", \"medicine\")\n",
- "g.edge(\"severeness\", \"survived\")\n",
- "g.edge(\"medicine\", \"survived\")\n",
- "g"
- ],
- "id": "15378e1f"
- },
- {
- "cell_type": "markdown",
- "metadata": {},
- "source": [
- "- To control for confounding, we need to condition on all of the\n",
- " common causes of the treatment and the outcome.\n",
- "\n",
- "## Confounding"
- ],
- "id": "acb3ed06-b9e0-4491-991d-f93271bd245f"
- },
- {
- "cell_type": "code",
- "execution_count": 26,
- "metadata": {
- "output-location": "column"
- },
- "outputs": [
- {
- "output_type": "display_data",
- "metadata": {},
- "data": {
- "image/svg+xml": 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zdHJva2U9ImJsYWNrIiBkPSJNMTk3LjExLC0yMTYuMDVDMTk0LjAyLC0yMDguMDYgMTkw\nLjI3LC0xOTguMzMgMTg2LjgzLC0xODkuNCIvPgo8cG9seWdvbiBmaWxsPSJibGFjayIgc3Ryb2tl\nPSJibGFjayIgcG9pbnRzPSIxODkuOTgsLTE4Ny44NiAxODMuMTIsLTE3OS43OSAxODMuNDUsLTE5\nMC4zOCAxODkuOTgsLTE4Ny44NiIvPgo8L2c+CjwvZz4KPC9zdmc+Cg==\n"
- }
- }
- ],
- "source": [
- "g = gr.Digraph()\n",
- "\n",
- "g.node(\"Family Income\")\n",
- "g.edge(\"Family Income\", \"Educ\")\n",
- "g.edge(\"Educ\", \"Wage\")\n",
- "\n",
- "g.node(\"SAT\")\n",
- "g.edge(\"SAT\", \"Educ\")\n",
- "\n",
- "g.node(\"Family Income\")\n",
- "g.edge(\"Family Income\", \"Wage\")\n",
- "\n",
- "g.edge(\"Intelligence\", \"SAT\")\n",
- "g.edge(\"Intelligence\", \"Wage\")\n",
- "g"
- ],
- "id": "aa4deaf1"
- },
- {
- "cell_type": "markdown",
- "metadata": {},
- "source": [
- "- Often, there are confounding variables that we cannot observe\n",
- " - For example, we cannot observe intelligence, but it is a common\n",
- " cause of both education (the treatment) and wages\n",
- "\n",
- "## Confounding"
- ],
- "id": "cd3ca0e8-9423-444e-ae8d-2c9c4feba5fd"
- },
- {
- "cell_type": "code",
- "execution_count": 27,
- "metadata": {
- "output-location": "column"
- },
- "outputs": [
- {
- "output_type": "display_data",
- "metadata": {},
- "data": {
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sbD0ibm9uZSIgc3Ryb2tlPSJibGFjayIgZD0iTTE5Ny4xMSwtMjE2LjA1QzE5NC4wMiwt\nMjA4LjA2IDE5MC4yNywtMTk4LjMzIDE4Ni44MywtMTg5LjQiLz4KPHBvbHlnb24gZmlsbD0iYmxh\nY2siIHN0cm9rZT0iYmxhY2siIHBvaW50cz0iMTg5Ljk4LC0xODcuODYgMTgzLjEyLC0xNzkuNzkg\nMTgzLjQ1LC0xOTAuMzggMTg5Ljk4LC0xODcuODYiLz4KPC9nPgo8L2c+Cjwvc3ZnPgo=\n"
- }
- }
- ],
- "source": [
- "g = gr.Digraph()\n",
- "\n",
- "g.node(\"Family Income\", style=\"filled\")\n",
- "g.edge(\"Family Income\", \"Educ\")\n",
- "g.edge(\"Educ\", \"Wage\")\n",
- "\n",
- "g.node(\"SAT\", style=\"filled\")\n",
- "g.edge(\"SAT\", \"Educ\")\n",
- "\n",
- "g.node(\"Family Income\", style=\"filled\")\n",
- "g.edge(\"Family Income\", \"Wage\")\n",
- "\n",
- "g.edge(\"Intelligence\", \"SAT\")\n",
- "g.edge(\"Intelligence\", \"Wage\")\n",
- "g"
- ],
- "id": "469aa63e"
- },
- {
- "cell_type": "markdown",
- "metadata": {},
- "source": [
- "- Often, there are confounding variables that we cannot observe\n",
- " - For example, we cannot observe intelligence, but it is a common\n",
- " cause of both education (the treatment) and wages\n",
- " - But we can use SAT as a **surrogate** or **proxy** for\n",
- " intelligence.\n",
- "\n",
- "# Selection\n",
- "\n",
- "## Selection\n",
- "\n",
- "- Selection bias often occurs when there is an unobserved variable\n",
- " that is a common cause of both the treatment and the selection into\n",
- " the sample, in other words, by conditioning on a variable that you\n",
- " shouldn’t."
