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MEMORY AND PROBABILITY
Pedro Bordalo John J. Conlon Nicola Gennaioli
Spencer Y. Kwon Andrei Shleifer
1
October 20, 2021
Abstract. People often estimate probabilities, such as the likelihood that an insurable risk will
materialize or that an Irish person has red hair, by retrieving experiences from memory. We
present a model of this process based on two established regularities of selective recall: similarity
and interference. The model accounts for and reconciles a variety of conflicting empirical
findings, such as overestimation of unlikely events when these are cued vs. neglect of non-cued
ones, the availability heuristic, the representativeness heuristic, conjunction and disjunction
fallacies, as well as over vs. underreaction to information in different situations. The model
makes new predictions on how the content of a hypothesis (not just its objective probability)
affects probability assessments by shaping ease of recall. We experimentally evaluate these
predictions and find strong experimental support.
1
Saïd Business School, University of Oxford, Harvard University, Bocconi University and IGIER, Harvard
University, and Harvard University. We are grateful to Ben Enke, Drew Fudenberg, Sam Gershman, Thomas
Graeber, Cary Frydman, Lawrence Jin, Yueran Ma, Fabio Maccheroni, Sendhil Mullainathan, Salvo Nunnari, Dev
Patel, Kunal Sangani, Jesse Shapiro, Josh Schwartzstein, Adi Sunderam, and Michael Woodford for helpful
comments. Julien Manili provided outstanding research assistance.
1. Introduction
It is well known that memory plays an important role in belief formation. Tversky and
Kahneman (1973) show that when instances of a probabilistic hypothesis are easier to recall, the
hypothesis is judged to be more likely, a finding they call the availability heuristic. When
prompted to think about an unlikely event, such as dying in a tornado, people overestimate its
frequency (Lichtenstein et al. 1978). They also attach a higher probability to an event if its
description is broken down into constituent parts, which facilitates retrieval of instances
(Fischoff et al. 1978 ). More broadly, beliefs depend on recalled personal experiences, such as
stock market crashes (Malmendier and Nagel 2011), and not just on statistical information.
It is also well known that beliefs depart from rationality in a variety of ways, which shape
economic behavior. Sometimes unlikely events are overestimated, as when consumers overpay
for insurance (Sydnor 2010; Barseghyan et al. 2013) or bet in long-shot lotteries (Chiappori et al.
2019). Other times, unlikely events are underestimated, as when investors neglect tail downside
risks (Gennaioli et al. 2012). Adding to the clutter, in finance there is extensive evidence of both
over and underreaction to news. Beliefs overestimate the future prospects of individual firms and
the aggregate market after periods of rapid earnings growth (Bordalo et al. 2019, 202 0 ), leading
to long-run return reversals, but underestimate the impact of other news, such as earnings
surprises, leading to price drifts and momentum (Chan et al. 1996 ; Bouchaud et al. 2019 ; Kwon
and Tang 2021). This bewildering diversity of biases has puzzled many economists and led
sceptics to minimize the direct evidence on beliefs and embrace rationality.
In this paper, we present a model of memory and belief formation and show that it helps
reconcile seemingly contradictory biases and generates new predictions. We test these
predictions in two new experiments and find empirical support.
This mechanism naturally accounts for biases typically explained with the availability
heuristic (Tversky and Kahneman 1973). For instance, we tend to overestimate cued unlikely
events both because the cue 𝐻 !
= “cause of death is flood” prompts selective recall of floods, as
well as because the residual hypothesis 𝐻 "
= “other causes of death” consists of a range of
dissimilar events which are therefore difficult to retrieve on the basis of that cue, allowing for
more interference. The model also explains why we neglect unlikely events when these are not
cued. When thinking about 𝐻 "
= “other causes of death” we do not think about the rare
“botulism”, because it is too dissimilar from typical causes of death. The mechanism also offers
a rationale for partition dependence: the probability estimate of the residual hypothesis increases
if the latter is formulated as 𝐻 "
$
= “cause of death is cancer, heart attack, or any other causes of
death” (Tversky and Koehler 1994). Intuitively, the finer partition singles out subgroups whose
elements are similar to each other but dissimilar to other experiences, which boosts the recall of
the residual hypothesis by reducing interference.
