View
142
Download
0
Category
Preview:
Citation preview
Work by Robert Spekkens and Chris Wood
arXiv:1208.4119
qplus.burgarth.de
presented to research group 6/13/2013 by Elie Wolfe
6/13/2013
arXiv:1208.4119
1
Algorithms exist to find all-possible causal structures for
a given probability distribution.
These algorithms are motivated by Machine Learning.
So, using formal methods, what are the causal structures
consistent with QM?
Spoiler: NONE!
6/13/2013
arXiv:1208.4119
2
Cause (influence) points to it’s effect.
A STRUCTURE is looser than a MODEL.
6/13/2013
arXiv:1208.4119
3
6/13/2013
arXiv:1208.4119
4
and independant
|
|
,
,
; 0
B b
A a
A B
P A B P A
P B A P B
P A B P A P B
H A B H A H B
A
I A
B
B
; ,
; | | | , |
1log
H | H , H H ;
x X
I A B H A H B H A B
I A B C H A C H B C H A B C
H X P xP x
X Y X Y Y X I X Y
and conditionally independant given
| , |
| , |
, | | |
, | | |
|
; | 0
B b
A a
A B C
P A B C P A C
P B A C P B C
P A B C P A C P B C
H A B C H A C H B
B
C
A
I A B C
C
1. No cycles. Influence cannot self-loop.
2. Reichenbach’s Common Cause Principle: If A and B
are not statistically independent then they must share
a common cause. In other words, A influences B, B
influences A, or C influences both A and B. In terms of
a graph: A must be connected to B by a causal path.
3. No fine tuning. If A and B are statistically
independent then they should not be causally
connected.
Additionally, every structure has “Instrumental
Inequalities” that restrict beyond just independence.
6/13/2013
arXiv:1208.4119
5
If not independent
If correlated then must have common cause
(exception example – conservation of momentum)
6/13/2013
arXiv:1208.4119
6
( ; ) 0I A B
C
A
B
A
B
B
A
OR OR
B
A
=
Called “Faithful” or “Stable” in the literature.
If have common cause then must be correlated
Because if B influences A then it should show up in the data.
Implausible that complex structure should be mistaken for
simple. Degrees of freedom argument!
(exception: binary one-time-pad)
6/13/2013
arXiv:1208.4119
7
A C AC BA B C AC ABC
AB A B
BC B C
A B
Such correlations can be found, but they require
interdependance of the conditional probabilities. The number
of C.P. labels is greater than the number of free variables
allowed by the C.I. data.
6/13/2013
arXiv:1208.4119
8
A B
6 free parameters
A B C AC BC ABC
AB A B
5 free parameters
A B C AC ABC
AB A B
BC B C
+4
+1
+1 +4
+1
+2
6/13/2013
arXiv:1208.4119
9
If the only way B experiences the influence of A is through some
intermediate variable(s) C then A and B are conditionally independent
given C.
implies
Markov condition: Causal structures imply that every variable X is conditionally independent of its nondescendants given its parents,
,
|
A C B C
A B C
X1
X4 X3
X5
X2 Nondescendants | ParentsX X X
Clearly structure implies CI, but no-fine-tuning means CI
implies structure as well. If an effect has multiple influences,
then information about the effect presumably adds
nontrivially to the to the mutual information of the influences.
RCCP: If A and B are correlated then A and B should
have a common cause.
NFT1: If A and B are independent then A and B
should not be modeled with a common cause.
NFT2: If A and B are independent given C then,
treating all ancestors of C as correlated, A and B should not
have a common cause path which bypasses C.
6/13/2013
arXiv:1208.4119
10
|S C T
Input: (S⊥C|T) ie. S & C must have common cause , but all causal paths
must not bypass T, including all of T’s ancestors, which we take to be all
correlated when T is given.
6/13/2013
arXiv:1208.4119
11
6/13/2013
arXiv:1208.4119
12
Input:
Output:
Input: Actually strength of the QM correlations
Output: NULL SET
The difference is that CI alone is incomplete input. (A candidate structure
implies Instrumental Inequalities. The instrumental inequalities of the
candidate structure above are none other than the Bell inequalities!)
6/13/2013
arXiv:1208.4119
13
(X ? Y ); (A ? Y jX); (B ? XjY )
A B
X Y ¸
“The instrumental inequality can, in a sense, be viewed as a generalization of Bell's inequality for cases where direct causal connection is permitted to operate between the correlated observables, X and Y:” -J. Pearl. Causality: Models, Reasoning, and Inference. Sec. 8.4.
X
A
Y
B
6/13/2013
arXiv:1208.4119
14
A B
X Y
¸
A B
X Y
¸
A B
X Y
¸ Su
pe
rlu
min
al
Ca
usa
tio
n
A B
X Y
¸
A B
X Y
¸
A B
X Y
¸ ¹
Less
th
an
Fre
e W
ill
A B
X Y
¸
A B
X Y
¸
A B
X Y
¸ ¹ Re
tro
ca
usa
tio
n
(w/o
c
yc
les)
Taking all 3 lemmas together cannot reproduce Bell-
inequality violating statistics, and yet they are real…
So what premise should we modify? Major open
question in Quantum Foundations.
6/13/2013
arXiv:1208.4119
15
1. Directed Acyclic Graphs labeled
with conditional probabilities
represent all causal models.
2. Reichenbach’s principle:
correlations must be explained by a
causal model
3. No fine-tuning of the causal model
• Can we reasonably question Reichenbach’s principle?
• Are there types of fine-tuning that aren’t so objectionable?
• Should we allow cycles in our causal models?
• Must we use conditional probabilities? (conditional density operator!)
The field of Causal Discovery Algorithms is hard pressed to move beyond only conditional independence analysis. True candidate testing, however, requires computation of the Instrumental Inequalities, which is NP hard, not unlike enumerating Bell Inequalities. Quantum Foundations has developed tools for assessing the possibility of local explanations of correlations. The techniques of QM are presumably useful then for developing Causal Discovery Algorithms!
This has applications to machine learning, medicine, genetics, economics…
New! Journal of Causal Inference. (How new? Volume 1, Issue 1 was first published 6/4/2013, merely 9 days ago!)
6/13/2013
arXiv:1208.4119
16
Recommended