- ],
- "id": "f96e3351-774a-4bd3-9d15-df4d578e4c2c"
- },
- {
- "cell_type": "code",
- "execution_count": 28,
- "metadata": {
- "output-location": "column"
- },
- "outputs": [
- {
- "output_type": "display_data",
- "metadata": {},
- "data": {
- "image/svg+xml": 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- }
- }
- ],
- "source": [
- "g = gr.Digraph()\n",
- "g.node(\"X\", style=\"filled\")\n",
- "g.edge(\"T\", \"X\")\n",
- "g.edge(\"T\", \"Y\")\n",
- "g.edge(\"Y\", \"X\")\n",
- "g.node(\"Investments\", \"Investments\", style=\"filled\")\n",
- "g.edge(\"Educ\", \"Investments\")\n",
- "g.edge(\"Educ\", \"Wage\")\n",
- "g.edge(\"Wage\", \"Investments\")\n",
- "g.node(\"Educ2\", \"Educ\")\n",
- "g.node(\"Wage2\", \"Wage\")\n",
- "g.node(\"Investments2\", \"Investments\")\n",
- "g.edge(\"Educ2\", \"Wage2\")\n",
- "g.edge(\"Wage2\", \"Investments2\")\n",
- "g.edge(\"Educ2\", \"Investments2\")\n",
- "g.node(\"CapitalGainsTax\", \"CapitalGainsTax\", style=\"filled\")\n",
- "g.edge(\"Investments2\", \"CapitalGainsTax\")\n",
- "g"
- ],
- "id": "b309f7ad"
- },
- {
- "cell_type": "markdown",
- "metadata": {},
- "source": [
- "## Selection\n",
- "\n",
- "- Selection bias can also occur when controlling for a **mediator**\n",
- " between the treatment and the outcome"
- ],
- "id": "ff865c1d-afed-4681-8abf-ce2f77418aa2"
- },
- {
- "cell_type": "code",
- "execution_count": 29,
- "metadata": {
- "output-location": "column"
- },
- "outputs": [
- {
- "output_type": "display_data",
- "metadata": {},
- "data": {
- "image/svg+xml": 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- }
- }
- ],
- "source": [
- "g = gr.Digraph()\n",
- "\n",
- "g = gr.Digraph()\n",
- "g.edge(\"T\", \"X\")\n",
- "g.edge(\"T\", \"Y\")\n",
- "g.edge(\"X\", \"Y\")\n",
- "g.node(\"X\", \"X\", style=\"filled\")\n",
- "\n",
- "g.edge('Educ', 'WhiteCollar')\n",
- "g.edge('Educ', 'Wage')\n",
- "g.edge('WhiteCollar', 'Wage')\n",
- "g.node('WhiteCollar', style=\"filled\")\n",
- "\n",
- "g"
- ],
- "id": "f5f9d54e"
- },
- {
- "cell_type": "markdown",
- "metadata": {},
- "source": [
- "# Choosing Covariates\n",
- "\n",
- "## Choosing Covariates\n",
- "\n",
- "- With these types of bias in mind, we can think about how to choose\n",
- " which covariates to condition on.\n",
- "\n",
- "- Notice that some controls will reduce bias, as in the case of\n",
- " confounding,\n",
- "\n",
- "- But *not all controls are good!* Some controls will actually create\n",
- " bias, as in the case of selection.\n",
- "\n",
- "- This means that we don’t want to just “throw the kitchen sink” at\n",
- " the problem, and include every variable we can think of.\n",
- "\n",
- "## Choosing Covariates\n",
- "\n",
- "- We want to choose covariates that will close any unblocked secondary\n",
- " dependence paths between the treatment and the outcome, but not open\n",
- " any new ones.\n",
- "\n",
- "- To do this, we can use the “front door criterion” and the “back door\n",
- " criterion”\n",
- "\n",
- "## The Front Door Criterion\n",
- "\n",
- "- The **front door criterion** is one way to isolate the effect of a\n",
- " treatment on an outcome, when there is a confounder and a mediator\n",
- " between the treatment and the outcome."
