SMHS AssociationCausality - Revision history

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‎References

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‎References

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‎Hill’s criteria for causality

Dinov: /* Halpern-Pearl’s Definition of Causality */

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‎Halpern-Pearl’s Definition of Causality

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‎Halpern-Pearl’s Definition of Causality

Dinov: /* Halpern-Pearl’s Definition of Causality */

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‎Halpern-Pearl’s Definition of Causality

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‎Halpern-Pearl’s Definition of Causality

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‎Halpern-Pearl’s Definition of Causality

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 * [http://ftp.cs.ucla.edu/pub/stat_ser/R266-part1.pdf   Causes and Explanations: A Structural-Model Approach-Part I: Causes]
 * [http://en.wikipedia.org/wiki/Bradford_Hill_criteria   Bradford Hill Criteria Wikipedia]
-* [http://www.nasonline.org/programs/sackler-colloquia/completed_colloquia/Big-data.html
+* [http://www.nasonline.org/programs/sackler-colloquia/completed_colloquia/Big-data.html Drawing Causal Inference from Big Data]

← Older revision		Revision as of 14:08, 28 April 2015
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	* [http://ftp.cs.ucla.edu/pub/stat_ser/R266-part1.pdf Causes and Explanations: A Structural-Model Approach-Part I: Causes]		* [http://ftp.cs.ucla.edu/pub/stat_ser/R266-part1.pdf Causes and Explanations: A Structural-Model Approach-Part I: Causes]
	* [http://en.wikipedia.org/wiki/Bradford_Hill_criteria Bradford Hill Criteria Wikipedia]		* [http://en.wikipedia.org/wiki/Bradford_Hill_criteria Bradford Hill Criteria Wikipedia]
		+	* [http://www.nasonline.org/programs/sackler-colloquia/completed_colloquia/Big-data.html
		+	Drawing Causal Inference from Big Data]

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 ===References===
-* [http://ftp.cs.ucla.edu/pub/stat_ser/R266-part1.pdf Causes and Explanation: A Structural-Model Approach / Joseph Halpern, Judea Pearl, 2001]
+* [http://en.wikipedia.org/wiki/Association_(statistics)  Association (statistics) Wikipedia]
-* [http://mirlyn.lib.umich.edu/Record/004088429 Causality : models, reasoning, and inference  Judea Pearl, J.]
+* [http://ftp.cs.ucla.edu/pub/stat_ser/R266-part1.pdf   Causes and Explanations: A Structural-Model Approach-Part I: Causes]
-* [http://mirlyn.lib.umich.edu/Record/012195186 Causality, probability, and time /  Kleinberg,S.]
+* [http://en.wikipedia.org/wiki/Bradford_Hill_criteria   Bradford Hill Criteria Wikipedia]

@@ Line 215: / Line 215: @@
 ===References===
-Joseph Halpern, Judea Pearl, 2001, Causes and Explanation: A Structural-Model Approach.
+* [http://ftp.cs.ucla.edu/pub/stat_ser/R266-part1.pdf Causes and Explanation: A Structural-Model Approach / Joseph Halpern, Judea Pearl, 2001]
 * [http://mirlyn.lib.umich.edu/Record/004088429 Causality : models, reasoning, and inference  Judea Pearl, J.]
-*[http://mirlyn.lib.umich.edu/Record/004088429 Causality : models, reasoning, and inference  Judea Pearl, J.]
+* [http://mirlyn.lib.umich.edu/Record/012195186 Causality, probability, and time /  Kleinberg,S.]
+* [http://en.wikipedia.org/wiki/Bradford_Hill_criteria Hill Criteria Wikipedia]

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 : The distinction between April showers being a cause of the fire (which they are not, according to our analysis) and April showers being a cause of a fire in June (which they are) is one that seems not to have been made in the discussion of this problem; nevertheless, it seems to us an important distinction.
-====Hill’s criteria for causality====
+====[http://en.wikipedia.org/wiki/Bradford_Hill_criteria Hill’s criteria for causality]====
 *''Strength:'' A small association does not mean that there is not a causal effect, though the larger the association, the more likely that is causal.
 *''Consistency:'' Consistent findings observed by different persons in different places with different sample strengthen the likelihood of an effect.

