Quantum Informatics for Cognitive, Social, and Semantic Processes: Papers from the AAAI Fall Symposium (FS-10-08)
Judged Probability, Unpacking Effect and Quantum Formalism
Riccardo Franco
One could ask why the quantum formalism, invented for
the physics of microscopic world, should be used also in
cognitive science. As already stressed in (Busemeyer et al.
2010), we note that we are not claiming that the brain is
a quantum computer. We are only using the mathematical apparatus of quantum mechanics to define formal and
abstract rules in order to compute the judged probabilities
and their combinations. The interesting point is that not
only such rules are able to produce good quantitative predictions for the unpacking effect, but also the basic principles of the quantum formalism (like the non-commutativity,
the collapse of the state vector after a measurement) seem to
clearly capture important basic features of human reasoning.
The paper is organized as follows: in section 2, we present
the unpacking effect and the most important experimental
results, in section 3 we present the quantum-like formalism
and its main predictions about unpacking effect, and in section 4 the conclusions are drawn.
Abstract
In this article we describe a cognitive heuristic known as the
unpacking effect by using a mathematical model, based on
the quantum formalism, already introduced for the conjunction fallacy. We present the basic postulates of such quantumlike model and we show that the presence of interference
terms is responsible of the unpacking effect. In particular, the
sign of the interference and its functional form are able to describe the experimental results about subadditivity, superadditivity and additivity. A comparison with previous models is
presented, as well as new experimental predictions, allowing
to conclude that this new formalism and the basic concepts
of quantum information processing provide a new promising
way to describe and understand human judgement and categorization.
1
Introduction
Since the first work of (Kahnemann et al. 1982) about
heuristic and biases in human probability judgements, a
large amount of experimental data and of new effects has
been found. At the same time, it can be noted that for all
such effects there is not a unique and coherent theory, but
many different models, which often are able to produce only
qualitative predictions.
The quantum cognition is a new area of research which
aims to cover this gap. In fact, it attempts to describe cognitive heuristics and other cognitive effects by using concepts and mathematical methods taken from the quantum
formalism. Also known as quantum decision theory, such
research area is developing with some difficulties, because
of its wide range of studied topics and of variants in the formalism, often focalized on a single effect at once. For a
list of references in this field, see (Franco 2009) and (Busemeyer et al. 2010). An attempt to design a common and general quantum-like formalism to describe a number of cognitive effects has been recently presented in (Busemeyer et al.
2010). This paper is based on the same general formalism
presented in (Busemeyer et al. 2010), and it shows that such
formalism is able to produce good qualitative and quantitative agreement with experimental data about the unpacking
effect.
2
Unpacking effect
The principle of description invariance is a basic normative
requirement for likelihood judgment which states that the
judged probability of an event should not depend on how
the event is described. Empirically, however, alternative descriptions of the same event can produce systematically different judgments: for instance, in (Rottenstreich & Tversky,
1997) respondents were asked to judge the probability that a
randomly selected death was a murder and obtained higher
probabilities when this event was described as homicide by
an acquaintance or stranger than when it was merely described as homicide. When distinct exemplars of homicide
were made salient, judged probability increased. This effect
is called unpacking effect, and in general it can present in the
form of superadditivity or subadditivity: when an event is
unpacked into two or more exhaustive and exclusive events,
we have superadditivity/subadditivity if the subjects judge
the unpacked event less/more likely than the packed event
respectively.
The unpacking can be experimentally performed in two
distinct ways: given the unpacked event and two exclusive
and exhaustive constituent hypotheses, we have 1) an implicit unpacking when the two hypotheses are judged together as a disjunction and 2) an explicit unpacking when
each hypothesis is judged separately.
c 2010, Association for the Advancement of Artificial
Copyright Intelligence (www.aaai.org). All rights reserved.
56
Subadditivity
sum of the unpacked events. We can also express equation
(1) in terms of conditional probabilities
Subadditivity has been observed in numerous studies reviewed by (Tversky & Koehler 1994) in many different areas, like medicine and games, where hypotheses were explicitly unpacked.