- ],
- "id": "91a52c2c-0721-4c2c-8b74-10d2893d6fab"
- },
- {
- "cell_type": "code",
- "execution_count": 30,
- "metadata": {
- "output-location": "column"
- },
- "outputs": [
- {
- "output_type": "display_data",
- "metadata": {},
- "data": {
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- }
- }
- ],
- "source": [
- "g = gr.Digraph()\n",
- "g.edge(\"LotsOfStuff\", \"Smoking\")\n",
- "g.edge(\"LotsOfStuff\", \"LungCancer\")\n",
- "g.edge(\"Smoking\", \"Tar\", style=\"dashed\")\n",
- "g.edge(\"Tar\", \"LungCancer\", style=\"dashed\")\n",
- "\n",
- "g"
- ],
- "id": "8e7a8b5b"
- },
- {
- "cell_type": "markdown",
- "metadata": {},
- "source": [
- "## The Front Door Criterion\n",
- "\n",
- "``` python\n",
- "g = gr.Digraph()\n",
- "g.edge(\"LotsOfStuff\", \"Smoking\")\n",
- "g.edge(\"LotsOfStuff\", \"LungCancer\")\n",
- "g.edge(\"Smoking\", \"Tar\", style=\"dashed\")\n",
- "g.edge(\"Tar\", \"LungCancer\", style=\"dashed\")\n",
- "\n",
- "g\n",
- "```\n",
- "\n",
- "\n",
- "\n",
- "- In this example, we want to know the effect of smoking on lung\n",
- " cancer, but we also know that there are lots of variables (like\n",
- " stress), that cause both the treatment and the outcome.\n",
- "\n",
- "- However, stress does not cause tar, so we can condition on tar.\n",
- "\n",
- " - This works by first measuring the effect of tar on lung cancer,\n",
- " and *then* the effect of smoking on tar.\n",
- "\n",
- "## The Back Door Criterion\n",
- "\n",
- "- It’s pretty rare that we’ll actually be able to use the front door\n",
- " criterion, because we usually don’t have a mediator that we can\n",
- " condition on.\n",
- "\n",
- "- Instead, we can close all of the “back door paths” from treatment to\n",
- " outcome. We have already seen an example of this."
- ],
- "attachments": {
- "causal_graphical_models_files/figure-ipynb/cell-32-output-1.svg": {
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- }
- },
- "id": "1ad7f459-51ec-43fd-bbbd-685f0e2dd2fb"
- },
- {
- "cell_type": "code",
- "execution_count": 32,
- "metadata": {},
- "outputs": [
- {
- "output_type": "display_data",
- "metadata": {},
- "data": {
- "image/svg+xml": 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ndDtTQVQ8L3RpdGxlPgo8cGF0aCBmaWxsPSJub25lIiBzdHJva2U9ImJsYWNrIiBkPSJN\nMTk3LjExLC0yMTYuMDVDMTk0LjAyLC0yMDguMDYgMTkwLjI3LC0xOTguMzMgMTg2LjgzLC0xODku\nNCIvPgo8cG9seWdvbiBmaWxsPSJibGFjayIgc3Ryb2tlPSJibGFjayIgcG9pbnRzPSIxODkuOTgs\nLTE4Ny44NiAxODMuMTIsLTE3OS43OSAxODMuNDUsLTE5MC4zOCAxODkuOTgsLTE4Ny44NiIvPgo8\nL2c+CjwvZz4KPC9zdmc+Cg==\n"
- }
- }
- ],
- "source": [
- "g = gr.Digraph()\n",
- "\n",
- "g.node(\"Family Income\", style=\"filled\")\n",
- "g.edge(\"Family Income\", \"Educ\")\n",
- "g.edge(\"Educ\", \"Wage\", style=\"dashed\")\n",
- "\n",
- "g.node(\"SAT\", style=\"filled\")\n",
- "g.edge(\"SAT\", \"Educ\")\n",
- "\n",
- "g.node(\"Family Income\", style=\"filled\")\n",
- "g.edge(\"Family Income\", \"Wage\")\n",
- "\n",
- "g.edge(\"Intelligence\", \"SAT\")\n",
- "g.edge(\"Intelligence\", \"Wage\")\n",
- "g"
- ],
- "id": "e27ceac1"
- },
- {
- "cell_type": "markdown",
- "metadata": {},
- "source": [