The same ideas illuminate biases in conditional probability assessments, many of which
are attributed to the representativeness heuristic (Kahneman and Tversky 1973). Consider base-
rate neglect. Given the data that a person is “shy and withdrawn”, it may be hard to think about
the hypothesis that he is 𝐻
!
= “a farmer”, because many farmers are not shy. Non-shy farmers
are irrelevant given the data “shy and withdrawn”, but they interfere with the recall of shy
farmers, reducing the assessment of that hypothesis. We show that this same mechanism can also
account for the conjunction fallacy (Tversky and Kahneman 1974, 1983).
Finally, these forces unify instances of over and underreaction to data in conditional
assessments. Overreaction takes the form of base rate neglect when the data is informative about
an unlikely cued hypothesis. Underreaction takes the form of aversion to extreme beliefs (Griffin
and Tversky 1992) when the data is informative about a very likely hypothesis, because the
unlikely yet cued alternative is relatively oversampled. When the hypothesis is moderately likely,
the model predicts overreaction if the data is quite informative about this hypothesis and
underreaction otherwise. This may help explain why overreaction in financial markets is
observed after informative histories such as rapid earnings growth but not when signals are weak,
such as a positive earnings surprise.
To test the predictions of our model, we introduce a novel experimental design in which
participants see 40 images that differ in content and in some cases also in color. Subjects then
assess the probability that a randomly selected image possesses a certain property. To do so, they
only need to recall what they saw. We manipulate the subjects’ database of experiences and the
cues they face when assessing a hypothesis. We also measure the recall of experiences. We find
support for our predictions that over and underestimation of unlikely events can be switched on
and off by modulating similarity and interference. We also generate over and underreaction to
data by varying the strength of the signal and the likelihood of the hypothesis. Across all
treatments, recall of experiences and probability judgments are strongly correlated.
Recent work in economics explores the role of memory in belief formation (Mullainathan
2002 ; Bordalo et al 2020; Wachter and Kahana 20 19 ; Enke et al. 2020). A few papers study this
phenomenon from the perspective of efficient information processing (Tenenbaum and Griffiths
2001; Dasgupta et al. 2020; Azeredo da Silveira et al. 2020; Dasgupta and Gershman 2021).
Compared to this work, we start with well-documented regularities in recall and interference, and
show that they unify the representativeness and availability heuristics (Tversky and Kahneman
1974): due to similarity and interference, representative experiences are more “available,” or
accessible, for recall. This approach micro-founds and generalizes previous formalizations of
In our running example, we consider a database of potential causes of death. Here a
subset of features captures different potential causes: 𝑓
!
may identify “car accident”, 𝑓
"
“flood”,
%
“heart attack”, etc. One feature, which we denote by 𝑓
&
, indicates whether the event was lethal
or not. There are superordinate features, such as 𝑓
&'!
=“disease”, 𝑓
&'"
=“natural disaster”, etc,
which take the value of 1 for the relevant subsets of possible death events. Experiences are
vectors of features. For instance, lethal heart attacks have 𝑓
!
"
%
&
&'!
= 1 and
&'"
= 0. Non-lethal heart attacks have the same feature values except for 𝑓
&
= 0. Additional
features may include the characteristics of people involved, such as their age or gender, or
contextual factors such as the time and emotion associated with the experience. The set of
features is sufficiently large that no two experiences are exactly identical.
We focus on the case in which the experiences in the database reflect the objective
frequency of events (that of different causes of death in our example). In principle, the database
could be person-specific (e.g., people from New York may hear of fewer experiences of death
from tornado than people from Des Moines), and could also be affected by repetition, rehearsal,
and prominence of events (e.g., people may hear of more experiences of airplane crashes than of
diabetes due to greater news coverage of the former). The database could also be influenced by
selective attention. A past smoker concerned with lung cancer could encode many events of this
disease (Schwartzstein 2014). We leave such extensions to future work.
