Solving the Flagpole Problem*
Abstract: In this paper I demonstrate that the causal structure of flagpole-like systems can be
determined by application of causal graph theory. Additional information about the ordering of
events in time or about how parameters of the systems of interest can be manipulated is not
needed.
1. Introduction
The length of a flagpole s shadow can be causally explained by reference to the solar altitude and the
flagpole s height, while neither the solar altitude nor the flagpole's height can be causally explained
by the remaining two parameters. This explanatory asymmetry is due to the fact that the solar
altitude a d the height of the flagpole are ausally rele a t to the le gth of the flagpole s shado
and not vice versa. But how can we know this? Since it was always a central concern of philosophers
to achieve the intended results without making more preconditions than absolutely necessary, this
question can be rephrased as follows: Is there a way to grasp the correct causal structure on the
basis of observation and without presupposing dispensable causal knowledge (this may be
knowledge that the cause occurs always earlier than its effect or knowledge about how to intervene
on specific parameters without directly manipulating some of the other parameters of the system
under investigation)? This yet unanswered and frequently discussed question (cf. Schurz 2001, for
instance) constitutes what I call the flagpole problem. In whichever way this problem might be
solved, it is a matter of fact that the strict correlations between the solar altitude, the height of the
flagpole, a d the le gth of the flagpole s shado alo e are ot suffi ie t for deter i i g the ausal
structure of the system – if any two of these three parameters are known, then the third one can be
computed, and thus there is no detectable asymmetry in terms of statistical regularities alone. In this
paper I will provide a new approach of how the flagpole problem can be solved: The underlying
causal structure (and hence, the causal asymmetry) of the flagpole scenario (and thus, the causal
structure of all flagpole-like systems) can be uniquely determined by application of causal graph
theory.
*
This is a draft paper. The final version of this paper is published under the following bibliographical data:
Gebharter, A. (2013). Solving the flagpole problem. Journal for General Philosophy of Science, 44(1), 63-67.
doi:10.1007/s10838-013-9208-6. The final publication is available at http://link.springer.com.
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The paper is structured in the following way: In section 2 I will introduce the basic terms and
principles of causal graph theory. In section 3 I will present a discovery algorithm for causal
structures whose correctness (within a causal graph framework) can be proven. In section 4 I will
demonstrate how the causal structure underlying the flagpole example can be uniquely determined
by application of this algorithm.
2. Causal graph theory
Causal graph theory, as it was developed by researchers like Judea Pearl (Pearl 2009; Pearl 1997),
Peter Spirtes, Clark Glymour, and Richard Scheines (Spirtes et al. 2000), is based on Hans
‘ei he
a h s idea of o
e ti g ausality
ith pro a ility ‘ei he
a h 19
; ‘ei he
a h 19 9 .
A causal graph G is an ordered pair V,E, where V is a non-empty and finite set of so-called vertices,
i.e., statistical variables representing events at a highly abstract level. Statistical variables shall be
referred to via upper-case letters X , Y , Z etc., while lower-case letters
,
, z etc. shall stand for
individual variables for specific values of X , Y , Z etc., respectively. Terms like X =
aria le X takes value
shall stand for
and are interpreted as (still very abstract) type-level events. E is an
asymmetric (and thus, irreflexive) relation among nodes in V (E V V . E(X,Y is a so-called
directed edge that can (and, in the following, will) be graphically represented via a causal arrow
( XY ). In causal graphs, terms like XY are i terpreted as X is a direct cause of Y . A chain of
causal unidirectional arrows from X to Y (e.g., XZ1Z2Z3Y) is called a directed path from X to Y
(XY). Given this characterization, XY turns out to be a special case of XY. Further causal
relations among variables in V a
e aptured ia the follo i g fa ily-terminology : Ele e ts of
the set of all variables Xi with XiY (Pa(Y)) are called parents of Y. Elements of the set of all variables
Xi with XiY (Anc(Y)) are called ancestors of Y. Elements of the set of all variables Yi with XYi
(Chi(X)) are called children of X. Elements of the set of all variables Yi with XYi (Des(X)) are called
descendants of X.
Whe it o es to ausal i fere e e ha e to disti guish et ee the true
ausal stru ture of the
world and a huge amount of man-made causal graphs intended to capture some interesting parts of
this highly complex causal structure. If there were a causal graph Gt = Vt,Et that would capture the
whole causal structure of the world, Vt would contain all in some way causally relevant variables and
thus, indeed, all variables that are common causes of any pair of variables in Vt. If we are looking at
any ordinary man-made causal graph G = V,E, o the other ha d, it is ot so lear that all true
common causes of pairs of variables in V are also included in V – a precondition needed for inferring
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causal structures in a more accurate way (see also section 4). It is because of this that we have to
distinguish between causally sufficient and insufficient variable sets V:
Causal Sufficiency: A variable set V is causally sufficient if a d o ly if all true
o
o
causes of all pairs of variables in V are also elements of V. (cf. Glymour et al. 1991)
Causal graphs can be connected to probability distributions via the following principle (cf. Spirtes et
al. 2000) going back to Reichenbach 1956:
Causal Markov Condition: A causal graph G = V,E and a probability distribution P over
V satisfy the Causal Markov Condition if and only if for every XV: X is probabilistically
independent of V \ Des(X) conditional on Pa(X).
