Causality

1
Causality

• Computational Systems Biology Lab
• Arizona State University
• Michael Verdicchio
• 26 March 2008
• With some slides and slide content from
• Judea Pearl, Chitta Baral, Xin Zhang

2
Talking Points

• Conditional Independence
• Definitions
• Interpretation
• Notation
• d-Separation
• Graphical Probabilistic Models
• Causal Graphs
• Causal Modeling Framework

3
Conditional Independence
4
Conditional Independence
5
Conditional Independence (Notation)
6
Talking Points

• Conditional Independence
• d-Separation
• Three rules
• Probabilistic implications
• Graphical Probabilistic Models
• Causal Graphs
• Causal Modeling Framework

7
d-Separation

• d-separation is a criterion for deciding, from a
  given a causal graph, whether a set X of
  variables is independent of another set Y, given
  a third set Z

8
d-Separation Rule 1

• Rule 1 Unconditional Separation
• Two nodes are d-connected if there is an
  unblocked path between them
• Path edges without directionality
• Unblocked no head-to-head arrows

9
d-Separation Rule 1

• One collider at t
• x-r-s-t unblocked, so x and t are d-connected
• t-u-v-y unblocked, so t and y are d-connected
• So are all the pairs, x-r, x-s, r-s, t-u, etc.
• x and y are not d-connected since we cant trace
  a path without hitting the collider hence they
  are d-separated
• So too are x-u, x-v, r-u, etc.

10
d-Separation Rule 2

• Rule 2 x and y are d-connected, conditioned on a
  set Z if there is a collider-free path between x
  and y that traverses no member of Z
• If no such path exists, we say that x and y are
  d-separated by Z
• We also say then that every path between x and y
  is "blocked" by Z

11
d-Separation Rule 2

• Let Z be the set r,v
• By Rule 2, x and y are d-separated by Z, along
  with x-s, u-y, s-u, etc.
• The path x-r-s is blocked by Z, along with u-v-y
  and s-t-u.
• Only s-t and u-t remain d-connected conditioned
  on Z
• The path s-t-u is also blocked Z since t is a
  collider, and is blocked by Rule 1

12
d-Separation Rule 3

• Rule 3 If a collider is a member of the
  conditioning set Z, or has a descendant in Z,
  then it no longer blocks any path that traces
  this collider
• Called the common effect of two independent
  causes explaining away one
• dead battery ? car wont start ? no gas

13
d-Separation Rule 3

• Let Z be the set r, p
• By Rule 3 s and y are d-connected by Z
• the collider at t has a descendant (p) in Z
• This unblocks the path s-t-u-v-y
• x and u are still d-separated by Z
• the linkage at t is unblocked
• but the one at r is blocked by Rule 2 (since r is
  in Z).

14
Theorem 1

• Probabilistic implication of d-separation

15
Theorem 2
16
Talking Points

• Conditional Independence
• d-Separation
• Graphical Probabilistic Models
• Bayesian Networks
• Causation vs. Correlation
• Causal Graphs
• Causal Modeling Framework

17
Graphical Probabilistic Models

• Provide convenient means of expressing
  substantive assumptions
• Facilitate economical representation of joint
  probability functions
• Facilitate efficient inferences from observations
• ? Bayesian Networks

18
Bayesian Networks

• Chain Rule of Probability
• With Markov Assumption

19
Causation vs. Correlation

• Correlation
• Rain and a falling barometer
• Rain does not cause barometer to fall
• Barometer falling does not cause rain
• Causation
• Rain causes mud
• Other events may also cause mud
• Denote causal relationship graphically with
  directed edges

20
Talking Points

• Conditional Independence
• d-Separation
• Graphical Probabilistic Models
• Causal Graphs
• Effects of intervention
• Causal Bayesian Networks
• Causal Stability
• Causal Modeling Framework

21
Directed (causal) Graphs

• A and B are causally independent
• C, D, E, and F are causally dependent on A and B
• A and B are direct causes C
• A and B are indirect causes D, E and F
• If C is prevented from changing with A and B,
  then A and B will no longer cause changes in D, E
  and F.

