A Rational Account of Anchor Effects in Hindsight Bias.

A Rational Account of Anchor Effects in Hindsight Bias
Samarie Wilson (samariew@princeton.edu)
Mechanical and Aerospace Engineering, Princeton University
Princeton, NJ 08540 USA
Somya Arora (somyaa@princeton.edu)
Department of Computer Science, Princeton University
Princeton, NJ 08540 USA
Qiong Zhang (qiongz@princeton.edu)
Princeton Neuroscience Institute, Princeton University
Princeton, NJ 08540 USA
Thomas L. Grifﬁths (tomg@princeton.edu)
Departments of Psychology and Computer Science, Princeton University
Princeton, NJ 08540 USA
Abstract
Hindsight bias is exhibited when knowledge of an outcome
(i.e., an anchor) affects subsequent recollections of previous
predictions (i.e., an estimate). Hindsight bias usually leads
to estimates being remembered as closer to the anchor than
they actually were. The exact amount of hindsight bias ex-
hibited depends on the anchor value and the anchor plausibil-
ity, with experimental results showing that hindsight bias is
elicited only when the anchor is perceived to be plausible. In
this paper we present a Bayesian model that captures the rela-
tionship between hindsight bias and anchor plausibility. This
model provides a rational account of hindsight bias by consid-
ering memory recall as a statistical problem, where the goal is
to reconstruct the original estimate using the anchor as new ev-
idence. Simulations show that the modeled trends align closely
with previously published human data.
Keywords: hindsight bias; rational analysis; Bayesian infer-
ence; anchor plausibility; prior
Introduction
Imagine you are at a soccer game with your friend. Before the
game begins, you predict that your favorite team is going to
win by 10 points. However, at the end of the game, they only
win by 2 points. While discussing the game with your friend
afterwards, you recall that you always knew it was going to be
a close game. This difference between the original estimate,
your prediction of a certain team winning by 10 points, and
your recalled estimate, the game being a close one, is called
hindsight bias – the recalled estimate is biased towards the
outcome, known as an “anchor” (Synodinos, 1986; Tversky
& Kahneman, 1974). Hindsight bias is also known as the
“knew it all along” effect (Roese, 2012).
Previous studies show that hindsight bias does not always
occur, but rather depends on the plausibility of the anchor
(Wegener, Petty, Detweiler-Bedell & Jarvis, 2001; Chapman
& Johnson, 1994). The plausibility can be manipulated in a
continuous manner by varying the distance between anchor
and the original estimate: an anchor is perceived as increas-
ingly implausible as this distance increases (Hardt & Pohl,
2003).Biases like the hindsight bias have typically been taken
as evidence of human irrationality, but recent work has sug-
gested that many apparently irrational biases can be made
sense of within rational models (Lieder, Grifﬁths, Huys &
Goodman, 2017; Feldman & Grifﬁths, 2007; Hemmer &
Steyvers, 2009, Parpart, Jones and Love, 2018; Sher, McKen-
zie, 2006). Here we pursue a similar approach, providing a
rational account of the hindsight bias.
We focus on explaining the plausibility effect. Speciﬁcally,
we focus on modeling the experimental results in Hardt &
Pohl (2003), as it provides continuous measures of anchor
distances, which allows for examining a range of values on
anchor plausibility. The exact relationship between hindsight
bias and anchor distance is shown in Figure 1. As we can see
in Figure 1, the extent an anchor inﬂuences memory depends
on how much it is inferred to be plausible: when the anchor
is considered to be plausible, the relationship between the
anchor distance and hindsight bias is directly proportional.
However, beyond a certain anchor distance, the anchor is con-
sidered to be more and more implausible, and the relationship
is inversely proportional. Previous modeling work by Pohl,
Eisenhauer & Hardt (2003) provides a mechanistic account
of hindsight bias, but no current model accounts for the ef-
fects of anchor plausibility.
In this paper, we deﬁne and evaluate a Bayesian model of
the relationship between hindsight bias and anchor plausibil-
ity. The paper is organized as follows. In the following sec-
tion, we provide a brief overview of existing work regarding
hindsight bias. Next, we develop the mathematical model for
how the anchor and perceived plausibility update the prior.
