Order Effects in Bayesian Updates

The application of Quantum Mechanic’s mathematical principles in areas outside of physics has been getting increasing attention in the scientific community in a field called Quantum Cognition \@BBOPcitep\@BAP\@BBN(Busemeyer & Bruza, 2012; Moreira, Tiwari, et al., 2020; Moreira, Hammes, et al., 2020)\@BBCP. These principles have been applied to explain paradoxical situations that cannot be easily explained through classical probability theory \@BBOPcitep\@BAP\@BBN(Moreira & Wichert, 2017b, 2015)\@BBCP. Quantum principles have also been adopted in many different domains, such as Cognitive Psychology \@BBOPcitep\@BAP\@BBN(Busemeyer et al., 2006; Pothos & Busemeyer, 2009; Pothos et al., 2013)\@BBCP, Economics \@BBOPcitep\@BAP\@BBN(Khrennikov, 2009; Haven & Khrennikov, 2013)\@BBCP, Biology \@BBOPcitep\@BAP\@BBN(Asano et al., 2012, 2015)\@BBCP, and Information Retrieval \@BBOPcite\@BAP\@BBNBruza et al. (2009, 2013)\@BBCP, to name a few.

Consider the following experimental scenario. A set of participants is required to answer three questions, $Q_{0}$, $Q_{1}$, and $Q_{2}$ consecutively. One set of participants is presented with the questions $Q_{0}\rightarrow Q_{1}\rightarrow Q_{2}$, while the other participants are presented with the same questions, but in reversed order, $Q_{0}\rightarrow Q_{2}\rightarrow Q_{1}$. Some researchers argue the above experimental contexts should attain similar outcomes for questions $Q_{1}$ and $Q_{2}$ since, in classical probability theory, the joint distribution between two events is commutative. In other words, it is argued that the expected probability outcomes should be $P(Q_{1},Q_{2}|Q_{0})=P(Q_{2},Q_{1}|Q_{0})$. We claim that this premise is a fallacy due to the following reasons:

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  There is no causality in probability theory. Probability theory describes the likelihood of outcomes of random phenomena. Since events are represented as sets, there is no notion of temporality (order) between events, and, consequently, probability theory, by itself, says nothing about causality. The joint $Q_{1}\cap Q_{2}$ means that both events occur, without any implications to time or cause & effect (order) between the events.

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  There is no temporality in probability theory. Order effects experiments imply the notion of time: one question needs to be asked after another. Again probability theory, due to its set-based foundation, nothing says about time. Conditional probability is simply a measure under additional information. It implies symmetry, and for that reason, it does not imply temporal order. Temporality is part of a model, and not part of probability theory itself.

A good set of tools to visualize these fallacies are Bayesian networks (BNs), which are models that represent a compact full joint probability distribution through a directed acyclic graphical structure. Its underlying mathematical principles are Bayes rule and conditional independence.

Given different full joint probability distributions, $P(Q_{0},Q_{1},Q_{2})$, each distribution will be associated to a specific configuration of random variables. Consider the Bayesian networks in Figure 1. They all represent the full joint probability distribution $P(Q_{0},Q_{1},Q_{2})$, but with different underlying probabilistic graphical models. It is not correct to say that the full joint probability distribution in Figure 1-(A) has to be same as in Figure 1-(B). First, because they represent different experimental scenarios. For that reason, they yield different measures over the sample spaces. Second, the priors affecting the random variables change in both scenarios. It is a fallacy to assume that these two models should represent the same experimental context.

For example, the equation expressing the full joint of the network in Figure 1-(A) is given by $P(Q_{0},Q_{1},Q_{2})=P(Q_{0})P(Q_{1}|Q_{0})P(Q_{2}|Q_{0},Q_{1})$. In the same way, the equation representing the network in Figure 1-(B) is given by $P^{\prime}(Q_{0},Q_{1},Q_{2})=P^{\prime}(Q_{0})P^{\prime}(Q_{2}|Q_{0})P^{\prime}(Q_{1}|Q_{0},Q_{2})$. Thus, it is a fallacy to assume that $P(Q_{0},Q_{1},Q_{2})=P^{\prime}(Q_{0},Q_{1},Q_{2})$.

In this paper, we argue that to model order effect between three questions, $Q_{0}$, $Q_{1}$, and $Q_{2}$, we need to have a causal reasoning network structure as depicted in Figure 1-(C). This means that order effects can be observed when:

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  $Q_{0}$ is correlated with $Q_{1}$ and $Q_{2}$, and

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  $Q_{1}$ and $Q_{2}$ are conditionally independent, given $Q_{0}$, $(Q_{1}\Perp Q_{2}|Q_{0})$.

These two conditions imply that, if there are no order effects, then the following equation should hold:

If Equation 3 is violated, then it suggests some correlation between $Q_{1}$ and $Q_{2}$, which may be derived from the degree of uncertainty that the decision-maker is experiencing. In the next section, we will use these notions to formulate a Bayesian update model to accommodate order effects.

