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Can Causal Models Include Variables with Their TimeDerivatives?
Naftali Weinberger
Tilburg Center for Logic, Ethics and Philosophy of Science
Time and Causality in the Sciences
June 8th, 2017
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Principle of the Common Cause
(1) iPad
Happiness
(2)
iPad
Happiness
Income
(3)
iPad
Happiness
Common Restriction: Does not
(generally)
apply
to
logically,
mereologically, or conceptually related
variables
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Variable Distinctness
 Distinctness
still assumed in contemporary
approaches (Causal Markov Condition,
Modularity)
 Central
Question: Are variables and their timederivatives distinct in the relevant respect?
 General
project: How do we deal with variables
that are ‘about’ other variables
X and chance of X
 Dispositional properties and their effects
 X and is perceived as X

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Overview
1.
Review a debate from the philosophy of
physics on the causal role of
instantaneous velocity
2.
Explain why similar issues arise in the
graphical causal modeling framework
and show how they can be resolved
3.
Draw conclusions about the semantics of
derivatives in causal models
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Instantaneous Velocity
 Textbook
Definition:
V0
Xt
δx
δt
v0
v0
0
time
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Is Instantaneous Velocity a Cause
Reductionism: instantaneous velocity in
physics is nothing more than the timederivative of distance
Question: Can the the reductionist account
for the (alleged) causal role of velocity as the
cause of the future trajectory of the object
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Lange’s Argument
a0
v-1 v0
v0
Xt
0

If Reductionism, then instantaneous velocity at t would be
partly constituted by events in the future, and V0 would not be
a cause of all points in its future trajectory

What if v0 were just the ‘derivative from below’?

If velocity (and acceleration) were defined from below, then
it could not also serve as an effect, and there could be no
causal chains
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Lange’s proposal
 V0 is
not reducible to any fact about the
object’s trajectory. If an object’s velocity is
V=v at t0 and the object continues to exist
with its trajectory undisrupted, then the
derivative from above at t0 will equal v
 His
proposal delivers the causal ordering:
ForceAccelerationVX
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Easwaran’s Response
a0
v0
Xt
0
 There
is a coherent reductionist picture
 Acceleration
at t=0 is the derivative from
above of the derivative from below of
displacement
 Acceleration
is only an effect, but it is
constituted by something (velocity) that can
serve as a cause
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To the Causal Models!
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From Metaphysics to Methods
 What
problems arise in trying to represent
variables and their derivatives in causal
models?
 Given
a causal model in which XY, it is
possible to change the value of Y via an
intervention that sets the value of X
 These
models use variables to represent
discrete events
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Interventions on X
 If V
is defined at t, then interventions on X
must also be interventions on V (if V is to
remain well-defined)
Xt
t
time
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Interventions on V
Vt
V’t
Xt
t
time
 An
intervention on V changes the velocity
from above. Now X is continuous, but not V.
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Lessons
 One
cannot intervene on a lower-order
derivative if there is a well-defined higherorder derivative at that time
 Even
though one cannot directly intervene
on lower-order derivatives, they still play an
indispensible role as inputs for the future
trajectory of the object
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Representation
 When
we include a variable with its
derivatives, we need new notation to represent
their relationship
A
V
X
 Including
a variable and its derivative does not
lead to problems if:
 One
cannot intervene on lower-order derivatives
 All effects of the highest-order derivative go via
the lower-order derivatives
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Derivatives in Causal Models
Existing modeling methods explicitly satisfy the
proposed constraint
Voortman, Dash and Druzdzel 2008
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Why Use Derivatives in a Model?
 By
including a variable with its timederivative, we decompose the evolution
of an entity into two components:
The state of the object at a time based on the
accumulation of prior changes
 The rate of change of the object which combined with the object’s present state determines its next one

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Derivatives Without Continuity
Use (the representational device developed for)
derivatives for fine-grained time scales s.t.:
Rate at which system is sampled >> rate at
which we can intervene
When we don’t use derivatives, we assume that:
Rate at which system is sampled << rate at
which we can intervene
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Acknowledgments
This work was supported by the Deutsche
Forschungsgemeinschaft (DFG) as part of the
priority program "New Frameworks of
Rationality" (SPP 1516).
My thoughts on this topic have been influenced
by ongoing discussions with the following
people: Karen Zwier, Shannon Nolen, Jonathan
Livengood and Adam Edwards
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Conclusions
 Lange
and Easwaran are correct that there are
problems in treating velocity both as a cause
and an effect
 Modeling
solution: model the highest-order
derivative as an effect and the rest as causes
and introduce notation to denote the relevant
constraints
 This
modeling device can be applied to model
the complex temporal dynamics even of noncontinuous systems
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Here’s What I’m Reading. Join Me!
Iwasaki, Y., & Simon, H. A. (1994). Causality and model abstraction.
Dash, D., & Druzdzel, M. (2001). Caveats for causal reasoning with equilibrium models.
Hausman et al. (2013). Systems Without a Graphical Causal Representation.
Voortman, M., Dash, D., & Druzdzel, M. J. (2012). Learning why things change: the
difference-based causality learner.
Lange, Marc. "How can instantaneous velocity fulfill its causal role?.”
Easwaran, Kenny. "Why physics uses second derivatives.”
Hoover, Kevin, (2015). The Ontological Status of Shocks and Trends in Economics”
Ismael, J. (2016). How physics makes us free.
Myrvold, Wayne. “Steps on the Way to Equilibrium”
Norton, John. (2016). The Impossible Process: Thermodynamic Reversibility”
Frisch, Matthias (2016). Causal Reasoning in Physics