Tetrad project

http://www.phil.cmu.edu/projects/tetrad/current.html
CAUSAL MODELS IN THE
COGNITIVE SCIENCES
Two uses

Causal graphical models used in:
 Practice/methodology
 Focus
on neuroimaging, but lots of other uses
 Framework
 Are
of cognitive science
for expressing human causal knowledge
human causal representations “just” these causal
graphical models?
 Also (but not today): Are other cognitive representations
“just” graphical models (perhaps causal, perhaps not)?
Learning from neuroimaging

Given neuroimaging data, what is the causal
structure inside the brain?
 Ignoring
differences in timescale, challenges in inverting
the hemodynamic response curve, etc.
??
Learning from neuroimaging

Big challenge: people likely have (slightly) different
causal structures in their brains
Full dataset is really from a mixed population!
 ⇒ “Normal” causal search falls apart
⇒

Idea: perhaps the differences are mostly in
parameters, not graphs
 Note
that “no edge” ≡ “parameter = 0”
IMaGES algorithm

Given data from individuals D1, …, Dn, the score for
graph G is computed by:
 Compute
ML estimate of parameters for Di
 Use that ML estimate to get BIC for Di
 Score for G is the average BIC over all datasets:
 Do
GES-style search over graphs (i.e., greedy edge
addition, then greedy edge removal)
IMaGES application

Standard causal search:
IMaGES:
Causal cognition



Causal inference: learning causal structure from a
sequence of cases (observations or interventions)
Causal perception: learning causal connections
through “direct” perception
Causal reasoning: using prior causal knowledge to
predict, explain, control your world
Descriptive theories (in 2000)

Paradigmatic causal inference situation:
A
set of binary potential causes: C1, …, Cn
 A known binary effect: E
 Minimal role for prior beliefs
 Observational
 Possible
data about variable values
formats include: sequential, list, or summary
Descriptive theories (in 2000)

Goal of theories: model (mean) “strength ratings” as
a function of the observed cases)
 Or

a series of (mean) ratings
Two theory-types: Dynamical vs. Long-run
 Dynamical
predict belief change after single cases
 Long-run predict stable beliefs after “enough time”
 Similar to algorithmic vs. computational distinction
Dynamical theories (in 2000)

Rescorla-Wagner (and variants)
 Associative
 Causal
strength for each cue (to the effect)
version: associative strengths are causal
 Schematic
form of R-W:
ΔVi = RateParams × (Outcome – Prediction)
 That
is, use error-correction to update the associative
strengths after each observed case
 Variant R-W models explain phenomena such as backwards
blocking by changing the prediction function
Long-run theories (in 2000)

In the long-run, causal strength judgments should be
proportional to the:
 Conditional
contrast (Conditional ΔP theory):
ΔPC.{X} = P (E |C & X ) – P (E |~C & X )
 Causal
strength estimate (Power PC):
pC = ΔPC.F / [1 – P (E |~C & F)]
 where
F is a “focal set” of relevant events
Dynamical & long-run theories

In the long-run, Rescorla-Wagner (and variants)
“converges” to conditional ΔP
 I.e.,


R-W is a dynamical version of conditional ΔP
Simple modification of the error-correction equation
converges to power PC
Primary debate (in 2000): which family of theories
correctly describes causal learning?
Parameter estimation

Connect causal models
and descriptive theories:
B
C1
…
wC
wC
Cn
is a constant background cause1 E n
 Limited correlations allowed between C1, …, Cn

B
wB
Additional restriction:
 Assume
we have: P(E) = f(wC1, …, wCn, wB), or more
precisely:
P(E | C1, …, Cn) = f(wC1, …, wCn, wB, C1, …, Cn)
Parameter estimation

Essentially every descriptive theory estimates the wparameters in this causal Bayes net
 Different
descriptive theories result from different
functional forms for P(E)

And all of the research on the descriptive theories
implies that people can estimate parameters in this
“simple” causal structure
Learning causal structure?