In (Fox & Tversky 1998), results clearly evidence subadditivity relevant to an explicit unpacking: participants were
asked to estimate the probability that a particular team, division or conference will win the NBA playoffs. At the time of
the study, eight teams remained: the sum of these probabilities is close to 1 for the two conferences, nearly 1.5 for the
four divisions, and more than 2 for the eight teams.
The evidence of subadditivity for implicit unpacking is
less strong: the previously cited example of homicide
unpacked into acquaintance or stranger (Rottenstreich &
Tversky, 1997) shows implicit subadditivity, but a similar
example where homicide is unpacked into daytime homicide
or nighttime homicide does not show subadditivity. Similarly, in (Fox & Tversky 1998) the implicit unpacking evidenced subadditivity only in the case of Eastern Conference
teams. In (Sloman et al. 2004) it has been noted that for categories with high typicality the subadditivity is very weak,
sometimes being replaced by additivity.
p(B) = p(B|A)p(A) + p(B|¬A)p(¬A)
(2)
where p(A) is the judged probability that hypothesis A happens according to the initial description and p(B|A) is the
probability relevant to B, given the hypothesis A. Equation (2) offers a simple description of the experimental situations previously described. For example, if we have the
category mammals that can hold their breath for more than
two minutes, we have for the unpacked situation that p(A) is
the probability that a randomly selected mammal is a whale,
while p(B|A) is the probability that, given a whale, such
mammal can hold its breath for more than two minutes.
We recognize in formula (2) the Bayes’ formula: the unpacking effect can be considered as a violation of Bayes’
formula: in fact, p(B) denotes the judged probability of
the packed event B, that is p(packed), while the right side
of equation (2) the sum of the constituent events, that is
p(unpacked). Thus the intensity of the unpacking effect will
be defined as
Δ = p(packed) − p(unpacked)
(3)
Positive Δ means superadditivity, while negative Δ subadditivity. Moreover, we can distinguish between implicit unpacking (where the hypotheses are presented into a disjunction) and explicit unpacking (where the hypotheses are presented separately).
Thus we can categorize the unpacking situations in terms
of equation (2) according to the probabilities p(B|A) and
p(A). The first, p(B|A), is the judged probability of B given
A and it can be considered, from the point of view of categories, as the representativity (or typicality) of category B
into category A. A high value of p(B|A) means that situation B is judged very typical when A is given. The second
probability p(A) is the judged probability of A according to
the initial description, and it is the weight of hypothesis A.
If we examine the experiments previously cited at the light
of such categorization, we find the following results:
a1) When p(B|A) is high, we have subadditivity. In other
words, unpacking into typical instances leads to subadditivity.
a2) For p(B|A) very high we have additivity (the unpacking effect is negligible). In (Sloman et al. 2004) such a fact is
described by noting that very representative exemplars lead
to additivity, while typical but not representative exemplars
lead to subadditivity.
b) When p(B|A) is low (atypical instances), we have superadditivity.
c) The unpacking effect, in case of subadditivity, is
stronger for explicit unpacking than for implicit unpacking.
Superadditivity
Superadditivity has been widely reported in (Sloman et al.
2004), where they showed that implicitly unpacking a category into atypical exemplars generally yield strong superadditivity. For example, experiment 1 considers the probability
of persons death causes, packed or implicitly unpacked into
typical/atypical hypotheses: the median judged probability
in the packed group was .55, in the typical group it was .60,
while in the atypical group .40.
Superadditivity for explicitly unpacked categories has
been observed also by (Macchi et al. 1999), where the
judgements for a partition sum less than 1.
In summary, there are experimental evidences for both
subadditivity and superadditivity, depending respectively
on the typicality/atypicality of the category exemplars presented in the unpacked version.
