INTERVENTIONS AND INFERENCE / REASONING. Causal models Recall from yesterday: Represent relevance using graphs Causal relevance ⇒ DAGs Quantitative.

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

Plant Growth

Light Bulb

Representing interventions

Example: Add a manipulation variable as a “cause”

Light Switch

Plant Growth

Current

Light Bulb

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

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

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

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

UnobservedFactors

Effect?

Why randomize?

Graphically,

Treatment

UnobservedFactors

Effect?

Why randomize?

Graphically,

Treatment

UnobservedFactors

Effect?

Why randomize?

Graphically,

Treatment

UnobservedFactors

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 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

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 In that case, learning Gas = Full changes the

likelihood that Plugs = Bad

And similarly if Car Starts → Emits Exhaust

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

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

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

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

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

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

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: 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)

Bayes (and Bayesianism)

Terminology:

Posteriordistribution

Likelihoodfunction

Priordistribution

Data 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) And P(M | E) is a term in the

Markov factorization!Exercise

FoodEaten

Weight

Metabolism

Looking ahead…

Have: Basic formal representation for causation Fundamental causal asymmetry (of

intervention) Inference & reasoning methods

Need: Search & causal discovery methods