Work by Robert Spekkens and Chris Wood

arXiv:1208.4119

qplus.burgarth.de

presented to research group 6/13/2013 by Elie Wolfe

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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!

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Cause (influence) points to it’s effect.

A STRUCTURE is looser than a MODEL.

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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.

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If not independent

If correlated then must have common cause

(exception example – conservation of momentum)

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( ; ) 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)

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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.

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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

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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.

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|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.

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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!)

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(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

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A B

X Y

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A B

X Y

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A B

X Y

¸ Su

pe

rlu

min

al

Ca

usa

tio

n

A B

X Y

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A B

X Y

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A B

X Y

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Less

th

an

Fre

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A B

X Y

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A B

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A B

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(w/o

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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.

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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!)

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