- "## Choosing Covariates: Two Examples"
- ],
- "id": "d89a3ecd-1a3d-4328-8b0f-0ee302d57c1d"
- },
- {
- "cell_type": "code",
- "execution_count": 33,
- "metadata": {
- "output-location": "column"
- },
- "outputs": [
- {
- "output_type": "display_data",
- "metadata": {},
- "data": {
- "image/svg+xml": "PD94bWwgdmVyc2lvbj0iMS4wIiBlbmNvZGluZz0iVVRGLTgiIHN0YW5kYWxvbmU9Im5vIj8+Cjwh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\nPgo8L3N2Zz4K\n"
- }
- }
- ],
- "source": [
- "g = gr.Digraph()\n",
- "g.edge(\"T\", \"Y\", style=\"dashed\")\n",
- "g.edge(\"X\", \"Y\")\n",
- "g"
- ],
- "id": "8138f6a2"
- },
- {
- "cell_type": "markdown",
- "metadata": {},
- "source": [
- "- Do we need to condition on $X$?\n",
- "\n",
- "## Choosing Covariates: Two Examples"
- ],
- "id": "7cbccc22-a866-405b-9e64-7285e9a06f16"
- },
- {
- "cell_type": "code",
- "execution_count": 34,
- "metadata": {
- "output-location": "column"
- },
- "outputs": [
- {
- "output_type": "display_data",
- "metadata": {},
- "data": {
- "image/svg+xml": "PD94bWwgdmVyc2lvbj0iMS4wIiBlbmNvZGluZz0iVVRGLTgiIHN0YW5kYWxvbmU9Im5vIj8+Cjwh\nRE9DVFlQRSBzdmcgUFVCTElDICItLy9XM0MvL0RURCBTVkcgMS4xLy9FTiIKICJodHRwOi8vd3d3\nLnczLm9yZy9HcmFwaGljcy9TVkcvMS4xL0RURC9zdmcxMS5kdGQiPgo8IS0tIEdlbmVyYXRlZCBi\neSBncmFwaHZpeiB2ZXJzaW9uIDIuNTAuMCAoMCkKIC0tPgo8IS0tIFBhZ2VzOiAxIC0tPgo8c3Zn\nIHdpZHRoPSIxMzRwdCIgaGVpZ2h0PSIxMTZwdCIKIHZpZXdCb3g9IjAuMDAgMC4wMCAxMzQuMDAg\nMTE2LjAwIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHhtbG5zOnhsaW5rPSJo\ndHRwOi8vd3d3LnczLm9yZy8xOTk5L3hsaW5rIj4KPGcgaWQ9ImdyYXBoMCIgY2xhc3M9ImdyYXBo\nIiB0cmFuc2Zvcm09InNjYWxlKDEgMSkgcm90YXRlKDApIHRyYW5zbGF0ZSg0IDExMikiPgo8cG9s\neWdvbiBmaWxsPSJ3aGl0ZSIgc3Ryb2tlPSJ0cmFuc3BhcmVudCIgcG9pbnRzPSItNCw0IC00LC0x\nMTIgMTMwLC0xMTIgMTMwLDQgLTQsNCIvPgo8IS0tIFQgLS0+CjxnIGlkPSJub2RlMSIgY2xhc3M9\nIm5vZGUiPgo8dGl0bGU+VDwvdGl0bGU+CjxlbGxpcHNlIGZpbGw9Im5vbmUiIHN0cm9rZT0iYmxh\nY2siIGN4PSIyNyIgY3k9Ii05MCIgcng9IjI3IiByeT0iMTgiLz4KPHRleHQgdGV4dC1hbmNob3I9\nIm1pZGRsZSIgeD0iMjciIHk9Ii04Ni4zIiBmb250LWZhbWlseT0iVGltZXMgTmV3IFJvbWFuLHNl\ncmlmIiBmb250LXNpemU9IjE0LjAwIj5UPC90ZXh0Pgo8L2c+CjwhLS0gWSAtLT4KPGcgaWQ9Im5v\nZGUyIiBjbGFzcz0ibm9kZSI+Cjx0aXRsZT5ZPC90aXRsZT4KPGVsbGlwc2UgZmlsbD0ibm9uZSIg\nc3Ryb2tlPSJibGFjayIgY3g9IjYzIiBjeT0iLTE4IiByeD0iMjciIHJ5PSIxOCIvPgo8dGV4dCB0\nZXh0LWFuY2hvcj0ibWlkZGxlIiB4PSI2MyIgeT0iLTE0LjMiIGZvbnQtZmFtaWx5PSJUaW1lcyBO\nZXcgUm9tYW4sc2VyaWYiIGZvbnQtc2l6ZT0iMTQuMDAiPlk8L3RleHQ+CjwvZz4KPCEtLSBUJiM0\nNTsmZ3Q7WSAtLT4KPGcgaWQ9ImVkZ2UxIiBjbGFzcz0iZWRnZSI+Cjx0aXRsZT5UJiM0NTsmZ3Q7\nWTwvdGl0bGU+CjxwYXRoIGZpbGw9Im5vbmUiIHN0cm9rZT0iYmxhY2siIHN0cm9rZS1kYXNoYXJy\nYXk9IjUsMiIgZD0iTTM1LjM1LC03Mi43NkMzOS43MSwtNjQuMjggNDUuMTUsLTUzLjcxIDUwLjA0\nLC00NC4yIi8+Cjxwb2x5Z29uIGZpbGw9ImJsYWNrIiBzdHJva2U9ImJsYWNrIiBwb2ludHM9IjUz\nLjIzLC00NS42NCA1NC43LC0zNS4xNSA0Ny4wMSwtNDIuNDQgNTMuMjMsLTQ1LjY0Ii8+CjwvZz4K\nPCEtLSBYIC0tPgo8ZyBpZD0ibm9kZTMiIGNsYXNzPSJub2RlIj4KPHRpdGxlPlg8L3RpdGxlPgo8\nZWxsaXBzZSBmaWxsPSJub25lIiBzdHJva2U9ImJsYWNrIiBjeD0iOTkiIGN5PSItOTAiIHJ4PSIy\nNyIgcnk9IjE4Ii8+Cjx0ZXh0IHRleHQtYW5jaG9yPSJtaWRkbGUiIHg9Ijk5IiB5PSItODYuMyIg\nZm9udC1mYW1pbHk9IlRpbWVzIE5ldyBSb21hbixzZXJpZiIgZm9udC1zaXplPSIxNC4wMCI+WDwv\ndGV4dD4KPC9nPgo8IS0tIFgmIzQ1OyZndDtZIC0tPgo8ZyBpZD0iZWRnZTIiIGNsYXNzPSJlZGdl\nIj4KPHRpdGxlPlgmIzQ1OyZndDtZPC90aXRsZT4KPHBhdGggZmlsbD0ibm9uZSIgc3Ryb2tlPSJi\nbGFjayIgZD0iTTkwLjY1LC03Mi43NkM4Ni4yOSwtNjQuMjggODAuODUsLTUzLjcxIDc1Ljk2LC00\nNC4yIi8+Cjxwb2x5Z29uIGZpbGw9ImJsYWNrIiBzdHJva2U9ImJsYWNrIiBwb2ludHM9Ijc4Ljk5\nLC00Mi40NCA3MS4zLC0zNS4xNSA3Mi43NywtNDUuNjQgNzguOTksLTQyLjQ0Ii8+CjwvZz4KPC9n\nPgo8L3N2Zz4K\n"