Additional queries:
 From
a “rational analysis” point-of-view:
 Can
people learn structure from interventions?
 Or from patterns of correlations?
 From
 Is
a “process model” point-of-view:
there a psychologically plausible process model of causal
graphical model structure learning?
Stick-Ball machine
Kushnir, T., Gopnik, A., Schulz, L., & Danks, D. 2003. Inferring hidden causes. Proceedings of
the 25th Annual Meeting of the Cognitive Science Society.
Stick-Ball machine
Kushnir, T., Gopnik, A., Schulz, L., & Danks, D. 2003. Inferring hidden causes. Proceedings of
the 25th Annual Meeting of the Cognitive Science Society.
Stick-Ball machine
Kushnir, T., Gopnik, A., Schulz, L., & Danks, D. 2003. Inferring hidden causes. Proceedings of
the 25th Annual Meeting of the Cognitive Science Society.
Stick-Ball machine
Kushnir, T., Gopnik, A., Schulz, L., & Danks, D. 2003. Inferring hidden causes. Proceedings of
the 25th Annual Meeting of the Cognitive Science Society.
Stick-Ball machine
Kushnir, T., Gopnik, A., Schulz, L., & Danks, D. 2003. Inferring hidden causes. Proceedings of
the 25th Annual Meeting of the Cognitive Science Society.
Experimental conditions

Two conditions with “identical” statistics
 Intervention
case
A
& B move together four times
 Intervene on A twice, B doesn’t move
 Intervene on B twice, A doesn’t move
 Pointing
control
A
& B move together four times
 A moves twice (point at it after), B doesn’t move
 B moves twice (point at it after), A doesn’t move
Kushnir, T., Gopnik, A., Schulz, L., & Danks, D. 2003. Inferring hidden causes. Proceedings of
the 25th Annual Meeting of the Cognitive Science Society.
Experimental logic

For causal models (& close-to-determinism):
Observation Intervention


AB
Correlated
B moves after A
AB
Correlated
A moves after B
AUB
Correlated
Neither moves
Uncorrelated
Neither moves
U1A BU2
Intervention case
Pointing control
Kushnir, T., Gopnik, A., Schulz, L., & Danks, D. 2003. Inferring hidden causes. Proceedings of
the 25th Annual Meeting of the Cognitive Science Society.
Experimental logic

Non-CGM causal inference theories make no prediction
for this case, as there is no cause-effect division

And on plausible variants that do predict, they predict no
difference between the conditions
Inference from interventions

Response percentages in each condition:
A causes B
B causes A
Common cause
Separate mechanisms
Intervention
Case
0
0
67
33
Pointing
Control
0
4
17
79
p<.001: each condition is different from chance
p<.01: conditions are different from each other
Kushnir, T., Gopnik, A., Schulz, L., & Danks, D. 2003. Inferring hidden causes. Proceedings of
the 25th Annual Meeting of the Cognitive Science Society.
Other learning from interventions

Learning from interventions
 Gopnik,
et al. (2004); Griffiths, et al. (2004); Sobel, et
al. (2004); Steyvers, et al. (2003)
 And many more since 2005

Planning/predicting your own interventions
 Gopnik,
et al. (2004); Steyvers, et al. (2003);
Waldmann & Hagmayer (2005)
 And many more since 2005
Learning from correlations

Lots of evidence that people (and even rats!) can
extract causal structure from observed correlations
 And
those structures are well-modeled as causal
graphical models

⇒ Lots of empirical evidence that we act “as if” we
are learning (approx. rationally) causal DAGs
Developing a process model

Process of causal inference is under-studied
 To

date, very few systematic studies
Ex: Shanks (1995)
Positive contingent
High P(E) non-contingent
Low P(E) non-contingent
Negative contingent
50
Mean
judgment
0
-50
5
10
15
20
Trials
25
30
35
40
Developing a process model