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 *AC1. $(M,\overrightarrow{u})⊨(\bar {X} ,\bar{x})\Lambda\phi$. That is, both $\overrightarrow{X}=\overrightarrow{x}$ and $\phi$ are true in the actual world.
-*AC2. There exists a partition $(\overrightarrow{Z},\overrightarrow{x})$ of $ν$ with $\overrightarrow{X} ⊆ \overrightarrow{ Z}$ and some settings $(\overrightarrow{x},\overrightarrow{ω})$ of the variable with $(\overrightarrow{X},\overrightarrow{W})$ such that if $(\overrightarrow{X},\overrightarrow{u})⊨Z=z^*$ for $Z in \overrightarrow{Z}$, then
+*AC2. There exists a partition $(\overrightarrow{Z},\overrightarrow{x})$ of $ν$ with $\overrightarrow{X} ⊆ \overrightarrow{ Z}$ and some settings $(\overrightarrow{x},\overrightarrow{ω})$ of the variable with $(\overrightarrow{X},\overrightarrow{W})$ such that if $(\overrightarrow{X},\overrightarrow{u})⊨Z=z^*$ for $Z \in \overrightarrow{Z}$, then
 :: (a) $(M,\overrightarrow{u})⊨[\overrightarrow{X} ← \overrightarrow{x},\overrightarrow{W} ← \overrightarrow{ω},\overrightarrow{Z'} ← \overrightarrow{z^*}]¬φ$. In words, changing $(\overrightarrow{X},\overrightarrow{W})$ from $(x,ω)$ to $(\overrightarrow{x'},\overrightarrow{ω'})$ changes $φ$ from the true to false.
 :: (b) $(M,\overrightarrow{u})⊨[\overrightarrow{X} ← \overrightarrow{x},\overrightarrow{W} ← \overrightarrow{ω},\overrightarrow{Z'} ← \overrightarrow{z^*}]φ$ for all subsets $\overrightarrow{Z'}$ of $\overrightarrow{Z}$. In words, setting $\overrightarrow{W}$ to $\overrightarrow{ω}$ should have no effect on $φ$ as long as $\overrightarrow{X}$ is kept at its current value $\overrightarrow{x}$, even if all the variables in an arbitrary subset of $\overrightarrow{Z}$ are set to their original values in the context $\overrightarrow{u}$.