- }
- }
- ],
- "source": [
- "g = gr.Digraph()\n",
- "g.edge(\"T\", \"Y\", style=\"dashed\")\n",
- "g.edge(\"X\", \"Y\")\n",
- "g"
- ],
- "id": "95b34e44"
- },
- {
- "cell_type": "markdown",
- "metadata": {},
- "source": [
- "- Do we need to condition on $X$?\n",
- " - No, not necessarily. There is no unblocked dependence path\n",
- " between $T$ and $Y$ that goes through $X$.\n",
- " - However, conditioning on $X$ will reduce the variance of our\n",
- " estimate!\n",
- " - If we didn’t condition on $X$, it would become part of our\n",
- " estimation error. But since $X \\perp T$, it won’t bias our\n",
- " estimate.\n",
- "\n",
- "## Choosing Covariates: Two Examples"
- ],
- "id": "484cf66c-df5a-4044-a0ec-96e6154b5c39"
- },
- {
- "cell_type": "code",
- "execution_count": 35,
- "metadata": {
- "output-location": "column"
- },
- "outputs": [
- {
- "output_type": "display_data",
- "metadata": {},
- "data": {
- "image/svg+xml": "PD94bWwgdmVyc2lvbj0iMS4wIiBlbmNvZGluZz0iVVRGLTgiIHN0YW5kYWxvbmU9Im5vIj8+Cjwh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\nCjwvZz4KPC9nPgo8L3N2Zz4K\n"
- }
- }
- ],
- "source": [
- "g = gr.Digraph()\n",
- "g.edge(\"R\", \"T\")\n",
- "g.edge(\"C\", \"T\")\n",
- "g.edge(\"C\", \"Y\")\n",
- "g.edge(\"T\", \"Y\", style=\"dashed\")\n",
- "g"
- ],
- "id": "67ddf57f"
- },
- {
- "cell_type": "markdown",
- "metadata": {},
- "source": [
- "- Suppose we don’t have any data on $C$. Can we identify the treatment\n",
- " effect?\n",
- " - Yes! If we look at the effect of $R$ on $Y$, we can see that it\n",
- " is unblocked.\n",
- " - Furthermore, since $R$ only affects $Y$ through $T$, the effect\n",
- " of $R$ on $Y$ is the same as the effect of $T$ on $Y$.\n",
- " - This is called an **instrumental variable**.\n",
- "\n",
- "# Factorizing the Joint Distribution\n",
- "\n",
- "## Factorizing the Joint Distribution\n",
- "\n",
- "- We have seen how these graphical models can be used to determine\n",
- " whether or not we can identify a treatment effect.\n",
- "\n",
- "- However, we can also use them as a computational tool, to help us\n",
- " find the simplest representation of the joint distribution of the\n",
- " data.\n",
- "\n",
- "- This is useful because it allows us to find the simplest model that\n",
- " is consistent with our assumptions about conditional independence.\n",
- "\n",
- "## Factorizing the Joint Distribution\n",
- "\n",
- "- To understand why this is useful, we’ll follow this example (from\n",
- " [Mark\n",
- " Paskin](http://ai.stanford.edu/~paskin/gm-short-course/lec2.pdf)).\n",
- "\n",
- "- Suppose we have a DGP with five variables:\n",
- "\n",
- " 1. $E\\in\\{true, false\\}$ - Has an earthquake happened? *Earthquakes\n",