Two variables (or sets of variables) X and Y are probabilistically independent conditional on a variable
(or a set of variables) M (X P Y|M) if and only if P(x|y,m) = P(x|m) for all values x, y and m of X, Y,
and M, provided P(y,m) > 0. In other words: Additional information about the actual value of Y would
not change the probability of any value x of X, given evidence M = m.
The Causal Markov Condition determines a set of conditional independencies which have to hold in
every probability distribution compatible with a given causal graph. In most cases, though, there is a
huge amount of causal graphs compatible with a given probability distribution. If, however, it can be
guaranteed that a given probability distribution captures all and only the conditional independencies
produced by the underlying causal structure, then the set of compatible causal graphs shrinks
drastically. A probability distribution that fulfills this condition is called faithful (cf. Spirtes et al.
2000):
Faithfulness Condition: A probability distribution P over a set of variables V is faithful if
and only if it captures all and only the (conditional) independencies produced by its
underlying causal structure.
3. The SGS algorithm
A multitude of more or less utile discovery algorithms for causal structures were developed over the
last decades (Pearl 2009; Spirtes et al. 2000; Glymour et al. 1991). One of the more intuitive of these
algorithms is the SGS algorithm, originally developed by Spirtes, Glymour, and Scheines (Spirtes et al.
2000, p. 82). Given a faithful probability distribution over a causally sufficient set of variables V, the
SGS algorithm uniquely determines the skeleton of the causal graph G = V,E (which correctly
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represe ts the true
ausal stru ture u derlyi g the orrelatio s a o g the aria les i V), and, in
addition, often uncovers a noteworthy amount of causal arrows between pairs of vertices of this
causal graph:
SGS algorithm:
Presupposition: A faithful probability distribution over a causally
sufficient variable set V.
Step 1: Form the complete undirected graph on the vertex set V.
Step 2: For all X,YV: If X and Y are probabilistically independent
conditional on any subset M of V \ {X,Y}, then remove the undirected
edge between X and Y in the graph resulting from step 1.
Step 3: For all X,Y,ZV: If X—Y—Z (or XY—Z) and X and Z are nonadjacent (not X—Z), then orient X—Y—Z (and accordingly XY—Z) as
XYZ in the graph resulting from step 2 if and only if X and Y are
probabilistically dependent conditional on every subset M of V \ {X,Y}
with YM.
Step 4: For all X,Y,ZV: If XY—Z and X and Z are non-adjacent, then
orient XY—Z as XYZ in the graph resulting from step 3. And: If
X—Y and XY, then orient X—Y as XY in the graph resulting from
step 3.
The correctness of the SGS algorithm for faithful probability distributions over causally sufficient
variable sets can be proven (Spirtes et al. 2000, p. 82). At this point we have all the formal tools to
dare tackle the flagpole problem.
4. Solving the flagpole problem
Remember the flagpole problem as introduced in section 1: The solar altitude (A = a) and the height
of the flagpole (H = h) are both causally relevant for the length of the flagpole s shado
L = l). But
how can we know this on the basis of observational evidence alone, i.e., without any additional
causal information?
The flagpole problem can be solved in the following way: If we take many random observations (we
look at many different flagpoles and not solely at a single one and take many different solar
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altitudes) and write down the observed values of A, H, and L over a sufficiently long period (e.g., a
year), we get a list of combinations of values of A, H, and L. By application of statistical methods, we
can get a faithful probability distribution P over V = {A,H,L} containing the following and only the
following conditional independence relation among variables in V: A P H| to get the right
dependence/independence relations in P it is important to measure A, H, and L at least
approximately at the same time).
That P will contain this and only this conditional independence relation – i.e., that P is faithful – can
be motivated via the following considerations: Specific solar altitudes are typically not accompanied
by specific heights of flagpoles and hence, there are no A-values a which have a probabilistic
influence on some H-values h. In addition, we can change H's value however we wish to without
changing the solar altitude. 1 Thus, the solar altitude A is without a doubt independent of the height
of the flagpole H (A P H|). But why are there no further independencies in the system under
investigation? If the sun is very low (A = low), then the probability for a longer flagpole shadow (L =
high) is increased – a certain range of L-values is probabilistically sensitive to a specific range of Avalues. “o the le gth of the flagpole s shado L is probabilistically dependent on the solar altitude A.