This sets us up to apply the formalization to simulate the rela-
tionship between hindsight bias and anchor distance as in Fig-
ure 1. Finally, we review the relationship between hindsight
bias and anchor plausibility, restate our assumptions, and lay
out future directions for this work.
Figure 1: Hindsight bias and its dependence on anchor dis-
tance as reproduced from Figure 4 of Hardt & Pohl (2003).
The magnitude of hindsight bias, shown on the vertical axis,
is the amount that the revised estimate moves from the origi-
nal estimate towards the provided anchor. Positive and nega-
tive labels indicate whether the anchor values were above or
below the original estimate, respectively.
Background: The Hindsight Bias in Hindsight
Hindsight Bias Effect
Hindsight bias occurs when people feel that they “knew it
all along” (Roese, 2012; Roese and V ohs, 2012). It is
elicited in human memory by the presentation of an an-
chor. Estimates are generally recalled to be closer to the an-
chor than they had been in reality (Christensen-Szalankski
& Beach, 1984), which has been postulated to take place ei-
ther during recollection or during reconstruction of memory
(Erdfelder, Brandt, & Broder, 2007). Meta-analysis across
many studies has shown that effects of hindsight bias can
be moderated by the type of anchor information presented
and the subject’s familiarity with the task. Observed in-
consistencies in the effects of hindsight bias suggest that
it may be a result of cognitive factors rather than motiva-
tional ones (Christensen-Szalanski & Willham, 1991). Over-
all, studies have underscored the robustness of hindsight bias
(Christensen-Szalankski & Willham, 1991). Therefore it is
worth exploring the manipulations that interrupt the effects
of anchor presentation on recollection of the original estimate
(Hawkins & Hastie, 1990).
Effect of Anchor Plausibility
Previous studies have shown that anchor plausibility is a func-
tion of the anchor’s agreement with the individual’s prior
knowledge of the relevant subject (Pohl, 1998; Strack &
Mussweiler, 1997). Intuitively, invalidating anchor data un-
dermines its functionality, thereby diminishing the hindsight
bias (Hasher, Attig & Alba, 1981). Determining an anchor
to be implausible is an example of such an invalidation. The
two factors that describe anchor plausibility are known to beknowledge quantity and knowledge precision (Pohl, 1998).
Studies on the effect of anchor plausibility on hindsight
bias have produced inconsistent ﬁndings. In one experiment,
Strack and Mussweiler (1997) found no differences between
the inﬂuences of plausible and implausible anchors. In an-
other, Chapman and Johnson (1994) found the anchoring ef-
fect of extremely implausible anchors to be insigniﬁcant.
Hardt and Pohl (2003) revisited anchor plausibility, focus-
ing on the idea that the subjective plausibility of anchors is
the key variable. Using measures of the extent to which par-
ticipants perceived continuously varying anchors as plausible,
they were able to produce more nuanced results than previous
studies. In their study, participants were given several ques-
tions (e.g., how old was Gandhi when he died?) and asked to
give the minimum and maximum values that they considered
to be plausible answers to each question, as well as an ex-
act estimate of the answer. After a week, participants were
brought back and presented with what they believed were
other participants’ estimates for each of the questions, before
recalling their own estimates from the week prior. The in-
ﬂuence of these presented estimates, (i.e., anchors), on the
recollection depended upon how plausible the anchors were
perceived to be by the participants.
Existing Modeling
Hindsight bias has been modeled by Pohl, Eisenhauer and
Hardt (2003) using the SARA model, a simpliﬁed and fo-
cused version of the associative memory model SAM (Search
of Associative Memory; Raaijmakers & Shiffrin, 1980;
Shiffrin & Raaijmakers, 1992). SARA is a process model de-
signed to better understand each process of the hindsight bias
phenomenon: estimate generation, anchor encoding, and es-
timate recollection. In this model, estimates are generated by
a probabilistic sampling process that samples knowledge rep-
resentations from long-term memory and compiles all these
representations to create an estimate. Presented anchors are
automatically encoded, and this encoding process strengthens
the representations that are most like the presented anchor.
