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Interventions and Inference / Reasoning
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Causal models
Recall from yesterday: Represent relevance using graphs Causal relevance ⇒ DAGs Quantitative component = joint probability
distribution And so clear definitions for independence &
association
Connect DAG & jpd with two assumptions: Markov: No edge ⇒ Independent given direct parents Faithfulness: Conditional independence ⇒ No edge
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Three uses of causal models Represent (and predict the effects of)
interventions on variables Causal models only, of course
Efficiently determine independencies I.e., which variables are informationally
relevant for which other ones? Use those independencies to rapidly
update beliefs in light of evidence
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Representing interventions
Central intuition: When we intervene, we control the state of the target variable And so the direct causes of the target
variable no longer matter But the target still has its usual effects
Directly applying current to the light bulb ⇒ light switch doesn’t matter, but the plant still grows
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Representing interventions
Formal implementation: Add a variable representing the
intervention, and make it a direct cause of the target
When the intervention is “active,” remove all other edges into the target
Leave intact all edges directed out of the target, even when the intervention is “active”
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Representing interventions
Example:
Light Switch
Plant Growth
Light Bulb
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Representing interventions
Example: Add a manipulation variable as a “cause”
Light Switch
Plant Growth
Current
Light Bulb
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Representing interventions
Example: Add a manipulation variable as a “cause”
that does not matter when it is inactive
Inactive Manipulation
Light Switch
Plant Growth
Current
Light Bulb
Inactive
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Representing interventions
Example: Add a manipulation variable as a “cause”
that does not matter when it is inactive When it is active,
Active Manipulation
Light Switch
Plant Growth
Current
Light Bulb
Inactive Manipulation
Light Switch
Plant Growth
Current
Light Bulb
Inactive
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Representing interventions
Example: Add a manipulation variable as a “cause”
that does not matter when it is inactive When it is active, break the incoming
edges, but leave the outgoing edges
Active Manipulation
Light Switch
Plant Growth
Current
Light Bulb
Inactive Manipulation
Light Switch
Plant Growth
Current
Light Bulb
Inactive
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Representing interventions
Straightforward extension to more interesting types of interventions Interventions away from current state Multi-variate interventions Etc.
Key: For all of these, the “intervention operator” takes a causal graphical model as input, and yields a causal graphical model as output “Post-intervention CGM” is an ordinary CGM
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Why randomize?
Standard scientific practice: randomize Treatment to find its Effects E.g., don’t let people decide on their own
whether to take the drug or placebo What is the value of randomization?
Randomization is an intervention ⇒ All edges into T will be broken, including from
any common causes of T and E! ⇒ If T E, then we must have: T → E
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Why randomize?
Graphically,
Treatment Effect?
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Why randomize?
Graphically,
Treatment
UnobservedFactors
Effect?
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Why randomize?
Graphically,
Treatment
UnobservedFactors
Effect?
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Why randomize?
Graphically,
Treatment
UnobservedFactors
Effect?
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Why randomize?
Graphically,
Treatment
UnobservedFactors
Effect?
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Three uses of causal models Represent (and predict the effects of)
interventions on variables Causal models only, of course
Efficiently determine independencies I.e., which variables are informationally
relevant for which other ones? Use those independencies to rapidly
update beliefs in light of evidence
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Determining independence
Markov & Faithfulness ⇒ DAG structure determines all statistical independencies and associations
Graphical criterion: d-separation X and Y are independent given S iff
X and Y are d-separated given S iffX and Y are not d-connected given S
Intuition: X and Y are d-connected iff information can “flow” from X to Y along some path
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d-separation
C is a collider on a path iff A → C ← B Formally:
A path between A and B is active given S iff Every non-collider on the path is not in S; and Every collider on the path is either in S, or else
one of its descendants is in S X and Y are d-connected by S iff there is an
active path between X and Y given S
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d-separation
Surprising feature being exploited here: Conditioning on a common effect induces an
association between independent causes Motivating example:
Gas Tank → Car Starts ← Spark Plugs Gas and Plugs are independent, but if we know
that the car doesn’t start, then they’re associated In that case, learning Gas = Full changes the
likelihood that Plugs = Bad
And similarly if Car Starts → Emits Exhaust
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d-separation
Algorithm to determine d-separation:1. Write down every path between X and Y
– Edge direction is irrelevant for this step– Just write down every sequence of edges
that lies between X and Y– But don’t use a node twice in the same path
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d-separation
Algorithm to determine d-separation:1. Write down every path between X and Y 2. For each path, determine whether it is
active by checking the status of each node on the path
– The node is not active if either:1. N is a collider + not in S (and no descendants of
N are in S); or2. N is not a collider and in S.3. I.e., “multiply” the “not”s to get the node status
1. Any node not active ⇒ path not active
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d-separation
Algorithm to determine d-separation:1. Write down every path between X and Y 2. For each path, determine whether it is
active by checking the status of each node on the path
3. Any path active ⇒ d-connected ⇒ X & Y associated No path active ⇒ d-separated ⇒ X & Y independent
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d-separation
Exercise and Weight given Metabolism? E → M → W
Blocked! M isan included non-collider
E → FE → W Unblocked! FE is
a non-included non-collider
⇒ E W | M
Exercise
FoodEaten
Weight
Metabolism
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d-separation
Metabolism and FE given Exercise? M → W ← FE
Blocked! W isa non-included collider
M ← E → FE Blocked! E is
an included non-collider
⇒ M FE | E
Exercise
FoodEaten
Weight
Metabolism
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d-separation
Metabolism and FE given Weight? M → W ← FE
Unblocked! W isan included collider
M ← E → FE Unblocked! E is
a non-included non-collider
⇒ M FE | W
Exercise
FoodEaten
Weight
Metabolism
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Updating beliefs
For both statistical and causal models, efficient computation of independencies ⇒ efficient prediction from observations Specific instance of belief updating Typically, “just” compute conditional
probabilities Significantly easier if we have (conditional)
independencies, since we can ignore variables
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Bayes (and Bayesianism)
Bayes’ Theorem: proof is trivial…
Interpretation is the interesting part: Let D be the observation and T be our
target variable(s) of interest ⇒ Bayes’ theorem says how to update our
beliefs about T given some observation(s)
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Bayes (and Bayesianism)
Terminology:
Posteriordistribution
Likelihoodfunction
Priordistribution
Data distribution
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Bayes and independence
Knowing independencies can greatly speed Bayesian updating
P(C | E, F, G) = [complex mess] Suppose C independent of F, G given E
⇒ P(C | E, F, G) = P(C | E) = [something simpler]
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Updating beliefs
Compute: P(M = Hi | E = Hi, FE = Lo) FE M | E ⇒
P(M | E, FE) = P(M | E) And P(M | E) is a term in the
Markov factorization!Exercise
FoodEaten
Weight
Metabolism
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Looking ahead…
Have: Basic formal representation for causation Fundamental causal asymmetry (of
intervention) Inference & reasoning methods
Need: Search & causal discovery methods