The DM forms beliefs about the relative frequency of two disjoint hypotheses 𝐻
!
and 𝐻
"
which are subsets of the database 𝐸. For instance, the DM may assess the frequency of death by
!
= “natural disaster” vs. 𝐻
"
= “all other causes”. These hypotheses partition the subset of
causes of death, identified by 𝑓
&
= 1 , on the basis of the “natural disaster” feature 𝑓
&'"
= 1 vs.
&'"
= 0. In the language of probability, the union of the hypotheses 𝐻 = 𝐻
!
"
is the sample
space over which the DM forms his subjective beliefs.
2
Appendix A3 extends the model to more
than two hypotheses. We refer to the sample space as the “relevant data” for the probabilistic
assessment. We refer to 𝐻
= 𝐸\𝐻, the set of experiences inconsistent with either hypothesis, as
the “irrelevant data”. Figure 1 depicts this decomposition of the memory database 𝐸.
Figure 1: Memory Database and Sample Space
As we describe in detail below, the DM makes his assessment by extracting a sample
from his database with replacement.
3
In a model in which sampling is only based on objective
frequency, the DM draws any given experience in 𝐸 with probability 1 /𝑁. He then estimates the
probability of 𝐻
as the relative frequency of this hypothesis among all draws belonging to the
relevant data 𝐻. In this model, due to the random nature of recall, the DM’s beliefs would be
noisy but average probabilistic assessments would be unbiased.
2
In a slight abuse of notation, we refer to 𝐻
!
both as a given hypothesis, e.g. “cause of death is flood”, and the
subset of experiences in 𝐸 consistent with hypothesis 𝐻 !
.
3
Sampling with replacement has two interpretations. The first is that the sample size is small relative to 𝑁. The
second is that repeated recall of certain events makes them more prominent in mind, affecting beliefs. This is
consistent with the fact that unique experienced events such as a stock market crash appear to persistently affect
beliefs (Malmendier and Nagel 2011). Sanborn and Chater (2016) allow for more structured Bayesian approaches to
frequency-based sampling, such as Markov Chain Monte Carlo. These, however, account neither for the role of
similarity nor for systematic violations of consistency such as disjunction and conjunction fallacies.
From Equation (1) it is automatic to define 𝑆
, the similarity between a single
experience 𝑒 and hypothesis 𝐻
. This is an important object, which we use to formalize how cued
recall affects memory sampling.
Assumption 1. Cued Recall : When cued with hypothesis 𝐻
, the probability 𝑟(𝑒, 𝐻
) that the
DM recalls experience 𝑒 is proportional to the similarity between 𝑒 and 𝐻
. That is,
+∈-
If similarity is constant, sampling is frequency-based—i.e., 𝑟(𝑒, 𝐻
) = 1 /𝑁. Compared
to this benchmark, the numerator of (2) captures the idea that sampling is shaped by similarity to
the cue 𝐻
. When thinking about deaths from 𝐻
= “natural disasters”, it is relatively easy to
recall deaths from floods, due to similarity. The denominator in (2) captures interference: all
experiences 𝑢 ∈ 𝐸 compete for retrieval, so they inhibit each other. Retrieval of 𝑒 is especially
inhibited by experiences that are similar to the cue. When thinking about 𝐻
= “natural disasters”,
deaths from tornadoes interfere with recall of deaths from floods, because the former are also
similar to 𝐻
Crucially, in the denominator of (2) interference also comes from experiences that are
inconsistent with the hypothesis 𝑢 ∉ 𝐻
. First, there is what we call “interference from the
alternative hypothesis”: when thinking about deaths from 𝐻
= “natural disaster”, the mind may
retrieve experiences of deaths from other causes such as “terrorist attacks” that belong to 𝐻 .
“other causes of death”. Second, there is what we call “interference from irrelevant data”: when
thinking about deaths from 𝐻
= “natural disasters”, the mind may retrieve instances of survival
in natural disasters. The latter experiences are inconsistent with either hypothesis and therefore
belong to the irrelevant data 𝐻 ≡ 𝐸\𝐻.