The same goes for the height of the flagpole H and the length of its shadow L: If the height of the
flagpole is very low (H = low), then the probability for a very short flagpole shadow (L = low) is
increased – a certain range of L-values is probabilistically sensitive to a specific range of H-values. So
the le gth of the flagpole s shado
L is probabilistically dependent on the height of the flagpole H.
The only remaining possible probabilistic independencies are those we could get among pairs of
variables in V by conditioning on the third variable in V. Since the value of any variable in V can be
computed given the actual values of the other two variables (see section 1), there are no such
probabilistic independencies within the flagpole example, and hence P is faithful.
It is now time to apply the SGS algorithm to our example: Let us therefore assume that the system V
= {A,H,L} is causally sufficient. Via applying step 1 to V and P we get the undirected graph consisting
of A—L—H and A—H; so every pair of vertices in V is connected via an undirected edge after step 1.
According to step 2, we have to remove the undirected edge between A and H due to the fact that
conditioning on the empty set makes A and H probabilistically independent. Since A and L are
probabilistically dependent conditional on H, L and H are probabilistically dependent conditional on
A, and A and L as well as L and H are probabilistically dependent conditional on the empty set, the
graph resulting from step 2 is A—L—H. With this intermediate result we have successfully revealed
1
Note: In section 1 I promised to solve the flagpole problem (i.e., to infer the causal structure of the flagpole
scenario on the basis of a given probability distribution) without reference to interventions. However,
interventions are not used in determining the causal structure of the flagpole scenario and thus, the use of
interventions in motivating the dependence/independence relations in P is harmless in regard to this promise.
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the skeleton of the much sought-after causal graph G = V,E that correctly represents the underlying
causal structure of the flagpole example. Since our graph A—L—H fulfills the antecedent-condition of
step 3 and since A and H become probabilistically dependent after conditioning on every subset M of
V \ {A,H} with LM (actually, there is only one such subset M, namely M = {L}), step 3 instructs us to
orient A—L—H as ALH. Since all undirected edges have already been removed/replaced by
causal arrows, step 4 can be skipped and the causal graph resulting from application of the SGS
algorithm turns out to be ALH.
5. Conclusion
In this paper I demonstrated an alternative solution to the flagpole problem. I gave a sketch of causal
graph theory, including its most important terms and principles. Afterwards I presented the SGS
algorithm which is custom-built to reveal causal structures on the basis of probability distributions
and whose correctness (within a causal graph framework) can be proven. I argued that there is one
and only one probabilistic independence within the flagpole example: The solar altitude A and the
height of the flagpole H are probabilistically independent (A P H|). A probability distribution (over
the system V = {A,H,L}), containing only the independence A P H|, is, according to the SGS
algorithm, compatible with only one causal graph: ALH. This causal graph mirrors our intuitions:
The solar altitude A and the height of a flagpole H are causally relevant for the length of the
flagpole s shado L (and not vice versa). This result can be generalized for any flagpole-like system V
(i.e., any causally sufficient system V containing exactly three variables X, Y, and Z with a faithful
probability distribution P over V with exactly one (conditional) independence relation X P Y|).
Acknowledgements: This work was supported by DFG, research unit Causation | Laws | Dispositions
| Explanation (FOR 1063). My thanks go to Stathis Psillos as well as to two anonymous referees for
their helpful comments and to my PhD advisor Gerhard Schurz for his great support. Thanks also to
Christian J. Feldbacher and Alexander G. Mirnig for constructive criticism of an earlier version of the
paper.
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6. References
Glymour, Clark, Spirtes, Peter, and Scheines, Richard. 1991. Causal Inference. Erkenntnis 35: 151-189.
Pearl, Judea. 2009. Causality. Cambridge: Cambridge University Press.
Pearl, Judea. 1997. Probabilistic Reasoning in Intelligent Systems. San Francisco: Morgan Kaufmann.
Reichenbach, Hans. 1956. The Direction of Time. Berkeley: University of California Press.
Reichenbach, Hans. 1949. The Theory of Probability. Berkeley: University of California Press.
Spirtes, Peter, Glymour, Clark, and Scheines, Richard. 2000. Causation, Prediction, and Search.
Cambridge: The MIT Press.
Schurz, Gerhard. 2001. Causal Asymmetries, Independent versus Dependent Variables, and the
Direction of Time. In Current Issues in Causation, ed. Wolfgang Spohn, Marion Ledwig, Michael
Esfeld, 47-67. Paderborn: Mentis.
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Alexander Gebharter