This makes these representations more likely to be recalled
in subsequent processes that use this knowledge to recon-
struct the original estimate. The reconstruction of the orig-
inal estimate is thus biased towards the anchor and represen-
tations similar to the anchor, producing hindsight bias. While
SARA successfully models the hindsight bias processes, it
does not capture the effect of anchor plausibility on hindsight
bias (Pohl, Eisenhauer & Hardt, 2003).
Hindsight as Bayesian Inference
Our model frames memory recall as a problem of statistical
inference. The goal is to reconstruct the original estimate us-
ing the noisy memory representation of the estimate and the
additional evidence from the anchor. It is rational to incor-
porate information from the anchor to the degree that it is
plausible.
To express this model formally, we introduce variables m,
sanda, where mis the original estimate, sis the memory of
the original estimate, and ais the presented anchor. The goal
is to reconstruct m, based on the evidence provided by the
noisy memory trace, s, and the anchor, a.sis assumed to be
centered around the true value of the original estimate, m:
p(sjm)N(m;s2
s) (1)
andp(m)is assumed to be a uniform distribution. The anchor
aprovides another source of information relevant to m, being
an answer to the estimation problem. If mis a good estimate
ofa, then the two should be related and p(ajm)will focus on
values close to m. Reconstructing mis then simply a matter
of Bayesian inference, with p(mja;s)µp(ajm)p(sjm)p(m).
However, there is a catch: this analysis assumes that ais
a plausible anchor. In practice, people infer whether an an-
chor is reasonable. To capture this we introduce a third vari-
able, z, to represent the plausibility of a. The implausible
case is denoted by z=0, and the plausible case is denoted
byz=1. The two categories of anchors can be captured by
deﬁning p(ajm;z)to be Gaussian distributions for each value
ofz. Intuitively, the larger the anchor distance, the lower
the perceived plausibility. Therefore the plausible anchors
are centered closer to m, whereas implausible anchors have
a larger variance, making p(ajm;z)look relatively uniform.
This results in the distributions shown in Figure 2, which can
be written as:
P(ajm;z)(
N(m;s2
0);z=0
N(m;s2
1);z=1(2)
where s2
0>>s2
1. This makes intuitive sense because in the
plausible case ( z=1), the distribution will have larger prob-
ability values near the mean of the memory, whereas for the
implausible case ( z=0), the distribution will be relatively
ﬂat. While s2
0can be inﬁnitely large, regardless what ques-
tion is answered; s2
1may be question dependent, related to
the typical range of plausible answers to that question. We
also assume that the prior probabilities assigned to the plau-
Figure 2: p(ajm;z)for values of a with s0=10 and s1=200.
Figure 3: Generative Model for how the prior of original es-
timate, m, is updated using the anchor ( a) and the plausibility
(z).
sible and implausible categories, p(z), are equal:
p(z=0) =p(z=1) =1
2: (3)
Given the aforementioned assumptions, we express the re-
lationships between these variables using a generative model.
Here the anchor, a, is produced either from a plausible source
or implausible source, z, both centered at m.sis a noisy rep-
resentation of the original estimate m. The inference of plau-
sibility and the incorporation of the anchor is determined by
the distribution of original estimate, m, giving our generative
model the structure shown in Figure 3.
Now that we have all our assumptions ﬁnalized for our
model, we can deﬁne how we will quantify hindsight bias.
The goal of our model is to determine how the anchor up-
dates the memory, p(mja;s), and to use this to compute the
hindsight bias. Once we have updated the memory, we can
take hindsight bias, Dh, to be the difference between the pos-
terior mean, E(mja;s), and the original estimate, m:
Dh=E(mja;s)m: (4)
If the plausibility of the anchor is known, it is straightforward
to infer mby computing p(mjs;z;a). However, in the case
we don’t know the plausibility of the anchor, we need to sum
over the possible values of zwhen computing p(mja;s):
p(mja;s) =å
zp(mjs;z;a)p(zja;s)(5)
where p(zja;s)reﬂects the inferred plausibility for a given an-
chor and p(mjs;z;a)is the posterior distribution over mcom-
puted by assuming that the anchor is either plausible or im-
plausible.