- " are unlikely*\n",
- " 2. $B\\in\\{true, false\\}$ - Has a burglary happened? *Burglaries are\n",
- " unlikely, but more likely than earthquakes*\n",
- " 3. $A\\in\\{true, false\\}$ - Is the alarm going off? *The alarm is\n",
- " triggered by both earthquakes and burglaries*\n",
- " 4. $J\\in\\{true, false\\}$ - Is John calling? *John calls when he\n",
- " hears the alarm, but he often misses it*\n",
- " 5. $M\\in\\{true, false\\}$ - Is Mary calling? *Mary calls when she\n",
- " hears the alarm, but she also calls to chat*\n",
- "\n",
- "## Factorizing the Joint Distribution\n",
- "\n",
- "- The goal is to compute $P(B|J=true)$ from the joint distribution\n",
- " $P(E,B,A,J,M)$. We’ll start by drawing the graphical model, to\n",
- " understand the conditional independence relationships."
- ],
- "id": "21ba1659-f150-4d64-9863-1a4edfdbb25d"
- },
- {
- "cell_type": "code",
- "execution_count": 36,
- "metadata": {},
- "outputs": [
- {
- "output_type": "display_data",
- "metadata": {},
- "data": {
- "image/svg+xml": "PD94bWwgdmVyc2lvbj0iMS4wIiBlbmNvZGluZz0iVVRGLTgiIHN0YW5kYWxvbmU9Im5vIj8+Cjwh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\n"
- }
- }
- ],
- "source": [
- "g = gr.Digraph()\n",
- "g.edge(\"E\", \"A\")\n",
- "g.edge(\"B\", \"A\")\n",
- "g.edge(\"A\", \"J\")\n",
- "g.edge(\"A\", \"M\")\n",
- "\n",
- "g"
- ],
- "id": "daf4e860"
- },
- {
- "cell_type": "markdown",
- "metadata": {},
- "source": [
- "## Factorizing the Joint Distribution"
- ],
- "id": "231e83d4-7bf6-4c3e-8cd0-cdac54856cb7"
- },
- {
- "cell_type": "code",
- "execution_count": 37,
- "metadata": {},
- "outputs": [
- {
- "output_type": "display_data",
- "metadata": {},
- "data": {
- "image/svg+xml": "PD94bWwgdmVyc2lvbj0iMS4wIiBlbmNvZGluZz0iVVRGLTgiIHN0YW5kYWxvbmU9Im5vIj8+Cjwh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\n"
- }
- }
- ],
- "source": [
- "g"
- ],
- "id": "d66dc034"
- },
- {
- "cell_type": "markdown",
- "metadata": {},
- "source": [
- "- In order to represent the full joint distribution, we could use the\n",
- " chain rule of probabilities:\n",
- "\n",
- "$$\n",
- "P(E,B,A,J,M) = P(E)P(B|E)P(A|E,B)P(J|E,B,A)P(M|E,B,A,J)\n",
- "$$\n",
- "\n",
- "- Q: How many probabilities would we need to compute to represent the\n",
- " joint distribution this way?\n",
- "\n",
- "## Factorizing the Joint Distribution"
- ],
- "id": "b6a3d727-cefc-42d2-8c77-66a63fbe796e"
- },
- {
- "cell_type": "code",
- "execution_count": 38,
- "metadata": {},
- "outputs": [
- {
- "output_type": "display_data",
- "metadata": {},
- "data": {
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- "source": [
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- "source": [
- "- In order to represent the full joint distribution, we could use the\n",
- " chain rule of probabilities:\n",
- "\n",
- "$$\n",
- "\\underbrace{P(E,B,A,J,M)}_{31} = \\underbrace{P(E)}_{1} \\underbrace{P(B|E)}_{2} \\underbrace{P(A|E,B)}_{4} \\underbrace{P(J|E,B,A)}_{8} \\underbrace{P(M|E,B,A,J)}_{16}\n",
- "$$\n",
- "\n",
- "- Q: How many probabilities would we need to store to represent the\n",