We are now ready to consider a formal description of unpacking effect by noting that all the experimental setups can
be described in the following general way: let B the unpacked event, and A, ¬A two exclusive and exhaustive hypotheses: then B can be considered as the union of B ∧ A
and B ∧ ¬A. For example, in the experiment in (Rottenstreich & Tversky, 1997) about homicide by acquaintance
or stranger, B is the event be killed and A is the event meet
an acquaintance. In experiment 3 of (Sloman et al. 2004)
about mammals that can hold their breath for more than two
minutes, we can say that A is the whale is a mammal. In
any case, we can call A the focal hypothesis, while ¬A the
residual hypothesis. In terms of the joint probabilities, we
have
p(B) = p(A, B) + p(¬A, B) ,
(1)
3
Quantum description of unpacking effect
We now present a formal description of unpacking effect,
based on the quantum formalism. We generically define as
events the categories or events A and B used in the unpacking experiments and presented in the previous examples, like
a mammal can hold its breath for more than two minutes, be
which clearly expresses the principle of description invariance: the probability of the packed event B is equal to the
57
killed, whale is a mammal, or Chicago wins the NBA playoffs. The following postulates, derived from (Busemeyer et
al. 2010) and translated in terms of unpacking effect experiments, can be easily recognized as an adaptation of the basic postulates of orthodox quantum mechanics to cognitive
systems. In fact, our idea is to use only the mathematical
apparatus of quantum mechanics and the basic concepts of
measurements and quantum contextuality.
Postulate 1 The interpretation of the given information before performing judgements defines a cognitive state represented statistically by a vector |x that lies within a high
dimensional vector space (its dimensionality depends on the
number of features activated in the subject). We can define
for such space a basis, formed by a set of mutually orthogonal and unit length spanning vectors.
Psychologically, each basis vector represents a unique combination of features which can potentially be activated.
be the description of an invented person called Linda in the
conjunction fallacy experiment (Franco 2009), or the number of teams in NBA playoff). We make events A and B
in correspondence to non-commuting projectors PA and PB
respectively. The negation of A is in correspondence to the
projector P¬A , where PA + P¬A = I, and I is the identity
operator in the vector space defined by the context. Since the
order of evaluation of projectors is important, we make the
hypothesis that the focus event A is considered before the
unpacked event B. For example, in the experiment about
homicide by acquaintance, the focus A is meet an acquaintance, and B is be killed. Such hypothesis is consistent with
known experimental situations, even if in general one could
create ad hoc experiments where A is forced to be evaluated
after B, leading to different results according to our model.
We consider now the unpacked events: consistently
with the order defined, we write the probabilities relevant to the constituent parts as p(A, B) = |PB PA |x|2
and p(¬A, B) = |PB P¬A |x|2 . We can rewrite them
as p(A, B) = p(A)p(B|A) (analogously for ¬A), where
p(A) = |PA |x|2 and p(B|A) = |PB PA |2 . It is important to
stress that p(A, B) and p(¬A, B) are relevant to an explicit
unpacking, where some subjects judge the unpacking A and
B and other subjects the unpacking ¬A and B. According to
Bayes’ formula (2), the classic probability theory equals the
sum p(A, B)+p(¬A, B) to p(B), which we will associate to
the explicit unpacking. However, in the quantum formalism
the probability of the unpacked event B is p(B) = |PB |x|2 ,
which can be rewritten as |PB (PA + P¬A )|x|2 , and, using
the distributivity property, as |PB PA + PB P¬A |x|2 . Now,
expanding the square modulus, we have an equation different from formula (2)
Postulate 2 Each event, like for example A is represented
by a subspace of the vector space, and each subspace has a
projector PA that is used to evaluate the event.
Postulate 3 The judged probability relevant to an event A
equals to the squared length of the projection of the state
vector onto the subspace representing the event, that is
|PA |x|2 .
Postulate 4 When the event A is considered true, the original state vector |x changes to a new conditional state vector
PA |x
|PA |x| , which is the projection onto the subspace representing event A, but now normalized to have unit length.
This postulate has important implications. The simple act of
considering an event as true or false changes the initial state
into the new conditional state vector through the projection.
Postulate 5 If two events A and B correspond to two projectors PA and PB which can be written with a unique common basis, then such events are said to be compatible. This
means that the two projectors are mutually commuting (i.e.
