Tuesday slides

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Transcript Tuesday slides

INTERVENTIONS AND
INFERENCE / REASONING
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
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
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
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”
Representing interventions

Example:
Light Switch
Light Bulb
Plant Growth
Representing interventions

Example:
 Add
a manipulation variable as a “cause”
Current
Light Switch
Light Bulb
Plant Growth
Representing interventions

Example:
 Add
a manipulation variable as a “cause” that does not
matter when it is inactive
Current
Light Switch
Inactive
Light Bulb
Plant Growth
Inactive Manipulation
Representing interventions

Example:
 Add
a manipulation variable as a “cause” that does not
matter when it is inactive
 When it is active,
Current
Light Switch
Inactive
Light Bulb
Plant Growth
Inactive Manipulation
Current
Light Switch
Light Bulb
Plant Growth
Active Manipulation
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
Current
Light Switch
Inactive
Light Bulb
Plant Growth
Inactive Manipulation
Current
Light Switch
Light Bulb
Plant Growth
Active Manipulation
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
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
⇒
Why randomize?

Graphically,
Treatment
?
Effect
Why randomize?

Graphically,
Treatment
?
Unobserved
Factors
Effect
Why randomize?

Graphically,
Treatment
?
Unobserved
Factors
Effect
Why randomize?

Graphically,
Treatment
?
Unobserved
Factors
Effect
Why randomize?

Graphically,
Treatment
?
Unobserved
Factors
Effect
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
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 iff
X 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
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
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

 And
In that case, learning Gas = Full changes the likelihood that
Plugs = Bad
similarly if Car Starts → Emits Exhaust
d-separation
Algorithm to determine d-separation:

Write down every path between X and Y
1.
–
–
–
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
d-separation
Algorithm to determine d-separation:

Write down every path between X and Y
For each path, determine whether it is active by
checking the status of each node on the path
1.
2.
The node is not active if either:
–
–
–
–
–
N is a collider + not in S (and no descendants of N are in S);
or
N is not a collider and in S.
I.e., “multiply” the “not”s to get the node status
Any node not active ⇒ path not active
d-separation
Algorithm to determine d-separation:

1.
2.
3.
Write down every path between X and Y
For each path, determine whether it is active by
checking the status of each node on the path
Any path active ⇒ d-connected ⇒ X & Y associated
No path active ⇒ d-separated ⇒ X & Y independent
d-separation

Exercise and Weight given Metabolism?
E
→M→W
Food
Eaten
 Blocked!
M is
an included non-collider
E
→ FE → W
FE is
a non-included non-collider
Exercise
Weight
 Unblocked!

⇒E
W|M
Metabolism
d-separation

Metabolism and FE given Exercise?
M
→ W ← FE
Food
Eaten
 Blocked!
W is
a non-included collider
M
← E → FE
E is
an included non-collider
Exercise
Weight
 Blocked!

⇒M
FE | E
Metabolism
d-separation

Metabolism and FE given Weight?
M
→ W ← FE
Food
Eaten
 Unblocked!
W is
an included collider
M
← E → FE
E is
a non-included non-collider
Exercise
Weight
 Unblocked!

⇒M
FE | W
Metabolism
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
Bayes (and Bayesianism)

Bayes’ Theorem: P (T | D) = P ( D | T ) P (T )
 proof

is trivial…
P ( D)
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)
Bayes (and Bayesianism)

Terminology:
Posterior
distribution
Likelihood
function
P ( D | T ) P (T )
P (T | D) =
P ( D)
Data distribution
Prior
distribution
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]
Updating beliefs

Compute: P(M = Hi | E = Hi, FE = Lo)
 FE
M|E ⇒
P(M | E, FE) = P(M | E)
Food
Eaten
 And
P(M | E) is a term in the
Markov factorization!
Exercise
Weight
Metabolism
Looking ahead…

Have:
 Basic
formal representation for causation
 Fundamental causal asymmetry (of intervention)
 Inference & reasoning methods

Need:
 Search
& causal discovery methods