- " joint distribution this way?\n",
- " - A: There are $2^5$ possible outcomes. We need 31 probabilities.\n",
- " (why not 32?)\n",
- "\n",
- "## Factorizing the Joint Distribution"
- ],
- "id": "9d3cdee7-3bea-439d-b745-1d4923ea95e8"
- },
- {
- "cell_type": "code",
- "execution_count": 39,
- "metadata": {},
- "outputs": [
- {
- "output_type": "display_data",
- "metadata": {},
- "data": {
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- "- However, we can use the conditional independence relationships to\n",
- " simplify this representation.\n",
- " - For example, we know that $P(B|E) = P(B)$, because $B$ and $E$\n",
- " are independent.\n",
- " - We also know that $P(A|E,B) = P(A|B)$, because $A$ is\n",
- " independent of $E$, given $B$.\n",
- "- This means that we can simplify the joint distribution to:\n",
- "\n",
- "$$\n",
- "\\underbrace{P(E,B,A,J,M)}_{10} = \\underbrace{P(E)}_1 \\underbrace{P(B)}_1 \\underbrace{P(A|E,B)}_4 \\underbrace{P(J|A)}_2 \\underbrace{P(M|A)}_2\n",
- "$$\n",
- "\n",
- "## Factorizing the Joint Distribution\n",
- "\n",
- "- Beyond causal inference, graphical models are a useful tool for\n",
- " computing all kinds of conditional probabilities.\n",
- "\n",
- "- This is useful in many inference problems, and in structural\n",
- " econometrics\n",
- "\n",
- " - The **likelihood function** is the conditional probability of\n",
- " the data, given the parameter values\n",
- " - In Bayesian inference the **posterior** is the conditional\n",
- " probability of the parameters, given the data (and our priors)\n",
- "\n",
- "## Variable Elimination\n",
- "\n",
- "- In order to simplify a conditional distribution, we can use\n",
- " **variable elimination**.\n",
- "\n",
- "- For example, let’s say we want to compute the probability of a\n",
- " burglary, given that John calls.\n",
- "\n",
- "$$\n",
- "\\begin{aligned}\n",
- "&p_{B \\mid J}(b, \\text { true }) \\propto \\sum_e \\sum_a \\sum_m p_{E B A J M}(e, b, a, \\text { true }, m) \\\\\n",
- "& =\\sum_e \\sum_a \\sum_m p_E(e) \\cdot p_B(b) \\cdot p_{A \\mid E B}(a, e, b) \\cdot p_{J \\mid A}(\\text { true }, a) \\cdot p_{M \\mid A}(m, a)\n",
- "\\end{aligned}\n",
- "$$\n",
- "\n",
- "- Then, we can reduce the computational complexity by eliminating\n",
- " variables one at a time, exploiting the distributive property of\n",
- " multiplication\n",
- " - $xy + xz = x(y + z)$.\n",
- "\n",
- "## Variable Elimination\n",
- "\n",
- "- Variable elimination works like this:\n",
- " - Repeat the following steps:\n",
- "\n",
- " 1. choose a variable to eliminate\n",
- " 2. push in its sum as far as possible\n",
- " 3. compute the sum, resulting in a new factor\n",
- "\n",
- "## Variable Elimination\n",
- "\n",
- "$$\n",
- "\\begin{aligned}\n",
- "&\\sum_e \\sum_a \\sum_m p_E(e) \\cdot p_B(b) \\cdot p_{A \\mid E B}(a, e, b) \\cdot p_{J \\mid A}(\\text { true }, a) \\cdot p_{M \\mid A}(m, a) \\\\\n",
- "& =\\sum_e \\sum_a p_E(e) \\cdot p_B(b) \\cdot p_{A \\mid E B}(a, e, b) \\cdot p_{J \\mid A}(\\text { true }, a) \\cdot \\sum_m p_{M \\mid A}(m, a) \\\\\n",
