Elsevier

Biosystems

Volume 190, April 2020, 104104
Biosystems

Extended Bayesian inference incorporating symmetry bias

https://doi.org/10.1016/j.biosystems.2020.104104Get rights and content

Abstract

We start by proposing a causal induction model that incorporates symmetry bias. This model has two parameters that control the strength of symmetry bias and includes conditional probability and conventional models of causal induction as special cases. We calculated the determination coefficients between assessments by participants in eight types of causal induction experiments and the estimated values using the proposed model. The mean coefficient of determination was 0.93. Thus, it can reproduce causal induction of human judgment with high accuracy. We further propose a human-like Bayesian inference method to replace the conditional probability in Bayesian inference with the aforementioned causal induction model. In this method, two components coexist: the component of Bayesian inference, which updates the degree of confidence for each hypothesis, and the component of inverse Bayesian inference that modifies the model of each hypothesis. In other words, this method allows not only inference but also simultaneous learning. Our study demonstrates that the method addresses unsteady situations where the target of inference occasionally changes not only by making inferences based on knowledge (model) and observation data, but also by modifying the model itself.

Introduction

As a cognitive bias observed in humans, the dispositions to infer “if Q, then P” and “if not P, then not Q” from “if P then Q” are well documented (Sidman and Tailby, 1982; Yamazaki, 2004; Markman and Wachtel, 1988). The former is termed symmetry bias (Takahashi et al., 2011) and the latter is termed mutual exclusivity bias (Markman and Wachtel, 1988). Consider the following example. We tend to infer “I will take you out if and only if you clean the room” and “If you don't clean the room, then I will not take you out” from “If you clean the room, then I will take you out.” Although these inferences are invalid according to classical logic, various people are inclined to make them regardless of age.

In the field of cognitive psychology, experiments on causal induction were carried out, seeking to identify how humans evaluate the strength of causal relations between two events from their co-occurrence. In the case of a regular conditional form such as “if p then q,” the degree of confidence for the statement is considered to be proportional to the conditional probability P(q|p), which is the probability of occurrence of q following the occurrence of p (Evans et al., 2003). On the other hand, in case of causal relations, it has been experimentally demonstrated that humans have a strong understanding of causal relation between a cause c and an effect e when P(c|e) is high, as well as when P(e|c) is high, where P(c|e) is a conditional probability of the antecedent occurrence of c, given the occurrence of e. Specifically, the causal strength humans express between c and e can be approximated well by the geometric mean of P(e|c) and P(c|e). This is called the dual-factor heuristics (DFH) model (Hattori and Oaksford, 2007). If the causal intensity between c and e is denoted as DFH(e|c), then DFH(e|c)=P(e|c)P(c|e)P(e|c). Here, note that DFH(c|e)=DFH(e|c), that is, the symmetric relationship holds.

Bayesian inference can be considered an algorithm for inferring the cause from the effect based on the notion of conditional probability. Bayesian inference speculates the hidden cause behind observations by retrospectively applying statistical inferences. The relation between Bayesian inference and brain function has been attracting attention in recent years in the field of neuroscience (Dehaene, 2014; Chater and Oaksford, 2008). In Bayesian inference, the degree of confidence in a hypothesis is updated based on a hypothetical model predefined in the form of conditional probability and current observational data. In other words, Bayesian inference is a process of narrowing down hypotheses (cause) to one that best explains observational data (effect).

Consider a situation where you estimate the emotions of others. When estimating the emotion of others, since we cannot directly observe private internal states, there is no way to estimate their emotion other than by using external clues (observational data) such as facial expression and tone of voice. In this case, emotion corresponds to a hypothesis (cause) in Bayesian inference in the sense that it is the inference target. In addition, the model corresponds to a probability distribution that represents which facial expression appears with what probability and in which emotion. For example, if you have a model such as “If she/he is pleased, there is an 80% chance that she/he will smile” and if you actually observe her/him smiling frequently, your confidence for the hypothesis that “She/he is pleased” will increase. In other words, by observing the smile (effect), you can guess the emotion of joy as being the cause of the smile.