@@ Line 76: / Line 76: @@
 In [http://bjps.oxfordjournals.org/content/56/4/843.short Causes and Explanations: A Structural-Model Approach], Halpern and Pearl gave a definition of actual causality based on the language of structural equations.
-Definition of '''Actual Cause (AC)''': $\overleftrightarrow {X} =\overleftrightarrow{x}$ is an actual cause of $\phi$ in $(M,\overleftrightarrow{u})$ if the following three conditions hold:
+Definition of '''Actual Cause (AC)''': $\overrightarrow {X} =\overrightarrow{x}$ is an actual cause of $\phi$ in $(M,\overrightarrow{u})$ if the following three conditions hold:
-*AC1. $(M,\overleftrightarrow{u})⊨(\bar {X} ,\bar{x})\Lambda\phi$. That is, both $\overleftrightarrow{X}=\overleftrightarrow{x}$ and $\phi$ are true in the actual world.
+*AC1. $(M,\overrightarrow{u})⊨(\bar {X} ,\bar{x})\Lambda\phi$. That is, both $\overrightarrow{X}=\overrightarrow{x}$ and $\phi$ are true in the actual world.
-*AC2. There exists a partition $(\overleftrightarrow{Z},\overleftrightarrow{x})$ of $ν$ with $\overleftrightarrow{X} ⊆ \overleftrightarrow{ Z}$ and some settings $(\overleftrightarrow{x},\overleftrightarrow{ω})$ of the variable with $(\overleftrightarrow{X},\overleftrightarrow{W})$ such that
+*AC2. There exists a partition $(\overrightarrow{Z},\overrightarrow{x})$ of $ν$ with $\overrightarrow{X} ⊆ \overrightarrow{ Z}$ and some settings $(\overrightarrow{x},\overrightarrow{ω})$ of the variable with $(\overrightarrow{X},\overrightarrow{W})$ such that
-if $(\overleftrightarrow{X},\overleftrightarrow{u})⊨Z=z^*$ for $Z ∈\overleftrightarrow{Z}$, then
+if $(\overrightarrow{X},\overrightarrow{u})⊨Z=z^*$ for $Z ∈\overrightarrow{Z}$, then
-:: (a) $(M,\overleftrightarrow{u})⊨[\overleftrightarrow{X} ← \overleftrightarrow{x},\overleftrightarrow{W} ← \overleftrightarrow{ω},\overleftrightarrow{Z'} ← \overleftrightarrow{z^*}]¬φ$. In words, changing $(\overleftrightarrow{X},\overleftrightarrow{W})$ from $(x,ω)$ to $(\overleftrightarrow{x'},\overleftrightarrow{ω'})$ changes $φ$ from the true to false.
+:: (a) $(M,\overrightarrow{u})⊨[\overrightarrow{X} ← \overrightarrow{x},\overrightarrow{W} ← \overrightarrow{ω},\overrightarrow{Z'} ← \overrightarrow{z^*}]¬φ$. In words, changing $(\overrightarrow{X},\overrightarrow{W})$ from $(x,ω)$ to $(\overrightarrow{x'},\overrightarrow{ω'})$ changes $φ$ from the true to false.
-:: (b) $(M,\overleftrightarrow{u})⊨[\overleftrightarrow{X} ← \overleftrightarrow{x},\overleftrightarrow{W} ← \overleftrightarrow{ω},\overleftrightarrow{Z'} ← \overleftrightarrow{z^*}]φ$ for all subsets $\overleftrightarrow{Z'}$ of $\overleftrightarrow{Z}$. In words, setting $\overleftrightarrow{W}$ to $\overleftrightarrow{ω}$ should have no effect on $φ$ as long as $\overleftrightarrow{X}$ is kept at its current value $\overleftrightarrow{x}$, even if all the variables in an arbitrary subset of $\overleftrightarrow{Z}$ are set to their original values in the context $\overleftrightarrow{u}$.
+:: (b) $(M,\overrightarrow{u})⊨[\overrightarrow{X} ← \overrightarrow{x},\overrightarrow{W} ← \overrightarrow{ω},\overrightarrow{Z'} ← \overrightarrow{z^*}]φ$ for all subsets $\overrightarrow{Z'}$ of $\overrightarrow{Z}$. In words, setting $\overrightarrow{W}$ to $\overrightarrow{ω}$ should have no effect on $φ$ as long as $\overrightarrow{X}$ is kept at its current value $\overrightarrow{x}$, even if all the variables in an arbitrary subset of $\overrightarrow{Z}$ are set to their original values in the context $\overrightarrow{u}$.
-*AC3. $\overleftrightarrow{X}$ is minimal; no subset of $\overleftrightarrow{X}$ satisfies conditions AC1 and AC2. Minimality ensures that only those elements of the conjunction $\overleftrightarrow{X}=\overleftrightarrow{x}$ that are essential for changing $φ$ in AC2(a) are considered part of a cause; inessential elements are pruned. The types of events that we allow as actual causes are ones of the form $X_1=x_1 ⋀ … ⋀X_k = x_k$, that is, conjunctions of primitive events; this is abbreviated as $X=\overleftrightarrow{x}$.
+*AC3. $\overrightarrow{X}$ is minimal; no subset of $\overrightarrow{X}$ satisfies conditions AC1 and AC2. Minimality ensures that only those elements of the conjunction $\overrightarrow{X}=\overrightarrow{x}$ that are essential for changing $φ$ in AC2(a) are considered part of a cause; inessential elements are pruned. The types of events that we allow as actual causes are ones of the form $X_1=x_1 ⋀ … ⋀X_k = x_k$, that is, conjunctions of primitive events; this is abbreviated as $X=\overrightarrow{x}$.