Similarity-based interference reflects the fact that we cannot fully control what we recall.
5
This is well-established in memory research going back to the early 20th century (Jenkins and
Dallenbach 1924; Keppel 1968; McGeoch 1932; Underwood 1957; Whitely 1927). For example,
recall from a target list of words suffers intrusions from other lists studied at the same time,
particularly for words that are semantically related to the target list, resulting in lower likelihood
of retrieval and longer response times (Shiffrin 1970; Lohnas et al. 2015). In the “fan effect”,
Anderson and Reder (19 99 ) show that concepts associated with more items are more difficult to
remember in response to any specific cue.
6
Our application of interference to probability estimates is new. We show that interference
from the alternative hypothesis and from irrelevant data have distinct implications, which
produce biases linked to the availability and representativeness heuristics.
For a given recall probability function 𝑟(𝑒, 𝐻
), we assume probability judgments are
formed according to the following two-stage sampling process:
Assumption 2. Sampling, Interference and Counting
Stage 1: [“Train of thought for 𝐻
”] Each hypothesis 𝐻
cues sampling of 𝑇 ≥ 1 experiences
from 𝐸 according to 𝑟(𝑒
/
). Denote by 𝑅
the number of successful recalls of all 𝑒 ∈ 𝐻
Stage 2: [Renormalization] The subjective probability of 𝐻
, denoted 𝜋N
, is the share of
successful counts for 𝐻
out of all successful counts for both hypotheses:
𝜋N(𝐻
. 0 ".
5
In our examples, interference comes from recalling a particular experience, but we do not claim that this process is
conscious. Recall failures may manifest as “mental blanks”, inability to recall anything when thinking about 𝐻 !
, or
as “intrusions”, namely recall of hypothesis-inconsistent experiences 𝑢 ∉ 𝐻 !
.
6
Two approaches have been proposed to account for this evidence: associative models (Anderson and Reder 1999)
and inhibition models (Anderson and Spellman 1995). These models do not incorporate interference from irrelevant
data. A robust phenomenon related to intrusions is that of “false memories” (Deese 1959; Roegiger and McDermott
1995). For example, recall from a list of words that are semantically related suffers intrusions from similar words
not on the list, e.g. mis-remembering milk from a list that includes butter, cheese, and white (Brown et al 2000).
We examine the two stages of Assumption 2 in turn. We first derive the probability of
recalling any experience 𝑒 ∈ 𝐻
when the DM thinks about 𝐻
. This is what psychologists call
retrieval fluency of hypothesis 𝐻
. Using Equation ( 2 ), we can show that this is given by:
) = B 𝑟(𝑒, 𝐻
1 ∈ 2 )
1 ∈ 2
)
+∈ 2 )
+∈ 2
4
+∈ 2
𝑆Q𝐻
.
R
⋅ 𝜋Q𝐻
.
R +
Q
R
Q
R
In Equation (4), 𝜋
/|𝐻| is the true relative frequency of 𝐻
in the relevant data 𝐻, i.e.,
the correct probability.
8
𝜋(𝐻) = |𝐻|/|𝐸| and 𝜋(𝐻
|/|𝐸| are respectively the frequency of
relevant and irrelevant data in the memory database 𝐸, with 𝜋Q𝐻R/𝜋(𝐻) denoting the relative
frequency of irrelevant data, or experiences 𝑒 ∉ 𝐻, to relevant data.
9
The term 𝑆
is the similarity of 𝐻
to itself. We call this the “self-similarity” of 𝐻
It captures the homogeneity of this hypothesis, namely the extent to which its experiences share
similar features. A tornado in Tulsa is fairly similar to a tornado in Little Rock, but neither is as
similar to an earthquake in California, which reduces the self-similarity of 𝐻 !
= “natural
disaster”. The “cross-similarity” terms 𝑆Q𝐻
.