PA PB = PB PA ). Viceversa, if two events must be evaluated using projectors relevant to two different bases, then
such events are said to be incompatible. Then the projectors
do not commutate, and the order of evaluation of the events
becomes important.
p(B) = p(A, B) + p(¬A, B) + I ,
(4)
where I is the interference term
I = 2cos(φ) p(A, B)p(¬A, B) .
(5)
The interference term I is the new part that modifies the classic Bayes’ formula, containing the quantum correction: the
factor cos(2φ) takes into account the relative phase of the
basis vectors for A and B, and it modulates the corrective
term, allowing for positive or negative interference. Positive
interference produces superadditivity, while negative interference produces subadditivity, and in case of strong negativity it can also lead to the conjunction fallacy (Franco
2009). Thus the presence of such interference term is a very
important fact, which is able to explain, together with the set
of postulates and its consequences, many cognitive heuristics (Busemeyer et al. 2010). In particular, such interference
term is able to explain with a single mechanism both superadditivity and subadditivity, differently from other explanations like support theory, as we will see.
In general, quantum theory does not give other interpretative rules about the sign of the interference term. However,
in the context of cognitive science, we can say that if the
transition from A to B is perceived as a decrease of typicality (since A is a typical exemplar), then the relative phase
factor cos(φ) is negative. On the contrary, if A is perceived
as an atypical exemplar of B, then the relative phase factor is
Compatibility requires using a higher dimensional space
to form all combinations of features, whereas incompatibility can make use of a lower dimensional representation by
changing perspectives. This intuition is in line with considerations about rational ignorance and quantum formalism
presented in (Franco 2007). Thus, incompatibility provides
an efficient and practical means for a cognitive system to
deal with all sorts and varieties of questions. We stress that
subjects can follow two different strategies to encode informations: one using low dimensional vector spaces, leading
to cognitive heuristics, connected with fast and intuitive reasoning, and another more rational, slow and correct, using
large vector spaces to encode separately different features.
At the light of the previous postulates, we can reanalyze
the unpacking experiments. First of all, we consider the initial informations encoded into vector |x (for example it can
58
positive. We can also say that the phase factor is a measure
of uncertainty increase/decrease, and it can be manipulated
with particular phase operators, well known in quantum information processing theory. These considerations allow to
obtain subadditivity/superadditivity in the correct cases, and
to compare directly the intensity of unpacking effect Δ defined in equation (3) with the interference term I of formula
(6). In particular, we consider the experimental facts a1),
a2), b) and c) described in the previous section.
a1) For typical instances, we have subadditivity, since the
interference term is negative. The intensity of unpacking
effect Δ assumes values from −0.1 to −0.24 in the experiment about NBA playoffs in (Fox & Tversky 1998), where
participants were asked to estimate the probability that a
particular team, division or conference will win (explicit
unpacking). It is important to note that the couples of
teams in the second column are related to the corresponding
division in the first column: the sum of the probabilities that
one of the two teams wins should be equal to the probability
that the corresponding division wins. Some of the median
judged probabilities for the target events are listed in the first
column (a particular division wins, unpacked description)
and in the second column (a particular team wins, explicitly
unpacked description), while in the last two columns the
values of Δ and I (the experimental and the predicted
unpacking effects respectively) are compared:
Winning division
Central 0.39
Atlantic 0.40
Pacific 0.25
Midwestern 0.40
Winning team
Chicago 0.40
Indiana 0.25
Orlando 0.35
NY
0.24
LA
0.05
Phoenix 0.30
SA
0.40
Huston 0.19
−Δ
-I
0.26
0.24
0.19
0.22
0.10
0.09
0.19
0.21
b) For atypical instances, we have superadditivity, since
the interference term is positive. Also in this case, we expect that strongly atypical exemplars reduce the superadditivity, as can be seen from figure (1). This prediction has
yet to be experimentally verified. However, in general it is
noted by (Sloman et al. 2004) that superadditivity effect is
greater in modulus than subadditivity effect. This can be explained by noting that for typical instances the conditional
probability is p(B|A) > 0.5 and in many cases it can be
near to 1, producing weak unpacking effect, as can be seen
in figure (1). On the contrary, for atypical instances we can
have p(B|A) < 0.5 (meaning anticorrelation) as well as
p(B|A) = 0.5 (meaning that we ignore if B is true or false,
given hypothesis A): in this last case, figure (1) depicts a
maximal unpacking effect.