Features of observed data
 Slow
convergence
 Pre-asymptotic “bump”

General considerations
 People
have memory/computation bounds
 Error-correction models (e.g., Rescorla-Wagner;
dynamic power PC) work well for simple cases
Bayesian structure learning

Three possible causal structures:
h+
B
h0
C
B
+ OR +
C
B
C
+ AND –
+
E

h–
E
E
Asymptotic prediction: Strength rating (wC) ∝
1
wC 
w Pw




h h , h0 , h 0
C
C
| h, D Ph | D dwC
Computed using Bayesian updating!
Bayesian dynamic learning

When presented with a sequence of data,
 After
each datapoint, update the structure and
parameter probability distributions (in the standard
Bayesian manner)
 Then use those posteriors as the prior distribution for the
next datapoint
 Repeat ad infinitum
Danks, D., Griffiths, T. L., & Tenenbaum, J. B. 2003. Dynamical causal learning. In Advances in
Neural Information Processing Systems 15.
Bayesian dynamic learning

Bayesian learning on the Shanks (1995) data
 Assume
effects rarely occur without the occurrence of
an observed cause
50
0
- 50
5
10
15
20
25
30
35
40
Danks, D., Griffiths, T. L., & Tenenbaum, J. B. 2003. Dynamical causal learning. In Advances in
Neural Information Processing Systems 15.
Side-by-side comparison
Shanks (1995):
Bayesian:
Positive contingent
High P(E) non-contingent
Low P(E) non-contingent
Negative contingent
50
50
Mean
judgment
0
0
-50
-50
5
10
15
20
Trials
25
30
35
40
5
10
15
20
25
30
35
Danks, D., Griffiths, T. L., & Tenenbaum, J. B. 2003. Dynamical causal learning. In Advances in
Neural Information Processing Systems 15.
40
Bayesian learning as process model

Challenges:
 All
of the terms in the Bayesian updating equation are
quite computationally intensive
 Number of hypotheses under consideration, and
information needs, grow exponentially with the number
of potential causes
 No clear way to incorporate inference to unobserved
causes
An alternate possibility

Constraint-based structure learning:
Given a set of independencies, determine the
causal Bayes nets that predict exactly those
statistical relationships
 Range
of algorithms for a range of assumptions
 Idea:
Use associationist models to make the necessary
independence judgments
Danks, D. 2004. Constraint-based human causal learning. In Proceedings of the 6th International
Conference on Cognitive Modeling (ICCM-2004).
An alternate possibility
Wellen, S., & Danks, D. (2012). Learning causal structure through local prediction-error learning.
In N. Miyake, D. Peebles, & R. P. Cooper (Eds.), Proceedings of the 34th annual conference of
the cognitive science society (pp. 2529-2534). Austin, TX: Cognitive Science Society.
Lingering problem

Pos connection for 1st 20 cases, then Neg connection
Lingering problem

Pos connection for 1st 20 cases, then Neg connection
Lingering problem

Pos connection for 1st 20 cases, then Neg connection
Causal inference summary

Very large literature over past 15 years showing
that our causal knowledge (from causal inference) is
structured like a causal DAG
 And
we learn (approx.) the right ones from data
 But we aren’t quite sure how we do it

And we do appropriate causal reasoning given that
causal knowledge
 As
long as we’re clear about what the knowledge is!
Causal perception


Paradigmatic case: “launching effect”
Similar perceptions/experiences for other causal
events (e.g., “exploding”, “dragging”, etc.)
 Including
social causal events (e.g., “fleeing”)
Causal perception

Driven by fine-grained spatiotemporal details,
including broader context
Causal perception vs. inference

Behavioral evidence that they are different
 Both

in responses & phenomenology
Neuroimaging evidence that they are different
 Different
brain regions “light up” in the different types
of experiments

Theoretical evidence that they are different
 “Best
models” of the output representations differ