R and 𝑆Q𝐻
, 𝐻R capture the similarities of 𝐻
to
the alternative hypothesis 𝐻
.
and to irrelevant data 𝐻, respectively. A death from flood in 𝐻
!
“natural disaster” is similar to a death from accidental drowning in 𝐻
"
, which raises 𝑆(𝐻
!
"
8
More precisely, 𝜋(𝐻
!
) is the probability of 𝐻
!
conditional on the relevant data 𝐻. To ease notation, we do not refer
to 𝜋(𝐻 !
) as 𝜋(𝐻
!
|𝐻), until we later study conditional beliefs in which the relevant data 𝐻 is restricted to a subset 𝐷.
9
Equation (4) highlights the key role of similarity in generating biased beliefs. If sampling is not distorted by
similarity, the DM either samples all data (which occurs when 𝑆:𝐻 !
, 𝐻
(
; = 𝑆(𝐻
!
, 𝐻
!
) = 𝑆:𝐻
!
, 𝐻;), or all relevant
data (which occurs when 𝑆:𝐻 !
, 𝐻
(
; = 𝑆(𝐻
!
, 𝐻
!
), 𝑆:𝐻
!
, 𝐻; = 0 ), with equal probability regardless of the cue. In
both cases, the expression in Equation (4) becomes proportional to 𝜋(𝐻 !
), which implies that beliefs are unbiased.
and it is also similar to the event of surviving a flood in 𝐻, which raises 𝑆Q𝐻
!
, 𝐻R. In Equation
( 4 ), such cross-similarity terms interfere, reducing retrieval fluency of 𝐻
Define 𝜔
as the ratio between retrieval fluency 𝑟
and the frequency-based
probability of drawing a member of the same hypothesis 𝐻
, i.e. 𝜔
| 2 )
|
6
Proposition 1 If 𝑆(𝐻
) = 𝑆Q𝐻
.
R = 𝑆Q𝐻
, 𝐻R , sampling is frequency based ( 𝜔(𝐻
) = 1 ). If
instead 𝑆
> maxY𝑆Q𝐻
.
R, 𝑆Q𝐻
, 𝐻RZ there is oversampling of 𝐻
given the cue 𝐻
) > 1 ). The extent of oversampling 𝜔(𝐻
) falls with:
1. the true frequency 𝜋(𝐻
) of 𝐻
- the strength of interference from 𝐻
.
and 𝐻 , measured by
782
)
, 2
:
7 ( 2
)
, 2
)
)
and
782
)
, 2 :
7 ( 2
)
, 2
)
)
respectively.
We provide the proof of Proposition 1 and all other proofs in Appendix A. As in
Equation (2), if similarity is constant, our model yields frequency-based sampling. If instead a
hypothesis is more similar to itself than to the rest of the database, which is often a valid
condition,
10
there is oversampling. When cued by 𝐻
= “natural disasters”, people scan their
memories for earthquakes, floods, etc., and so oversample these events relative to their true
likelihood. If all hypotheses being considered have high self-similarity, they are all
oversampled.
11
Henceforth, we assume 𝑆
> max[𝑆Q𝐻
.
R, 𝑆
.
Importantly, Property 1 says that oversampling is especially severe when thinking about
objectively unlikely hypotheses. People rarely experience floods and earthquakes compared to
heart attacks or accidents, so cueing 𝐻 !
= “natural disasters” greatly boosts their retrieval.
10
This condition can be violated if 𝐻
has two opposite clusters and 𝐻
,
is in the middle. Consider a database with
two generic features, and suppose that the DM assesses hypotheses 𝐻
≡ {( 1 , 0 ), ( 0 , 1 )} and 𝐻
,
≡ {( 1 , 1 )}. Here
members of 𝐻
disagree along all features, while 𝐻
,
agrees with one of them, so 𝑆(𝐻
, 𝐻
) < 𝑆(𝐻
, 𝐻
,
).
11
When varying the similarities 𝑆:𝐻
!
, 𝐻
(
;, 𝑆
( 𝐻
!
, 𝐻
!
) , and 𝑆
( 𝐻
!