c) In case of subadditivity, Δ is lower for implicit unpacking than for explicit unpacking. For example, in (Sloman et
al. 2004) are reported mean values of Δ for implicit unpacking with a maximum of −0.05. Coherently with quantum
principles, we expect that the order of evaluation is important, and we make the hypothesis that the focus event A is
considered first, followed by B. Thus the disjunction can be
considered as the sum of two processes: the first, where A
is judged true or false while B is true, and the second process, where the couples (A, B) and (¬A, B) are considered
separately. Thus the quantum-like description of the implicit
unpacking is the sum
u[p(A, B) + p(¬A, B) + I] + v[p(A, B) + p(A, ¬B)], (6)
with the two positive weights u + v = 1. In other words,
the implicit disjunction can be described as an incoherent
(from a quantum point of view) mixture of two distinct measurement processes modulated by weights u and v. This
quantum-like description of the disjunction allows to explain
the experimental fact c), since the effect of I is now modulated by the weights u, v, wakening the subadditivity effect
for implicit unpacking.
We can note in the previous table a good agreement of
the model predictions to the experimental observation (I is
very similar to Δ), with a phase factor suitable adjusted to
cos(φ) = −0.3. In particular, we can see that, when one of
the two unpacked situations has low probabilities, then the
unpacking effect has lower intensity, and that the interference terms behaves similarly. Moreover, the quantum-like
model entails the additional prediction that the unpacking
effect is weaker when the judged probability of A according
to the initial description is very high or very low. This can
be clearly seen from figure (1), where the maximum value
of interference is always obtained for p(A) = 0.5. This prediction has yet to be experimentally verified.
a2) Very representative exemplars lead to additivity. This
can be explained by noting that high typicality means that
p(A, B) high, and in particular p(B|A) near to 1. Figure
(1) clearly shows that the quantum-like model takes into account such experimental fact, where a value of p(B|A) =
0.99 produces a maximal interference value of about -0.05.
From a quantum point of view, this fact simply means that
projectors PA and PB define almost parallel rays in the
space vector and thus they commute, leading to additivity.
Existing models for unpacking
One of the first and most important models which attempt
to describe unpacking effect is support theory (Tversky
& Koehler 1994): according to such theory, the probability is not attached to events, but rather to descriptions of
events, called hypotheses. Support theory assumes that each
hypothesis B has a nonnegative support value s(B) corresponding to the strength of the evidence for this hypothesis. The judged probability p(B) that the focal hypothesis
B rather than ¬B holds, assuming that one and only one of
them obtains, is given by
p(B) =
s(B)
s(B) + s(¬B)
(7)
The theory assumes that unpacking the focal hypothesis B
into disjoint components B1 , B2 increases its support, and
the sum of the support of the hypotheses is at least as large
as the support of their disjunction
s(B) ≤ s(B1 ∨ B2 ) ≤ s(B1 ) + s(B2 ) ,
59
(8)
two:
1) unpacking a category into a small number of typical exemplars should generally yield additivity. The judged probability of the packed category should approximately equal
the judged probability of a typical unpacking because the
packed category is already assumed to be interpreted in
terms of typical exemplars
2) unpacking a category into atypical exemplars should generally yield superadditivity.
Such interpretation is quite coherent respect to the experimental data, but it does not produce quantitative predictions.
Finally we cite a formal model called vignette memory
model, designed to describe cognitive heuristics and in particular the conjunction fallacy. Surprisingly, some concepts
of such model are similar to those of quantum formalism, as
evidenced in (Busemeyer et al. 2010), even if there are no
specific predictions for the unpacking effect.