To understand the main idea, let us think of the well-known Clinton/Gore order-effect example, which corresponds to an order effect experiment originally conducted by \@BBOPcitet\@BAP\@BBNMoore (2002)\@BBCP. Consider the following two questions that were used in the experiment:

$Q_{1}$:

Is [former US president Bill] Clinton honest and trustworthy?

$Q_{2}$:

Is [former US vice-president Al] Gore honest and trustworthy?

Experimental data shows that we get disparate results if we ask the above questions in different orders (Table 1) \@BBOPcite\@BAP\@BBNWang et al. (2014)\@BBCP. Namely, if we ask $Q_{1}$ before $Q_{2}$, the probability that $Q_{1}$ is answered in the positive is $53\%$. However, if we ask $Q_{1}$ after $Q_{2}$, i.e., if we change the order, $Q_{1}$ is answered positively $59\%$ of the time. Similarly, $Q_{2}$ gets $75\%$ if asked first and $65\%$ if asked after $Q_{1}$. The change in probabilities of $Q_{1}$ and $Q_{2}$ with the order they are answered is known as the order effect.

How can we understand the order effect from a Bayesian point of view? A simple Bayesian model, such as the one presented by \@BBOPcite\@BAP\@BBNTrueblood & Busemeyer (2011)\@BBCP, cannot account for order effects. As discussed previously, the reason is simple: from a set-theoretic point of view, the conjunction $A\cap B$ is the same as $B\cap A$, so there is no order in it.

So, does that mean that it is impossible to obtain order effects with Bayesian probabilities? The answer is no. To see this, consider the following circumstance. Imagine we ask a participant a question they are not sure of its answer. After some reflection, they may answer yes to it. However, this does not mean they are sure of their answer. For example, if after a moment of pondering, they answer “yes” to the question, they weighed the evidence for and against it and found a preponderance toward yes. Thus, we can think of the process of answering a question as a reflection on our beliefs. It is a type of mini-experiment. As such, after this mini-experiment, their beliefs may change. Consequently, they need to update their prior. Being asked and answering a question updates the prior to a posterior.

To implement the above idea to produce an order effect, we follow the belief update described in \@BBOPcite\@BAP\@BBNMorris (1974, 1977); de Barros (2013)\@BBCP. Imagine a participant being asked the two questions, $Q_{1}$ and $Q_{2}$. Before answering $Q_{1}$, the participant has an unconscious prior joint probability $P(Q_{1},Q_{2})$. Once they reflect on $Q_{1}$, their prior is updated to a posterior $P^{\prime}(Q_{1},Q_{2})$. This new posterior is the one used to compute the answer to $Q_{2}$. If the prior joint probability has $Q_{1}$ and $Q_{2}$ correlated (as is likely the case of Gore and Clinton), then the expectation for $Q_{2}$ will likely be different. The same would happen if $Q_{2}$ were asked first, with a new posterior $P^{\prime\prime}(Q_{1},Q_{2})$ giving rise to a different expectation for $Q_{1}$.

Let us describe mathematically the above intuition. We start with a participant, Alice, who will be asked to decide between the two questions, $Q_{1}$ and $Q_{2}$. Alice starts the experiment with a prior joint probability distribution $P(Q_{1}=x;Q_{2}=y|\kappa)$ conditioned on Alice’s current knowledge $\kappa$, where $Q_{1}$ and $Q_{2}$ are $0/1$-valued random variables representing Alice’s two possible answers, “yes” ($1$) and “no” ($0$). We will denote $P(Q_{1}=y)$ as $P(q_{1})$ and the complementary probability, $P(Q_{1}=n)$ as $P(\overline{q}_{1})$. Prior to being asked a question, Alice is unaware of her belief for each answer, encoded by the prior. It is not until she is asked that, by thinking about her belief, Alice comes up with one of two possible answers and perhaps a measure of her belief. Thus, Alice’s choice of “yes” or “no” and her reflection requires a belief update. According to Bayes’s theorem, Alice’s prior became the posterior $P^{\prime}(Q_{1}=x,Q_{2}=y|Q_{1}=z)$. Assuming a simple linear likelihood function, the updated posterior for the order $Q_{1}$ and then $Q_{2}$ becomes

where we are using the simplifying notation $P^{\prime}(Q_{1}=x,Q_{2}=y|Q_{1}=z)=P^{\prime}(x,y|z)$ when there is no ambiguity in meaning, and the subscript $O_{1}$ for the order $Q_{1}$ first. Similarly, for the order $Q_{1}\rightarrow Q_{2}$ we get

for the inverse order, with $Q_{2}$ first.

For simplicity, let us assume that the prior has well the following expectations.

$a$ and $b$ are the probabilities of $Q_{1}$ and $Q_{2}$ as true when asked first, respectively, and $c$ is related to their correlation, as it equals their joint moment. For instance, $c=1$ if both $Q_{1}$ and $Q_{2}$ are true.

