I explore ways to combine complex network science with the Rubin model of conference inference. In broad strokes, I discuss the difference between exogenous shocks and endogenous process, and how granularity in time can be used to tease causality out of a complex system.
2. Why causality?
Causal relations are scientifically interesting
because when exposed, they are:
● a reliable mechanism
● that supports intervention
By understanding causality, we can predict and
control. This is the interest behind causal
knowledge.
3. Why this talk?
● A “working group” talk - I don’t claim to be an
expert
● I’m working through research problems that
are challenging to me
● I’m presenting this work in progress both to
inform and solicit feedback
● You are welcome to do the same with your
work!
5. Potential outcomes
“The causal effect of a treatment on a single
individual or unit of observation is the
comparison (e.g., difference) between the
value of the outcome if the unit is treated
and the value of the outcome if the unit is not
treated.” (Angrist, Imbens, and Rubin, 1996)
6. Potential outcomes
“The causal effect of a treatment on a single
individual or unit of observation is the
comparison (e.g., difference) between the
value of the outcome if the unit is treated
and the value of the outcome if the unit is not
treated.” (Angrist, Imbens, and Rubin, 1996)
7. Potential outcomes
Average effect: (10+9+7+12)/4 = 9.5
Controlled Outcome - Yi
(0)
Treated Outcome - Yi
(1)
Causal Effect of
Treatment Yi
(1) - Y(0)
Alice 20 30 10
Bob 15 24 9
Cathy 10 17 7
David 22 34 12
8. Potential outcomes
You only ever see some of these. This has been called the
Fundamental Problem of Causal Inference
Controlled Outcome - Yi
(0)
Treated Outcome - Yi
(1)
Causal Effect of
Treatment Yi
(1) - Y(0)
Alice 20 30 10
Bob 15 24 9
Cathy 10 17 7
David 22 34 12
9. Potential outcomes
Stable Unit Treatment Value Assumption
"the [potential outcome] observation on one
unit should be unaffected by the particular
assignment of treatments to the other units"
10. Potential outcomes
Controlled Outcome - Yi
(0) Treated Outcome - Yi
(1) Causal Effect of Treatment
Yi
(1) - Y(0)
Alice 20 30 10
Bob, if
Alice in
not
treated
15 24 9
Bob, if
Alice is
treated
18 29 11
Cathy 10 17 7
David 22 34 12
11. Potential outcomes
So for every unit, we have to map out all the
variables that can have an effect on the
potential outcomes.
Spouse treated
Unit treated
Outcome
12. Potential outcomes
So for every unit, we have to map out all the
variables that can have an effect on the
potential outcomes.
A great tool for this: Pearl’s causal networks.
14. Note
There are differences between Pearl and
Rubin’s frameworks but their core concepts are
compatible,
so says Andrew Gelman, 2009:
http://andrewgelman.com/2009/07/07/more_on_pearls/
15. Causal networks
N.B.
An interesting thing about causal networks is
that the conditional probability distributions can
be arbitrarily complex.
Also, variables need not just be whole numbers
or scalars. They could be a matrix.
16. Complex networks
Now I’m going to talk about networks that are
not causal networks.
Sometimes in the literature these are called
complex networks.
20. Complex networks
There are lots of different kinds of networks
observed in nature and society.
They can differ substantially in their emergent
properties.
21. Complex networks
def: emergent property
“An emergent property is a property which a collection or
complex system has, but which the individual members do
not have. A failure to realize that a property is emergent,
or supervenient, leads to the fallacy of division.”
24. How to make a graph
Different processes for generating graphs have
result in graphs with different properties.
25. How to make a graph
● Erdős–Rényi (ER) model: G(n,p): Create n
nodes and create edges with probability p
● Barabási–Albert (BA) model:
○ Begin with a fully connected network of m0
nodes
○ Each new node is connected to m (< m0
) node
probability proportional to degree ki
of each node i
26. How to predict a graph
The distribution of degree of an Erdős–Rényi
graph is binomial.
The distribution of degree of a Barabási–
Albert (BA) graph is scale-free/power-law.
28. Bayes Theorem
Recall Bayes Theorem:
P(H|D) ∝ P(D|H) P(H)
If we can show a difference in the
likelihood
of data under different hypotheses, we can
learn something
32. What process created this graph?
● We can in principle statistically distinguish
hypotheses about generative processes
based on emergent properties.
● Is this a causal inference?
34. Ways to model this
All the logic of graph generation is buried in the
conditional probability function P(B|A).
A
Process
B
Graph
35. Ways to model this
All the logic of graph generation is buried in the
conditional probability function P(B|A).
“Logical causation” allows no intervention!
A
Process
B
Graph
36. Ways to model this
The Barabási–Albert (BA) model suggests this
interpretation of graph changing over time.
You can model time series data like this.
B0
B1
B2
B3
B4
37. Ways to model this
Now we can model a stable transition function
within the graph and external causes.
B0
B1
B2
B3
B4
C
38. Ways to model this
In other words,
endogenous process + exogenous shocks
B0
B1
B2
B3
B4
C
39. Tools in the toolbox
If we want to understand the effect of a kind of
exogenous shock,
and we know the endogenous process,
then we can look for natural experiments that
expose the treated outcome.
40. Tools in the toolbox
To know the endogenous process, then
compute the likelihood of emergent properties.
B0
B1
B2
B3
B4
EEE E E
41. Problems
● The vastness of the hypothesis space of
graph generation processes
● Related: how do you choose a prior?
42. Problems
● Computing the likelihood of a graph’s
emergent properties given a particular
generation process is
○ tricky math
○ maybe computationally hard
43. Dissatisfactions
● We have continued to bury much of the
mechanism of interest in the conditional
probability function.
● Suppose we want to cash this out as a finer-
grained mechanism that supports finer
interventions
● Can we think at multiple levels of abstraction
at once?