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, TreatmentEffect ?

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 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: – 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. 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E → M → W Blocked! M is an included non-collider  E → FE → W Unblocked! FE is a non-included non-collider  ⇒ E W | M Exercise Food Eaten Weight Metabolism

d-separation  Metabolism and FE given Exercise?  M → W ← FE Blocked! W is a non-included collider  M ← E → FE Blocked! E is an included non-collider  ⇒ M FE | E Exercise Food Eaten Weight Metabolism

d-separation  Metabolism and FE given Weight?  M → W ← FE Unblocked! W is an included collider  M ← E → FE Unblocked! E is a non-included non-collider  ⇒ M FE | W Exercise Food Eaten 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: Posterior distribution Likelihood function Prior distribution 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 Food Eaten Weight Metabolism

Looking ahead…  Have:  Basic formal representation for causation  Fundamental causal asymmetry (of intervention)  Inference & reasoning methods  Need:  Search & causal discovery methods