- "& =\\sum_e \\sum_a p_E(e) \\cdot p_B(b) \\cdot p_{A \\mid E B}(a, e, b) \\cdot p_{J \\mid A}(\\text { true }, a) \\cdot \\psi_A(a) \\\\\n",
- "& =\\sum_e p_E(e) \\cdot p_B(b) \\cdot \\sum_a p_{A \\mid E B}(a, e, b) \\cdot p_{J \\mid A}(\\text { true }, a) \\cdot \\psi_A(a) \\\\\n",
- "& =\\sum_e p_E(e) \\cdot p_B(b) \\cdot \\psi_{E B}(e, b) \\\\\n",
- "& =p_B(b) \\cdot \\sum_e p_E(e) \\cdot \\psi_{E B}(e, b) \\\\\n",
- "& =p_B(b) \\cdot \\psi_B(b)\n",
- "\\end{aligned}\n",
- "$$\n",
- "\n",
- "## Variable Elimination\n",
- "\n",
- "- That’s a lot of math! But let’s focus on the first step: $$\n",
- " \\begin{aligned}\n",
- " & p_{B \\mid J}(b, \\text { true }) \\propto \\\\\n",
- " &\\underbrace{\\underbrace{\\sum_e \\sum_a \\sum_m}_{2^3 = 8\\text{ iterations}} \\underbrace{p_E(e) \\cdot p_B(b) \\cdot p_{A \\mid E B}(a, e, b) \\cdot p_{J \\mid A}(\\text { true }, a) \\cdot p_{M \\mid A}(m, a)}_{4\\text{ multiplications}}}_{8*4=32\\text{ multiplications }+7\\text{ additions}=39\\text{ total operations}}\n",
- " \\end{aligned}\n",
- " $$\n",
- "\n",
- "## Variable Elimination\n",
- "\n",
- "- That’s a lot of math! But let’s focus on the first step: $$\n",
- " \\begin{aligned}\n",
- " & p_{B \\mid J}(b, \\text { true }) \\propto \\\\\n",
- " &\\underbrace{\\underbrace{\\sum_e \\sum_a \\sum_m}_{2^3 = 8\\text{ iterations}} \\underbrace{p_E(e) \\cdot p_B(b) \\cdot p_{A \\mid E B}(a, e, b) \\cdot p_{J \\mid A}(\\text { true }, a) \\cdot p_{M \\mid A}(m, a)}_{4\\text{ multiplications}}}_{8*4=32\\text{ multiplications }+7\\text{ additions}=39\\text{ total operations}} \\\\\\\\\n",
- " & =\\underbrace{\\underbrace{\\sum_e \\sum_a}_{2^2 = 4\\text{ iterations}} \\underbrace{p_E(e) \\cdot p_B(b) \\cdot p_{A \\mid E B}(a, e, b) \\cdot p_{J \\mid A}(\\text { true }, a) \\cdot }_{4\\text{ multiplications}} \\underbrace{\\sum_m p_{M \\mid A}(m, a)}_{1\\text{ addition}}}_{4*(4\\text{ multiplications} + 1\\text{ addition}) + 3\\text{ additions} = 23\\text{ total operations}} \\\\\n",
- " \\end{aligned}\n",
- " $$\n",
- "\n",
- "## Variable Elimination\n",
- "\n",
- "- Variable elimination is an algorithm that exploits our conditional\n",
- " independence assumptions to reduce the computational complexity of\n",
- " conditional probabilities.\n",
- "\n",
- "- In this case, we were able to reduce the number of operations from\n",
- " 39 to 23 operations in just one step, each further step will\n",
- " continue to reduce the complexity.\n",
- "\n",
- "- This is a very simple example, because we only have 5 variables. In\n",
- " practice, we might have hundreds or thousands of variables, and the\n",
- " computational complexity can become very large.\n",
- "\n",
- "- Systematic patterns of conditional independence can be exploited to\n",
- " reduce the complexity of inference problems in some large models.\n",
- "\n",
"## Credits\n",
"\n",
"This lecture draws heavily from [Causal Inference for the Brave and\n",
@@ -1836,7 +967,12 @@
"As well as [The Effect: Chapter 7 - Drawing Graphical\n",
"Diagrams](https://theeffectbook.net/) by Nick Huntington-Klein."
],
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