We can rewrite the posterior distribution by applying
Bayes’ rule:
p(mjs;z;a)µp(ajm;z)p(sjm)p(m) (6)
Given that p(ajm;z)andp(sjm)are normally distributed as
seen from Equations 1 and 2 and p(m)being uniform, we can
write the posterior p(mjs;z;a)as
p(mjs;z=0;a)N
s2
s
s2s+s2
0a+s2
0
s2s+s2
0s;1
s2s+1
s2
01!
(7)
p(mjs;z=1;a)N
s2
s
s2s+s2
1a+s2
1
s2s+s2
1s;1
s2s+1
s2
11!
:
(8)
Finally, the two components of the posterior are weighted by
p(zja;s)in Equation 5. We can apply Bayes’ rule to get:
p(zja;s)µp(ajz;s)p(z) (9)
We also have
p(ajz=0;s)N(s;s2
s+s2
0) (10)
and
p(ajz=1;s)N(s;s2
s+s2
1) (11)
which are derived by incorporating the additive noise from
Equation 2 into the normal distribution in Equation 1.
Given a value of an anchor, we can calculate the exact val-
ues of p(ajz=0;s)andp(ajz=1;s)using Equations 10 and
11. Combining them with Equation 9 and the property that
p(z=0ja;s)andp(z=1ja;s)sum to 1, we can ﬁnd the exact
values of p(z=0ja;s)andp(z=1ja;s). By substituting them
into Equation 5, together with Equation 7 and 8, we obtain the
updated memory, p(mja;s). Now that we understand how the
anchor updates the memory, p(mja;s), we can compute the
hindsight bias using Equation 4.
Simulations
Having derived a Bayesian model of the effects of plausibility
on hindsight bias, we now show that this model can reproduce
the relationship between anchor distance and the magnitude
of this bias shown in Figure 1. A foundation of the model is
the relationship between the plausible and implausible distri-
butions ( s0ands1). It is important that s0>>s1. Plausible
anchors are centered closely around the original estimate, m,
and implausible anchors are further away from it.
In order to simulate hindsight bias, which can be written as
E(mja;s)m, where p(mja;s) =åzp(mjs;z;a)p(zja;s), we
must ﬁrst ﬁnd p(mjs;z;a), which is the the posterior proba-
bility characterizing one’s best estimate of the true state of
mgiven noisy memory content sand new evidence a, condi-
tioned on whether the new evidence is plausible or not z. Fig-
ure 4 shows how p(mjs;z;a)is dependent on anchor plausibil-
ity. Intuitively, when the anchor is considered to be plausible,
p(mjs;z=1;a), memory is biased toward the anchor, which
in this case we ﬁxed at a=120; when the anchor is consid-
ered to be implausible, p(mjs;z=0;a)is centered around the
noisy memory content s=60 and not biased by the anchor
(we took s=60 for this example, assuming the memory trace
is veridical). This effect is evident from Equations 7 and 8.
Given s2
0>>s2
1, this leads to the anchor abeing weightedmore when the anchor is plausible ( z=1), whereas abeing
weighted less when the anchor is not plausible ( z=0).
Figure 4: Simulation of p(mjs;z;a)for a ﬁxed anchor value,
a=120. Dotted lines represent the mean of the memory, s,
and the anchor value, a.
For each anchor value, we can obtain p(z=0ja;s)and
p(z=1ja;s). Figure 5 shows the distribution of p(zja;s)for
different values of the anchor, a. Consistent with Equation
10 and 11, when the anchor is farther from the noisy mem-
ory,s, it is more likely to be implausible, which explains the
high values at the edges for the black curve, p(z=0ja;s)and
the low values near the center. Similarly, when the anchor is
closer to the noisy memory, s, it is more likely to be inferred
as plausible, z=1, explaining the high values in the center
and low values near the edges of the gray curve.
