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If $$p(x,y) = p(x)p(y)$$, then this distribution still factorizes over the graph $$y \rightarrow x$$, since we can always write it as $$p(x,y) = p(x\mid y)p(y)$$ with a CPD $$p(x\mid y)$$ in which the probability of $$x$$ does not actually vary with $$y$$. However, we can construct a graph that matches the structure of $$p$$ by simply removing that unnecessary edge. +**Framework for determining d-separation** {% Citation: Pieter Abbeel's YouTube video on d-separation https://www.youtube.com/watch?v=yDs_q6jKHb0&t=1040s%} +Given a query $$X_1 \perp X_2 \mid X_3, X_4$$ +- Shade all evidence nodes ($$X_3, X_4$$) +- For all paths from $$X_1$$ to $$X_2$$, take triplets along each path and using the 4 rules described above, determine if the path is active or inactive. +- If any path is active it is not guaranteed that $$X_1 \perp X_2 \mid X_3, X_4$$ is true. +- If all paths are inactive (one of the triplets along the path violates the 4 rules) then the query is true. + +**Example of Applying this Framework** + +{% include marginfigure.html id="dp2" url="assets/img/dsep3.png" description="In this example, $$Y$$ and $$Z$$ are not $$d$$-separated given $$X$$." %}{% include marginfigure.html id="dp1" url="assets/img/dsep3.png" description="Here we shade out our evidence node as described in the framework to determine if triples are active or inactive." %} + +We consider the query $$Y \perp Z \mid X$$ +- There are two paths from $$Y to $$Z. ($$Y, X, V, T, Z$$ and $$Y, W, V, T, Z$$) +- Considering Path 1 $$Y, X, V, T, Z$$, the first triple is $$Y, X, V$$. +- This is an inactive triple since $$X is observed, which means that Path 1 is inactive. +- Along Path 2, the first triple is $Y, W, V$$ which is active. +- The second triple is $$W, V, T$$ which is active and finally the third triple $$V, T, Z$$ is also active. +- Since there is an active path between $$Y$$ and $$Z$$ the independence statement is not guaranteed to be true. + + ### The representational power of directed graphs This raises our last and perhaps most important question: can directed graphs express all the independencies of any distribution $$p$$? More formally, given a distribution $$p$$, can we construct a graph $$G$$ such that $$I(G) = I(p)$$? From d9081943517105f4690d2f153b7bb61ad8b9220d Mon Sep 17 00:00:00 2001 From: Anirudh Joshi Date: Sun, 15 Mar 2020 22:56:56 -0700 Subject: [PATCH 2/2] Formatting change --- representation/directed/index.md | 11 ++++++----- 1 file changed, 6 insertions(+), 5 deletions(-) diff --git a/representation/directed/index.md b/representation/directed/index.md index f9bbe10e..07c5a466 100644 --- a/representation/directed/index.md +++ b/representation/directed/index.md @@ -102,7 +102,8 @@ In other words, all the independencies encoded in $$G$$ are sound: variables tha In a way this is almost a trivial statement. If $$p(x,y) = p(x)p(y)$$, then this distribution still factorizes over the graph $$y \rightarrow x$$, since we can always write it as $$p(x,y) = p(x\mid y)p(y)$$ with a CPD $$p(x\mid y)$$ in which the probability of $$x$$ does not actually vary with $$y$$. However, we can construct a graph that matches the structure of $$p$$ by simply removing that unnecessary edge. -**Framework for determining d-separation** {% Citation: Pieter Abbeel's YouTube video on d-separation https://www.youtube.com/watch?v=yDs_q6jKHb0&t=1040s%} +**Framework for determining d-separation** {% include sidenote.html note="Taken from Pieter Abbeel's YouTube video on d-separation https://www.youtube.com/watch?v=yDs_q6jKHb0&t=1040s" %} + Given a query $$X_1 \perp X_2 \mid X_3, X_4$$ - Shade all evidence nodes ($$X_3, X_4$$) - For all paths from $$X_1$$ to $$X_2$$, take triplets along each path and using the 4 rules described above, determine if the path is active or inactive. @@ -111,13 +112,13 @@ Given a query $$X_1 \perp X_2 \mid X_3, X_4$$ **Example of Applying this Framework** -{% include marginfigure.html id="dp2" url="assets/img/dsep3.png" description="In this example, $$Y$$ and $$Z$$ are not $$d$$-separated given $$X$$." %}{% include marginfigure.html id="dp1" url="assets/img/dsep3.png" description="Here we shade out our evidence node as described in the framework to determine if triples are active or inactive." %} +{% include marginfigure.html id="dp2" url="assets/img/dsep3.png" description="In this example, $$Y$$ and $$Z$$ are not $$d$$-separated given $$X$$." %} We consider the query $$Y \perp Z \mid X$$ -- There are two paths from $$Y to $$Z. ($$Y, X, V, T, Z$$ and $$Y, W, V, T, Z$$) +- There are two paths from $$Y$$ to $$Z$$. ($$Y, X, V, T, Z$$ and $$Y, W, V, T, Z$$) - Considering Path 1 $$Y, X, V, T, Z$$, the first triple is $$Y, X, V$$. -- This is an inactive triple since $$X is observed, which means that Path 1 is inactive. -- Along Path 2, the first triple is $Y, W, V$$ which is active. +- This is an inactive triple since $$X$$ is observed, which means that Path 1 is inactive. +- Along Path 2, the first triple is $$Y, W, V$$ which is active. - The second triple is $$W, V, T$$ which is active and finally the third triple $$V, T, Z$$ is also active. - Since there is an active path between $$Y$$ and $$Z$$ the independence statement is not guaranteed to be true.