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Bayes Net Perspectives on Causation and Causal Inference

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Example Problems

- Genetic regulatory networks
- Yeast – ~5000 genes, ~2,500,000 potential edges

A gene regulatory network in mouse embryonic stem cells http://www.pnas.org/content/104/42/16438/F3.expansion.html

Causal Models → Predictions

- Probabilistic – Among the cells that have active Oct4 what percentage have active Rcor2?

- Causal – If I experimentally set a cell to have active Oct4, what percentage will have active Rcor2?

Causal Models → Predictions

- Counterfactual – Among the cells that did not have active Oct4 at t-1, what percentage would have active Rcor2 if I had experimentally set a cell to have active Oct4 at t-1?

Data → Causal Models

- Large number of variables
- Small observed sample size

- Overlapping variables
- Small number of experiments
- Feedback
- Hidden common causes
- Selection bias
- Many kinds of entities causally interacting

Outline

- Bayesian Networks
- Search
- Limitations and Extensions of Bayesian Networks
- Dynamic
- Relational
- Cycles
- Counterfactual

- Search

- Limitations and Extensions
- Dynamic
- Relational

- Cycles
- Counterfactual

SES

SEX PE CP

IQ

SES – Socioeconomic Status

PE – Parental Encouragement

CP – College Plans

IQ – Intelligence Quotient

SEX– Sex

- The vertices are random variables.
- All edges are directed.
- There are no directed cycles.

- Search

- Limitations and Extensions
- Dynamic
- Relational

- Cycles
- Counterfactual

SES

SEX PE CP

IQ

SES

SEX PE CP

IQ

SES

SEX PE CP

IQ

Independent, identically distributed

- Search

- Limitations and Extensions
- Dynamic
- Relational

- Cycles
- Counterfactual

SES

SEX PE CP

IQ

- P(SES,SEX,PE,IQ,CP) =

P(SEX)P(SES)P(IQ|SES)

P(PE|SES,SEX,IQ)

P(CP|PE,SES,IQ)

- If
- then P factors according to G
- G represents all of the distributions that factor according to G

- Search

- Limitations and Extensions
- Dynamic
- Relational

- Cycles
- Counterfactual

- X is independent of Y conditional on Z (denoted IP(X,Y|Z)) iff P(X|Y,Z) = P(X|Z).
- IP(CP,SEX|{SES,IQ,PE}) iff P(CP|{SES,IQ,PE,SEX}) = P(CP|{SES,IQ,PE})

- Search

- Limitations and Extensions
- Dynamic
- Relational

- Cycles
- Counterfactual

SES

SEX PE CP

IQ

- If for every P that factors according to G, IP(X,Y|Z) holds, then GentailsI(X,Y|Z).
- Examples: G entails
- I(IQ,SEX|∅)
- I(IQ,SEX|SES)
- Can read entailments off of graph through d-separation

- Search

- Limitations and Extensions
- Dynamic
- Relational

- Cycles
- Counterfactual

SES

SEX PE CP

IQ

- X d-separated from Y conditional on Z in G iff G entails X independent of Y conditional on Z
- D-separation between X and Y conditional on Z holds when certain kinds of paths do notexist between X and Y

- D-connection (the negation of d-separation) between X and Y conditional on Z holds when certain kinds of paths do exist between X and Y

- Search

- Limitations and Extensions
- Dynamic
- Relational

- Cycles
- Counterfactual

SES

SEX PE CP

IQ

- A node X is active on a path UconditionalonZ iff
- X is a collider (→ X ←) and there is a directed path from X to a member of Z or X is in Z; or
- X is not a collider and X is not in Z.

- SES → IQ → PE ← SEX is a path U.
- PE is active on U conditional on {CP, IQ}.
- IQ is inactive on U conditional on {CP, IQ}.

- Search

- Limitations and Extensions
- Dynamic
- Relational

- Cycles
- Counterfactual

SES

SEX PE CP

IQ

- A path U is active conditional onZ iff every vertex on U is active relative to Z.
- X is d-connectedto Y conditional onZ iff there is an active path between X and Y conditional on Z.