Figure 5: Simulation of p(zja;s)for values of a. While the an-
chor is implausible, it is more likely to have a value far from
the mean of the expectation. When the anchor is plausible, it
is more likely to be close to this mean.
Because our posterior distribution is a Gaussian mixture
model with different means and variances, E(mja;s)is sim-
ply the mean of the distributions p(mjs;z;a)weighted by the
probabilities p(zja;s). We calculated the analytical mean by
computing the mean of each category, p(mjs;z=0;a)and
p(mjs;z=1;a), and averaging by p(z=0ja;s)and p(z=
1ja;s)respectively.
To create the ﬁnal curves, the values of hindsight bias,
E(mja;s)m, were graphed against the magnitude of the
anchor distance,jamjin Figure 6. A graph of hindsight
bias versus anchor values is also included in Figure 7. Fig-
ure 6 provides a close match to the desired curve in Figure
1, capturing the inverse U-shape relationship between anchor
distance and hindsight bias. The model does not show a dif-
ference between negatively directed anchors and positively
directed anchors as seen from the experiment in Figure 1. We
will discuss this discrepancy as part of the future work.
Figure 6: Hindsight bias ( Dh=E(mja;s)m) and its de-
pendence on anchor distance ( jamj) as predicted by the
Bayesian model for m=s=60. The model captures the in-
verse U-shape relationship between anchor distance and hind-
sight bias in Figure 1.
Figure 7: Hindsight bias and its dependence on speciﬁc an-
chor values as predicted by the Bayesian model. Note that the
hindsight bias is negative when the anchor is less than m, pos-
itive when the anchor is greater than m, and zero when equal
tom.
Discussion
The results presented in this paper establish that the effect of
anchor plausibility on hindsight bias can be explained as theconsequence of optimally solving the statistical problem of
memory reconstruction using additional knowledge from the
observed anchor. Hardt and Pohl (2003) hypothesized that the
U-shaped relationship between anchor distance and amount
of hindsight bias is the result of two independent processes.
More precisely, the impact of the anchor increases with an-
chor distance, but the probability of a biased reconstruction
depends on anchor plausibility. However, we proposed that
the impact from anchor distance and the impact from anchor
plausibility are not two independent processes; our computa-
tional model captures how they jointly affect the amount of
hindsight bias. The amount of hindsight bias exhibited de-
pends on both anchor distance and the extent the anchor is
perceived to be plausible.
The current work closely relates to the existing literature
on modeling the effects of category structure on perception.
In the model presented by Huttenlocher, Hedges and Ve-
vea (2000), people used category structure to compensate for
uncertainty in memory of sizes; in Feldman and Grifﬁths’s
(2007) model, category structure is used to correct for uncer-
tainty in speech signals. The key difference between these
models and our own is that in our model, the inference of
category is from the additional evidence of the anchor rather
than from the original stimuli.
The proposed model simulates the qualitative patterns of
the inverse U-shaped relationship between anchor distance
and amount of hindsight bias. Hindsight bias increases
as the anchor distance increases, as long as the anchor is
considered to be plausible; However, as anchor distance
keeps increasing, the anchor is considered to be more and
more implausible, hindsight bias starts decreasing. One
direction for future work is to evaluate the model’s ﬁt to
individual subject data. Another direction for future work
entails explaining the additional ﬁnding that hindsight bias
was larger for negatively directed anchors than for positively
directed anchors in Figure 1. Hardt and Pohl suggested
that the direction effect is potentially contributed to by
the distribution of the estimates. When the distribution is
skewed, anchors in one direction will become unacceptable
sooner than anchors with the same amount of deviation
in the opposite direction (Hardt & Pohl, 2003). Further
experimental and modeling research is needed to verify this
hypothesis.
Acknowledgements
This work was supported by a C.V . Starr Fellowship awarded
to Q.Z.
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Samarie Wilson
Somya Arora
Qiong Zhang
Thomas L
Griffiths