Attention should be paid to the following two points. First, in general, in order to estimate the cause more accurately, it is better to observe more data. However, this can be said only if it is ensured that the observational data originate from the same cause. Emotions are not always constant: it is a variable that changes from time to time. For targets such as emotion, inference must be drawn as quickly as possible within a short period. Here a trade-off appears between the accuracy of estimation and the follow ability to changes. The second is that it is not possible to have an emotion model of the person that you meet for the first time, in advance. In such a situation, it is necessary that the model is learned simultaneously with the inference being drawn. If this model is wrong, the correct reasoning for the emotion of the person cannot be acquired.

Arecchi (2011) proposed the concept of the inverse Bayesian inference where the model is modified according to circumstances. Gunji et al. (2018) and Horry et al. (2018) formulated the inverse Bayesian inference and demonstrated that animal herding and human decision-making can be satisfactorily modeled by combining Bayesian inference and inverse Bayesian inference. This framework can be said to simultaneously perform both Bayesian inference that picks up the optimal hypothesis from the predefined set of hypotheses, and inverse Bayesian inference that creates or modifies the models of hypotheses based on observational data. The latter can be said to be a learning of the model, rather than an inference.

First, we propose a causal induction model that incorporates symmetry bias by parametrizing the mixing rate of P(e|c) and P(c|e), i.e., the strength of symmetry bias. Second, we propose a human-like Bayesian inference where the conditional probability schema in Bayesian inference is replaced with the causal induction model proposed above. We term the inference “extended Bayesian inference” in the sense that it includes Bayesian inference as a special case where the strength of symmetry bias is zero. Finally, we conducted the simulation, derived from the problem of inference the probability of getting heads in the course of repetitive coin toss and verified the validity of the extended Bayesian inference.

Section snippets

Causal induction models

This section describes simple causal induction models that infer the strength of a causal relation between c and e from four pieces of co-occurrence concerning the cause candidate c and the effect event e (Table 1). The most representative model of causal induction is the ΔP model (Jenkins and Ward, 1965). It takes the difference between the conditional probability of occurrence of e given the occurrence of c and the conditional probability of occurrence of e given non-occurrence of c as an

Evaluation of descriptive validity of extended confidence model

We derived the values of α and m that best fit the causal strength judged by humans using equation (13) and the eight types of experimental data shown in the literature (Hattori and Oaksford, 2007; Anderson and Sheu, 1995; Buehner et al., 2003; Lober and Shanks, 2000; White, 2003). Generally, in a simple causal induction experiment, participants are given four types of co-occurrence information concerning the cause c and the effect e (Table 1). Then, they are asked to subjectively assess the

Discussion and conclusions

We first proposed extended confidence model as a causal induction model. Then, we formulated an inference model that incorporates the said causal induction model into Bayesian inference. We noticed that this inference model necessarily involves the inverse Bayesian inference, which allows for the handling of unsteady situations where the inference target changes from time to time by the adjustment of the hypothesis model itself. Finally, we demonstrated how this model can work well with unknown

Conflicts of interest

The authors report no financial interests or potential conflicts of interest.

Author contributions

S. S. conceived the extended confidence model and the extended Bayesian inference, conducted the meta-analysis and the simulation, wrote the manuscript. U. C. revised it. T.T collected the data for the meta-analysis. N. M., K. S., U. C., T. T., P·Y.G., Y. N. and S. M. contributed to the interpretation of the study findings. All authors participated in the editing and revision of the final version of the manuscript.

Acknowledgements

This research is partially supported by the Center of Innovation Program from the Japan Science and Technology Agency, Japan and Grant Number JP16K01408 from the Japan Society for the Promotion of Science, Japan.

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