- SES → IQ → PE ← SEX is inactive conditional on{CP, IQ}.
- SES is d-connected to SEX conditional on {CP, IQ} because SES → PE ← SEX is active conditional on {CP, IQ}

- Search

- Limitations and Extensions
- Dynamic
- Relational

- Cycles
- Counterfactual

- If conditional independence relation I is not entailed by G, then I may hold in some (but not every) distribution P that factors according to G.

SES

SEX PE CP

IQ

- Example: There are P and P’ that factor according to G such that ~IP(SES,CP|∅) and IP’(SES,CP|∅). P’ is said to be unfaithful to G.

- Search

- Limitations and Extensions
- Dynamic
- Relational

- Cycles
- Counterfactual

- An ideal manipulationassigns a density to a set X of properties (random variables) as a function of the values of a set Z of properties (random variables)
- Directly affects only the variables in X
- Successful
- Example – randomized experiment

- Search

- Limitations and Extensions
- Dynamic
- Relational

- Cycles
- Counterfactual

SES

SEX PE CP

IQ

- There is an edge SES → CP in Gbecause there are two ways of manipulating {SES,SEX,IQ,PE} that differ only in the value they assign to SES that changes the probability of CP.

Stable Unit Treatment Value Assumption

- Search

- Limitations and Extensions
- Dynamic
- Relational

- Cycles
- Counterfactual

SES

SEX PE CP

IQ

- A set S of variables is causally sufficient if there are no variables not in S that are direct causes of more than one variable in S.
- S = {SES,IQ} is causally sufficient.
- S = {SES,PE,CP} is not causally sufficient.

- Search

- Limitations and Extensions
- Dynamic
- Relational

- Cycles
- Counterfactual

SES

SEX PE CP

IQ

- In a population Pop with distribution P and causal graph G, if V is causally sufficient, P(V) factorsaccording to G.
- P(SES,SEX,PE,IQ,CP) =

P(SEX)P(SES)P(IQ|SES)

P(PE|SES,SES,IQ)

P(CP|PE,SES,IQ)

- Search

- Limitations and Extensions
- Dynamic
- Relational

- Cycles
- Counterfactual

SES

SEX PE CP

IQ

P(SES,SEX,PE=1,IQ,CP||PE=1) =

P(SEX)P(SES)P(IQ|SES) * 1 * P(CP|PE,SES,IQ) =

P(SES,SEX,PE=1,IQ,CP)/P(PE|SEX,SES,IQ)

- Search

- Limitations and Extensions
- Dynamic
- Relational

- Cycles
- Counterfactual

- Looks for set of DAGs (possibly with latent variables and selection bias) that entail all and only the conditional independence relations that hold in the data according to statistical tests.

- Search

- Limitations and Extensions
- Dynamic
- Relational

- Cycles
- Counterfactual

- Two DAGs G1 and G2 are Markov equivalent when they contain the same variables, and for all disjoint X, Y, Z, X is entailed to be independent from Y conditional on Z in G1 if and only if X is entailed to be independent from Y conditional on Z in G2

- Search

- Limitations and Extensions
- Dynamic
- Relational

- Cycles
- Counterfactual

SES

SEX PE CP

IQ

SES

SEX PE CP

IQ

DAG G’

DAG G

- Search

- Limitations and Extensions
- Dynamic
- Relational

- Cycles
- Counterfactual

SES

SEX PE CP

IQ

- In a population Pop with causal graph G and distribution P(V), if V is causally sufficient, IP(X,Y|Z) only if G entails I(X,Y|Z).
- ~IP(SES,CP|∅) because I(SES,CP|∅)is not entailed by G
- +…

- Search

- Limitations and Extensions
- Dynamic
- Relational

- Cycles
- Counterfactual

SES

SEX PE CP

IQ

- Causal Faithfulness is too strong because
- can prove consistency with assumptions about fewer conditional independencies
- is unlikely to hold, especially when there are many variables.

- Causal Faithfulness is too weak because it is not sufficient to prove uniform consistency (put error bounds at finite sample sizes.)