From Eq. 6-8, one can compute the remaining values of the full joint distribution by solving the following linear equation system:

From Eq. 9, we can find that $P(q_{1},q_{2})=c$, $P(q_{1},\overline{q}_{2})=a-c$, $P(\overline{q}_{1},q_{2})=b-c$, and $P(\overline{q}_{1},\overline{q}_{2})=1-a-b+c$. By substituting the values of Eq. 4 by the outputs of our linear solver, then we obtain Eq. 10 from which it is straightforward to show that there is an order effect. For instance, for $Q_{1}$ first, the probability of a $true$ answer is $a$, but if one asks $Q_{2}$, after the update, the probability of being $true$ is

We can see from the above equations that the updated probability for $Q_{2}$, $P^{\prime}(q_{2}|Q_{1})$, is not the same as it was before, $P(q_{2})$. What are the conditions for the probability to be the same, i.e., for not having an order effect? No order effect can be verified if the following conditions are satisfied:

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  Both questions are statistically independent, $Q_{1}\Perp Q_{2}$, and the following conditions are satisfied:

  • –

    $P^{\prime}(q_{2}|Q_{1})=P(q_{2})$, the belief of answering ”true” to the second question independently of having knowledge about the first question.

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    $P^{\prime}(q_{1}|Q_{2})=P(q_{1})$, the belief of answering ”true” to the first question independently of having knowledge about the second question.

If the above conditions are not satisfied, then we have an order effect. Figure 2 illustrate an order effect in the Bayesian update from $P(q_{1})$ to $P^{\prime}(q_{2}|Q_{1})$ (Equation 10).

One possibility to not have an order effect is for $a=1/2$. This possibility is intuitive since, if we have no opinion on Clinton, it stands to reason that what a random choice will not influence how we think of Gore. The other solution, $c=a.b$, is also intuitive (and it is represented in Figure 3): it means that our prior opinion of Gore and Clinton are statistically independent. Note that independence between questions is one of the properties that were highlighted in Eq. 3, as a necessary condition for order effects to not occur. Therefore whatever we learn about our beliefs about Clinton, they will not affect our beliefs about Gore.

With the above example, we can understand how the update leads to the order effect. We start with a prior, say one that has Clinton at 40% and Gore at 70% as our belief that they are honest and trustworthy. Let us assume that, because we know Gore was Clinton’s vice president, there is a correlation between our belief’s about Clinton and Gore. By reflecting on Clinton and thinking, “no, we do not think he is honest and trustworthy, and we think that the likelihood of trustworthy is just 40%”, our prior updates and our belief about Gore changes.

What about the QQ equality, a measure of how well the order effect matches quantum predictions. We can also compute the QQ equality for this system, and we obtain the following prediction.

We can see that the QQ equality is satisfied when $a=b$, $b=1-a$, or when both Clinton and Gore are statistically independent. The latter case is not interesting since it also implies there is no order effect. Interestingly, Bayesian updates do not predict the QQ equality, but it provides conditions for which the QQ equality is satisfied. Furthermore, it allows us to try and find experimental situations where QQ is maximally violated.

Order effect has been studied extensively in the literature. It is the most striking example of an experimental outcome that fits well with quantum models while challenging traditional cognitive models. Furthermore, quantum models predict some constraints for order effects, most notably the $QQ$ equality.

In this paper, we examined the order effect as a Bayesian update phenomenon. We first showed that, by thinking about questions in terms of a Bayesian Network, one could realize that some traditional models, such as those discussed by \@BBOPcite\@BAP\@BBNTrueblood & Busemeyer (2011)\@BBCP, fail because they do not account for temporal processes. Notably, they do not incorporate changes in the probability distribution during a Bayesian update. To address this challenge, we constructed an explicit Bayesian update model where each question can be thought of as a mini-experiment where the respondent reflects on their beliefs. For this model, we showed that order effects appear, and they have a simple cognitive explanation for their existence: the respondent’s prior belief that two questions are correlated.

Similar to the quantum order effect model, our Bayesian model allows us to make several predictions. First, we see certain conditions on the prior that limit the existence of order effects. For example, if $Q_{1}$ and $Q_{2}$ are statistically independent, one should not expect order effects. These conditions allow us to think about possible experiments where order effects will be strong or non-existent. Second, we show that, for our model, the $QQ$ equality is not necessarily satisfied. This does not mean that $QQ$ will not be satisfied for a particular experiment. It only means that $QQ$ requires further assumptions (i.e., certain symmetry conditions). Third, like the quantum model, the Bayesian model does not predict the exact replicability of questions. However, contrary to the quantum model, the more questions are asked, the more the posterior will converge, stabilizing the answer. Finally, the Bayesian model has the advantage of having fewer parameters than the quantum model. For the simplest order-effect quantum model, a minimum of four parameters is necessary. For our Bayesian model, three parameters describe the joint probability distributions entirely. Whether the Bayesian model is better than the quantum model is a question to be settled empirically, which we are planning to address as future work.

This work was supported by Centre for Data Science First Byte Funding Program at Queensland University of Technology (QUT) and by QUT’s Women in Research Grant Scheme.