22
Causal Graphs

• DAGs as conditional independence carriers does
  not imply causation
• So build DAG models around causal rather than
  associative information
• If conditional independence judgments are
  byproducts of stored causal relationships, then
  representing these causal relationships directly
  would be a more natural and more reliable way of
  expressing what we know or believe about the
  world. --Judea Pearl

23
Causal Graphs

• Another advantage
• Probabilities do not predict effects of
  interventions
• Causal networks can be oracles for interventions
• Example turn the sprinkler on remove edges

24
Causal Graphs

• The ability of causal graphs to predict the
  effects of actions requires a stronger set of
  assumptions in network construction
• These assumptions must rest on causal knowledge,
  not just associational
• These assumptions are encapsulated in the
  following definition of Causal Bayesian Networks

25
Definition
26
Definition
27
Definition
28
Causal Stability

• We now have a semantic basis for things like
  causal effects or causal influence
• Causal relationships are ontologicaldescribing
  objective physical constraints in the our world
• Probabilistic relationships are
  epistemicreflecting what we know or believe
  about the world
• Thus, causal relationships should remain
  unaltered as long as no changes have happened in
  the environment -- even when knowledge about the
  environment changes

29
Causal Stability

• For example, causal statement S1
• turning the sprinkler on would not affect the
  rain
• Versus probabilistic statement S2
• the state of the sprinkler is independent of (or
  unassociated with) the state of the rain
• The network above shows two ways S2 can become
  false, but S1 remains true

30
Talking Points

• Conditional Independence
• d-Separation
• Graphical Probabilistic Models
• Causal Graphs
• Causal Modeling Framework
• Causal Structure
• Causal Model
• IC Algorithm

31
Causal Structure
32
Causal Model
33
Causal Model

• Once a causal model M is formed, it defines a
  joint probability distribution P(M) over the
  variables in the system
• This distribution reflects some features of the
  causal structure
• Each variable must be independent of its
  grandparents, given the values of its parents
• We may recover the topology D of the DAG, from
  features of the probability distribution

34
IC algorithm (Inductive Causation)

• IC algorithm (Pearl)
• Based on variable dependencies
• Find all pairs of variables that are dependent of
  each other (applying standard statistical method
  on the database)
• Eliminate (as much as possible) indirect
  dependencies
• Determine directions of dependencies

35
Comparing abduction, deduction and induction
A gt B A --------- B

• Deduction major premise All balls in the
  box are black
• minor premise
  These balls are from the box
• conclusion
  These balls are black
• Abduction rule All balls
  in the box are black
• observation
  These balls are black
• explanation These balls
  are from the box
• Induction case These
  balls are from the box
• observation
  These balls are black
• hypothesized rule All
  ball in the box are black

A gt B B ------------- Possibly A
Whenever A then B but not vice versa -------------
Possibly A gt B
Induction from specific cases to general
rules Abduction and deduction both from
part of a specific case to other part of
the case using general rules (in different ways)
Source from httpwww.csee.umbc.edu/ypeng/F02671/le
cture-notes/Ch15.ppt
36
IC Algorithm (Contd)

• Input
• P a stable distribution on a set V of
  variables
• Output
• A pattern H(P) compatible with P
• Pattern is a partially directed DAG
• some edges are directed and
• some edges are undirected

37
IC Algorithm Step 1

• For each pair of variables a and b in V, search
  for a set Sab such that (a-b Sab) holds in P
  in other words, a and b should be independent in
  P, conditioned on Sab .
• Construct an undirected graph G such that
  vertices a and b are connected with an edge if
  and only if no set Sab can be found.

38
IC Algorithm Step 2

• For each pair of nonadjacent variables a and b
  with a common neighbor c, check if c? Sab.
• If it is, then continue
• Else add arrowheads at c
• i.e a? c ? b

39
Example
40
IC Algorithm Step 3

• In the partially directed graph that results,
  orient as many of the undirected edges as
  possible subject to two conditions
• The orientation should not create a new
  v-structure
• The orientation should not create a directed
  cycle

41
Rules required to obtain a maximally oriented
pattern

• R1 Orient b c into b?c whenever there is an
  arrow a?b such that a and c are non adjacent

42
Rules required to obtain a maximally oriented
pattern

• R2 Orient a b into a?b whenever there is a
  chain a?c?b

43
Rules required to obtain a maximally oriented
pattern

• R3 Orient a b into a?b whenever there are two
  chains ac?b and ad?b such that c and d are
  nonadjacent

44
Rules required to obtain a maximally oriented
pattern

• R4 Orient a b into a?b whenever there are two
  chains ac?d and c?d?b such that c and b are
  nonadjacent

c
a
d
d
c
b
45
Next Time

• Using a variant of the IC algorithm to learn
  causal gene regulatory networks
• Again, thanks to Chitta Baral, Andrew Moore and
  Xin Zhang, all of whom got most of their stuff
  from Judea Pearl