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- Limitations and Extensions
- Dynamic
- Relational

- Cycles
- Counterfactual

- Is pointwise consistent: As sample size → ∞, P(error in output pattern) → 0.
- Can be applied to distributions where tests of conditional independence are known
- Can be applied to hidden variable models (and selection bias models)

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- Limitations and Extensions
- Dynamic
- Relational

- Cycles
- Counterfactual

- There is no reliable way to set error bounds on the pattern without making stronger assumptions.
- Can only get set of Markov equivalent DAGs, not a single DAG
- Doesn’t allow for comparing how much better one model is than another
- Need to assume some version of Causal Faithfulness Assumption

- Search

- Limitations and Extensions
- Dynamic
- Relational

- Cycles
- Counterfactual

- Depending on the parametric family, a DAG can entail constraints that are not conditional independence constraints
- Assuming linearity and non-Gaussian error terms, if a distribution is compatible with X → Y it is not compatible with X ← Y, even though they are Markov equivalent.

- Search

- Limitations and Extensions
- Dynamic
- Relational

- Cycles
- Counterfactual

- Assign score to graph and sample based on
- maximum likelihood of data given graph
- simplicity of model
- Do search over graph space for highest score

- Search

- Limitations and Extensions
- Dynamic
- Relational

- Cycles
- Counterfactual

- Get more information about graph
- Additive noise models, unique DAG
- Doesn’t rely on binary decisions
- Local mistakes don’t propagate

- Search

- Limitations and Extensions
- Dynamic
- Relational

- Cycles
- Counterfactual

- Often slower to calculate or not known how to calculate exactly if include
- unmeasured variables
- selection bias
- unusual distributions
- Search over graph space is often heuristic

- Search

- Limitations and Extensions
- Dynamic
- Relational

- Cycles
- Counterfactual

- If measure same variable at different times, then the samples from the variable are not i.i.d.
- Solution: index each variable by time (time series)

- Search

- Limitations and Extensions
- Dynamic
- Relational

- Cycles
- Counterfactual

- Make a template for the causal structure that can be filled in with actual times

Xt-2Xt-1Xt

Yt-2Yt-1Yt

- Continuous time or differential equations?
- Continuous time or differential equations?

- Search

- Limitations and Extensions
- Dynamic
- Relational

- Cycles
- Counterfactual

parent-of

parent-of

parent-of

PopulationSES

SEX PE CP

IQ

SES

SEX PE CP

IQ

SES

SEX PE CP

IQ

- Search

- Limitations and Extensions
- Dynamic
- Relational

- Cycles
- Counterfactual

parent-of

parent-of

parent-of

PopulationSES

SEX PE CP

IQ

- Not i.i.d. distribution
- Violations of SUTVA
- Causal relations between relations (e.g. sibling causes rivalry)

- Search

- Limitations and Extensions
- Dynamic
- Relational

- Cycles
- Counterfactual

- A manipulation assigns a density to
- a set of properties or relations
- at a set of times (measurable set of times T)
- for a set of units
- as a function of the values of
- a set of properties of relations
- at a set of times (measurable set of times T)
- for a set of units

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- Limitations and Extensions
- Dynamic
- Relational

- Cycles
- Counterfactual

parent-of

parent-of

Extended Factorization AssumptionAlice&Jim

SES

SEX PE CP

IQ

Sue

Bob

P([Alice&Jim.SES, Sue.SEX,Sue.PE, Sue.IQ, Sue.CP,

Alice&Jim.SES, Bob.SEX,Bob.PE, Bob.IQ, Bob.CP) =

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- Limitations and Extensions
- Dynamic
- Relational

- Cycles
- Counterfactual

P(Sue.SEX) P(Alice&Jim.SES)P(Sue.IQ|Alice&Jim

.SES) P(Sue.PE|Alice&Jim.SES,Sue.SEX, Sue.IQ) P(Sue.CP|Sue.PE, Alice&Jim.SES, Sue.IQ)

P(Bob.SEX) P(Alice&Jim.SES) P(Bob.IQ|Alice&Jim.SES) P(Bob.PE|Alice&Jim.SES, Bob.SEX, Bob.IQ) P(Bob.CP|Bob.PE, Alice&Jim.SES, Bob.IQ)

- Search

- Limitations and Extensions
- Dynamic
- Relational

- Cycles
- Counterfactual

SES

SEX PE CP

IQ

- Equilibrium values of PE and CP cause each other.
- Average of values of PE and CP while reaching equilibrium influence each other.
- Mixture of PE→ CP and CP→ PE

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