4
Conclusions
We conclude that the quantum-like description is able to
produce quantitative predictions coherent with experimental
data about unpacking effect. In particular, we stress that the
positivity/negativity of the interference term can take into
account superadditivity/subadditivity effects. Moreover, the
functional form of the interference term is compatible with
experimental observations.
We note that the presence of alternative theories which evidence some features in common with quantum formalism
is of course encouraging. However, such theories are often
ad-hoc models to explain particular effect, while the quantum formalism is a much more general theory, with a well
known internal formal consistence. Thus in our opinion is
preferable to design a theory making use only of standard
quantum formalism to describe cognitive heuristics.
This article is another step towards a consistent and coherent formulation of quantum cognition theory, based on the
general formalism defined in (Busemeyer et al. 2010) based
on projectors. We evidence that such formalism is rich and
consistent enough to produce a theory of probability judgements.
Figure 1: Interference term I for different values of p(A)
and different conditional probabilities p(B|A), leading to
subadditiviy and superaditivity.
Support theory thus takes into account the subadditivity
effects by imposing a particular behavior to the support function: in this respect, the support theory does not explain fully
the subadditivity. Moreover, we are interested in evaluating the difference between the judged packed and unpacked
probabilities, which we call the intensity of the unpacking
effect. The experimental data in (Fox & Tversky 1998) evidence that the minimum value of unpacking strength is in the
case of the Pacific division, where we have P (A, B) = 0.05
and P (¬A, B) = 0.30, while P (B) = 0.25, giving an unpacking intensity of Δ = −0.1. This seems to suggest that
the unpacking strength is low when at least one of the two
probabilities are low. The support theory does not give any
prediction about this suggestion. In other words, the theory does not predict in an exact way when the unpacking
strength is high or low, unless defining particular functional
forms for the support. Most importantly, support theory
is unable to explain superadditivity, as already stressed by
(Sloman et al. 2004). We finally note that in (Bordley 1998)
there is an attempt to show that in particular cases the functional form of support theory can be similar to the quantumlike predictions.
Another existing model for the unpacking effect is provided by the narrow interpretation conjecture (Sloman et
al. 2004), according to which categories are represented via
typical instances and unpacked instances serve as the focus
of judgment. The consequences of such assumptions are
References
Bordley R. F. (1998). Quantum Mechanical and Human Violations of Compound Probability Principles: Toward a Generalized
Heisenberg Uncertainty Principle. Operations Research, Vol. 46,
No. 6., pp. 923-926.
Busemeyer J. R., Photos E. M. and Franco R. (2010). A quantum
theoretical explanation for probability judgment errors. Submitted
to Psych. Rev. Lett. ;
http://mypage.iu.edu/jbusemey/quantum/PJudge6.pdf.
Fox, C. R., & Tversky, A. (1998). A belief-based account of decision under uncertainty. Management Science, 44, 879895.
Franco R. (2007) Quantum mechanics and rational ignorance,
arXiv:physics/0702163
Franco R. (2009). Conjunction fallacy and interference effects.
Journal of Mathematical Psychology, 53(5):415 – 422, 2009. Special Issue: Quantum Cognition.
Kahneman, D., Slovic, P., & Tversky, A., (1982). Judgment under
uncertainty: Heuristics and biases: Cambridge University Press.
60
Macchi, L. ,Osherson, D. & Krantz, D. H. (1999). A note on superadditive probability judgment. Psychological Review, 106, 210214.
Rottenstreich, Y., & Tversky, A. (1997). Unpacking, repacking,
and anchoring: advances in support theory. Psychological Review,
104, 406-415.
Sloman, S.A., Rottenstreich, Y., Wisniewski, E., Hadjichristidis,
C., & Fox, C. (2004). Typical versus atypical unpacking and superadditive probability judgment. Journal of Experimental Psychology: Learning, Memory, & Cognition, 30, 573- 582.
Tversky, A., & Koehler, D. J. (1994). Support theory: A nonextensional representation of subjective probability. Psychological Review, 101, 547567.
61