, 𝐻
@ ) , we hold the true frequencies 𝜋
( 𝐻
!
) constant.
Equation (6) shows that the retrieval fluency of different hypotheses also shapes the
variability of beliefs. We defer the analysis of this aspect to Section 5. Here it suffices to say that,
in general, when two hypotheses are easy to recall—i.e., when both 𝑟
!
and 𝑟
"
are high—
the variability of beliefs declines, because the DM benefits from a larger sample size. In Section
5 we test this and other predictions about noise in probabilistic assessments.
3. Judgment Biases
We next examine probabilistic assessments. Section 3.1 deals with interference from the
alternative hypothesis, which yields several biases related to the availability heuristic. Section
3.2 incorporates interference from irrelevant data, and shows that it accounts for biases related to
the representativeness heuristic. Section 3.3 shows that these two forces can unify over and
underreaction of beliefs to data.
3.1 Biases due to Interference from the Alternative Hypothesis
To study interference from the alternative hypothesis, we focus on the case in which the
database 𝐸 coincides with the relevant data for assessing 𝐻 !
and 𝐻
"
(or equivalently that
similarity falls very sharply when moving outside 𝐻). In our example, this means that the DM
only samples causes of death and there is no intrusion from unrelated events. Furthermore, we
assume that 𝑇 is high enough that average odds are characterized by Equation (5).
Lichtenstein et al (1978) document the overestimation of cued low probability events,
such as death from botulism or a flood, and underestimation of cued and likely causes such as
heart disease. The average assessed odds in Equation ( 5 ) produce this phenomenon.
Proposition 3 The estimate 𝜋N
!
increases in the objective frequency 𝜋
!
. Overestimation,
i.e., 𝜋N
!
!
, occurs if and only if 𝜋
!
∗
, where threshold 𝜋
∗
is defined by:
∗
∗
!
"
!
!
!
"
"
"
If both hypotheses are equally self-similar, 𝑆
!
!
"
"
, then 𝜋
∗
Overestimation of an unlikely hypothesis is due to cued recall of its instances. When
thinking about 𝐻 !
= “floods”, the DM selectively retrieves deaths due to floods. When thinking
about 𝐻 "
= “other causes”, he retrieves other causes of death. Oversampling occurs for both
hypotheses but, as shown in Proposition 1, it favors the less likely one, which would otherwise
be less sampled. When the hypotheses are equally self-similar, 𝑆(𝐻
!
!
"
"
), there is
smearing toward 50:50, whereby probability assessments are attenuated to 50%. In particular, a
hypothesis is overestimated if and only if it is less than 50% likely, and probability assessments
exhibit insensitivity to true frequency.
Kahneman and Tversky’s (1979) probability weighting function also features
insensitivity to true frequency in the domain of objective probabilities in lottery choice. Our
model applies instead to the construction of subjective probabilities of cued hypotheses from
experience, when objective probabilities are not given.
12
In this context, similarity-based recall yields a fundamental new principle: it is the
content of events in the database, and not only their frequency, that shapes biases. A first
implication of this principle is that, in sharp contrast with KT’s probability weighting function,
unlikely events are prone to be neglected when they are not directly cued.
12
Recent work founds this function based on the salience of lottery payoff (Bordalo et al. 2012), noisy perception of
numerical probabilities (Khaw et al. 2020, Frydman and Jin 2020), and cognitive uncertainty (Enke and Graeber
2019 ). In the appendix, we show that recall is strongly correlated with a measure of subjective uncertainty.
When 𝐻 !
becomes more self-similar, it is easier to recall. As a result, it is less likely that, when
thinking about it, recall slips to its alternative hypothesis 𝐻 "
. According to Equation (5), this
increases the estimation of 𝐻 !
, even if its objective probability stays constant.
One implication of this result is that cued unlikely events are prone to overestimation not
only due to their low probability, but also because they are often more self-similar than their
alternative. When cued by 𝐻
!
