04.Bayesian networks(W5)

What's causality

X causes Y iff (if noly if) there is some manipulation of X leading to a change in the probability distribution of Y. (Judea Pearl, 2000; Neapolitan, 2003)

How to discover causality?

The gold standard method: A randomized controlled experiment (RCE).

The reasons for the unavailability of the experiments.

Infeasible
Unethical
Expensive

What's the IDA(Intervention calculus when the DAG is Absent)

The purpose of IDA in this context is to provide a methodological framework for inferring causality in situations where a DAG cannot be fully constructed.

Steps for IDA

1. Use the PC algorithm to learn the CPDAG(Completed Partially Directed Acyclic Graph).
2. Infer the causal effects, e.g. eff(X2, Y), in all DAGs in the equivalent class
3. Find the lower bound of causal effects.

Causal inference - IDA

Causal inference

Local structure learning algorithms

• A major weakness of PC algorithm
  • Time complexity is exponential to the number of variables.
• Fix a variable and find parent and child nodes of the node.
  • Both PC-Simple and Hiton-PC have polynomial time complexity with the number of variables.

1. PC algorithm

PC Algorithm

Steps of Example

Initial: Let the $Z$ be the target, $k=0, PC^0 = {A,B,C,D,E,F,G,H}$.

1. Test the independence between $Z$ with each variable.
   • ${H,D,E}$ are independent of Z
   • $k=1, PC^1 = {A,B,C,D,E,F,G,H}$
2. Test the independence between $Z$ with each variable given another variable.
   • $G$ is indenpendent of $Z$ given $F$.
   • $k=2, PC^2 = {A,B,C,F}$
3. Test the independence between $Z$ with each variable given another two variables.
   • $A$ is independent of $Z$ given ${B,C}$
   • $k=3, PC^3 = {B,C,F}$
4. Since $|PC^3| = k (k=3)$, the program is terminated
   $PC = {B,C,F}$

2. Hiton-PC algorithm

Hiton-PC Algorithm

Steps of Example

Initial: Let the $Z$ be the target, $max_k = 2$, $OPEN = [B,C,A,G,F], PC={}$
OPEN sorted in descending order of association strength

1. Set $OPEN = [C,A,G,F], PC={B}$

2. Test the independence between $C$ with $Z$ given by $B$

   • It's not independence.
   • $OPEN = [A,G,F], PC={B,C}$

3. Test the independence between $A$ with $Z$ given by the combination from $PC$.

   • $A$ is independent of $Z$ given by ${B,C}$
   • Remove $A$ from $OPEN$
   • $OPEN = [G,F], PC={B,C}$

4. Test the independence between $G$ with $Z$ given by the combination from $PC$.

   • It's not independence.
   • $OPEN = [F], PC={B,C,G}$

5. Test the independence between $F$ with $Z$ given by the combination from $PC$.

   • It's not independence.
   • $OPEN = [], PC={B,C,G,F}$

6. Test the independence between item from $PC$ (Pick up item from left to right) given by the combination of other right items.

   • 1. test the independence between $B$ with $Z$ given by combination of ${C,G,F}$
   • 2. test the independence between $B$ with $Z$ given by combination of ${G,F}$
   • 3. test the independence between $B$ with $Z$ given by combination of ${F}$
   • $G$ is indenpendent given by $F$
     $PC = {B,C,F}$

Cohort study

Cohorts

Cohorts: share common characteristics but exposed or not exposed.
Determine how the exposure causes an outcome.
Measure: $\text{odds ratio} = \frac{(a/b)}{(c/d)}$

	Diseased	Healthy
Exposed	a	b
Not Exposed	c	d

Limitations of cohort study

• Need to know a hypothesis beforehand
• Domain experts determine the control variables.
• Collect data and test the hypothesis.
• Not for data exploration.

We need:

• Given a dataset without any hypotheses.
• An automatic method to find and validate hypotheses.
• For data exploration.

Matches

References

1. Week 5 Slides from Thuc (SP52023)
2. 贝叶斯理论：Data Science必备技能，数据分析师为你揭开 Bayes 统计的神秘面纱（第426期