 *Example: Suppose that there was a heavy rain in April and electrical storms in the following two months; and in June the lightning took hold. If it hadn’t been for the heavy rain in April, the forest would have caught fire in May. The question is whether the April rains caused the forest fire. According to a naive counterfactual analysis, they do, since if it hadn’t rained, there wouldn’t have been a forest fire in June.
-This is unacceptable. A good enough story of events and of causation might give us reason to accept some things that seem intuitively to be false, but no theory should persuade us that delaying a forest’s burning for a month (or indeed a minute) is causing a forest fire.
+: This is unacceptable. A good enough story of events and of causation might give us reason to accept some things that seem intuitively to be false, but no theory should persuade us that delaying a forest’s burning for a month (or indeed a minute) is causing a forest fire.
-In our framework, as we now show, it is indeed false to say that the April rains caused the fire, but they were a cause of there being a fire in June, as opposed to May. This seems to us intuitively right. To capture the situation, it suffices to use a simple model with three endogenous random variables:
+: In our framework, as we now show, it is indeed false to say that the April rains caused the fire, but they were a cause of there being a fire in June, as opposed to May. This seems to us intuitively right. To capture the situation, it suffices to use a simple model with three endogenous random variables:
 *AS for “April showers”, with two values—0 standing for did not rain heavily in April and 1 standing for rained heavily in April;
@@ Line 98: / Line 98: @@
 *And F for “fire”, with three possible values: 0 (no fire at all), 1 (fire in May), or 2 (fire in June).
-We do not describe the context explicitly, either here or in the other examples. Assume its value is such that it ensures that there is a shower in April, there are electric storms in both May and June, there is sufficient oxygen, there are no other potential causes of fire (like dropped matches), no other inhibitors of fire (alert campers setting up a bucket brigade), and so on. That is, we choose so as to allow us to focus on the issue at hand and to ensure that the right things happened (there was both fire and rain).
+: We do not describe the context explicitly, either here or in the other examples. Assume its value is such that it ensures that there is a shower in April, there are electric storms in both May and June, there is sufficient oxygen, there are no other potential causes of fire (like dropped matches), no other inhibitors of fire (alert campers setting up a bucket brigade), and so on. That is, we choose so as to allow us to focus on the issue at hand and to ensure that the right things happened (there was both fire and rain).
-Avoiding writing out the details of the structural equations—they should be obvious, given the story (at least, for the context $\overleftrightarrow{u}$; this is also the case for all the other examples in this section. The causal network is simple: there are edges from AS to F and from ES to F. It is easy to check that each of the following hold.
+: Avoiding writing out the details of the structural equations—they should be obvious, given the story (at least, for the context $\overrightarrow{u}$; this is also the case for all the other examples in this section. The causal network is simple: there are edges from AS to F and from ES to F. It is easy to check that each of the following hold.
-*AS = 1 is a cause of the June fire (F = 2) (taking $\overleftrightarrow{W}={ES}$ and $\overleftrightarrow{Z}={AS,F}$) but not of fire ($F=2 ⋁ F=1$).
+*AS = 1 is a cause of the June fire (F = 2) (taking $\overrightarrow{W}={ES}$ and $\overrightarrow{Z}={AS,F}$) but not of fire ($F=2 ⋁ F=1$).
 *ES = (1,1) is a cause of the both $F=2$ and ($F=1 ⋁ F=2$). Having electric storms in both May and June caused there to be a fire.
 *$AS=1∧ES=(1,1)$ is not a cause of $F=2$, because it violates the minimality requirement of AC3; each conjunct alone is a cause of $F=2$. Similarly, $AS=1∧ES=(1,1)$ is not a cause of (F=1⋁F=2).
-The distinction between April showers being a cause of the fire (which they are not, according to our analysis) and April showers being a cause of a fire in June (which they are) is one that seems not to have been made in the discussion of this problem; nevertheless, it seems to us an important distinction.
+: The distinction between April showers being a cause of the fire (which they are not, according to our analysis) and April showers being a cause of a fire in June (which they are) is one that seems not to have been made in the discussion of this problem; nevertheless, it seems to us an important distinction.
 ====Hill’s criteria for causality====