= “flood”, it is easy to imagine instances of this disaster, because
they are fairly similar to each other. By contrast, the alternative 𝐻
"
= “causes other than flood”
is very heterogeneous, and contains causes of deaths similar to floods, like other natural disasters
or accidental drownings. This creates strong interference for 𝐻 "
, hindering its assessment.
This idea explains a striking experiment by Tversky and Kahneman (1983). A group of
subjects was asked to assess the share of 𝐻 !
= “words ending with n” in a certain text.
Another group of subjects was asked to assess the probability of 𝐻
!
$
= “words ending with _ing”.
Remarkably, subjects attached a lower probability to 𝐻 !
than to 𝐻
!
$
, despite the lower objective
frequency of the latter. Similarity accounts for this phenomenon. Intuitively, instances of 𝐻
!
$
“words ending with _ing” share many features, such as being gerunds, denoting similar activities,
etc, which makes it easy to bring many examples to mind. In contrast, 𝐻 !
= “words ending with
n” additionally includes many words which do not share these features (and which often do
not share many features with each other). This reduction in self-similarity makes it harder to
recall words in 𝐻
!
, even though it is a superset of 𝐻
!
13
A third implication, following from Corollaries 1 and 2, is partition dependence. The total
likelihood of death is estimated to be lower for “natural causes” than for “cancer, heart attack or
other natural causes” (Tversky and Koehler 1994). In our model, this phenomenon arises because
13
To check this intuition, we ran a simple online survey. Respondents indeed rate randomly generated groups of
_ing words as being more similar to each other than groups of n words. Results are available on request.
partitioning a hypothesis into more specific sub-events increases its overall self-similarity,
reducing interference. To see this, consider again a DM assessing the same hypothesis 𝐻
"
, but
now 𝐻
"
is explicitly partitioned into 𝐻
"!
and 𝐻
""
. For the purpose of Proposition 4 , assume the
subsets are equally: i) likely, 𝜋(𝐻
"!
""
), ii) self-similar , 𝑆(𝐻
"!
"!
""
""
) , and
iii) cross-similar, 𝑆
"!
!
""
!
14
We then obtain:
Proposition 4 Partitioning the alternative hypothesis 𝐻 "
into 𝐻
"!
and 𝐻
""
is equivalent to
changing the self-similarity of 𝐻 "
while leaving its true frequency unchanged. The self-similarity
of 𝐻 "
increases if and only if:
"!
"!
"!
""
In this case, partitioning 𝐻 "
reduces the estimate of 𝐻
!
, the more so the higher is
7
( 2
.-
, 2
.-
)
7 ( 2
.-
, 2
..
)
The assessment of 𝐻
!
= “flood” is reduced when its alternative is specified as 𝐻
"!
“natural causes” and 𝐻
""
= “non-natural causes other than flood”, compared to when it is
specified as 𝐻
"
= “causes other than flood”. Cueing 𝐻
"!
and 𝐻
""
fosters retrieval when thinking
about alternatives to flood, which reduces the assessment of 𝐻 !
= “flood”. Several famous
studies show that the probability assigned to an event decreases if its alternative is partitioned
(Benjamin 2019).
15
In Tversky and Koehler’s (1994) “Support Theory”, this phenomenon arises
because people evaluate events using a sub-additive “support function”. In our model, partition
dependence comes from similarity in recall.
16
14
These conditions nest three hypotheses (𝐻
, 𝐻
,+
, 𝐻
,,
) in the binary hypotheses case, connecting to Proposition 2.
15
For example, Fischhoff et al. (1978) famously show that when assessing the cause of a car’s failure to start,
mechanics judge “ignition” more likely when alternative causes were partitioned into “ignition”, “fuel”, “other”.
16
In contrast, death by “pneumonia, diabetes, cirrhosis or any other disease” is estimated to be less likely than death
by “any disease” (Sloman et al. 2004). This is consistent with a natural extension of our model in which atypical
cues such as “cirrhosis” focus attention on a narrow subset, which interferes with the retrieval of more common
diseases. A similar pattern occurs in free recall tasks (Slamecka 1968; Sanborn and Chater 2016).