@@ Line 76: / Line 76: @@
 In [http://bjps.oxfordjournals.org/content/56/4/843.short Causes and Explanations: A Structural-Model Approach], Halpern and Pearl gave a definition of actual causality based on the language of structural equations.
-Definition of Actual Cause: $\overrightarrow {X} ,\overrightarrow {x}$ is an actual cause of $*φ* in *(M,\overrightarrow {u})* $ if the following three conditions hold:
+Definition of '''Actual Cause (AC)''': $\overleftrightarrow {X} =\overleftrightarrow{x}$ is an actual cause of $\phi$ in $(M,\overleftrightarrow{u})$ if the following three conditions hold:
+*AC1. $(M,\overleftrightarrow{u})⊨(\bar {X} ,\bar{x})\Lambda\phi$. That is, both $\overleftrightarrow{X}=\overleftrightarrow{x}$ and $\phi$ are true in the actual world.
-*AC1. $(M,\overrightarrow{u})⊨(\overrightarrow {X} ,\overrightarrow {x})\Lambda\phi$. (That is, both $X ⃗=x ⃗ $ and $\phi$ are true in the actual world.
+*AC2. There exists a partition $(\overleftrightarrow{Z},\overleftrightarrow{x})$ of $ν$ with $\overleftrightarrow{X} ⊆ \overleftrightarrow{ Z}$ and some settings $(\overleftrightarrow{x},\overleftrightarrow{ω})$ of the variable with $(\overleftrightarrow{X},\overleftrightarrow{W})$ such that
-*AC2. There exists a partition $(Z ⃗,x ⃗) $ of $ν$ with $X ⃗⊆ Z ⃗ $ and some settings $(x ⃗,ω ⃗)$ of the variable with $(X ⃗,W ⃗) $such that if $(X ⃗,u ⃗ )⊨Z=z^*$ for $Z ∈Z ⃗$, then
+if $(\overleftrightarrow{X},\overleftrightarrow{u})⊨Z=z^*$ for $Z ∈\overleftrightarrow{Z}$, then
-**(a) (M,u ⃗ )⊨[X ⃗←x ⃗,W ⃗←ω ⃗,(Z^' ) ⃗←(z^* ) ⃗]¬φ. In words, changing (X ⃗,W ⃗) from (x,ω) to ((x') ⃗,(ω') ⃗) changes φ from the true to false.
+**(a) $(M,\overleftrightarrow{u})⊨[\overleftrightarrow{X} ← \overleftrightarrow{x},\overleftrightarrow{W} ← \overleftrightarrow{ω},\overleftrightarrow{Z'} ← \overleftrightarrow{z^*}]¬φ$. In words, changing $(\overleftrightarrow{X},\overleftrightarrow{W})$ from $(x,ω)$ to $(\overleftrightarrow{x'},\overleftrightarrow{ω'})$ changes $φ$ from the true to false.
 **(b) (M,u ⃗ )⊨[X ⃗←x ⃗,W ⃗←ω ⃗,(Z^' ) ⃗←(z^* ) ⃗]φ for all subsets (Z') ⃗ of Z ⃗. In words, setting W ⃗ to ω ⃗ should have no effect on φ as long as X ⃗ is kept at its current value x ⃗, even if all the variables in an arbitrary subset of Z ⃗ are set to their original values in the context u ⃗.
 *AC3. X ⃗ is minimal; no subset of X ⃗ satisfies conditions AC1 and AC2. Minimality ensures that only those elements of the conjunction $X ⃗=x ⃗ $ that are essential for changing φ in AC2(a) are considered part of a cause; inessential elements are pruned.