Subjective probability assessment in decision analysis: partition dependence and bias toward the ignorance prior.

1. Introduction

Decision and risk analysis models often require assessment of subjective probabilities for uncertain events, such as the failure of a dam or a rise in interest rates. Spetzler and Stael Von Holstein (1975) were the first to describe practical procedures for eliciting subjective probabilities from experts. Their procedures are still in use, largely unchanged, as reflected in work by Clemen and Reilly (2001), Cooke (1991), Keeney and von Winterfeldt (1991), Merkhofer (1987), and Morgan and Henrion (1990).

Human limitations of memory and information processing capacity often lead to subjective probabilities that are poorly calibrated or internally inconsistent, even when assessed by experts (see, e.g., Kahneman et al. 1982, Gilovich et al. 2002). In this paper, we study a particular bias in probability assessment that arises from the initial structuring of the elicitation. At this stage, the analyst, sometimes with the assistance of an expert, identifies relevant uncertainties and the specific events for which probabilities will be judged. Although existing probability assessment protocols provide guidance on important steps in the elicitation process (e.g., identifying and selecting experts, training experts in probability elicitation, and the probability assessment itself), little attention has been given to the choice of events to be assessed.

Analysts typically assume that the particular choice of events into which the state space is partitioned does not affect the assessed probability distribution over states. Unfortunately, our experimental results demonstrate that this assumption is unfounded: assessed probabilities can vary substantially with the partition that the analyst chooses. We refer to this phenomenon as partition dependence (see also Fox and Rottenstreich 2003). It is more general than the pruning bias documented in the assessment of fault trees by Fischhoff et al. (1978), in which particular causes of a system failure (e.g., reasons why a car might fail to start) are judged more likely when they are explicitly identified (e.g., dead battery or ignition system) than when pruned from the tree and relegated to a residual catchall category ("all other problems"). Most previous investigators have interpreted pruning bias as an availability or salience effect: when particular causes are singled out and made explicit rather than included implicitly in a catchall category, people are more likely to consider those causes in assessing probability; according to Fischhoff et al. (1978), "what is out of sight is also out of mind" (p. 333).

Our goal in this paper is to extend the investigation of pruning bias from fault trees to the more general problem of probability assessment of event trees. Our studies suggest that the traditional availability-based account does not fully explain pruning bias or the more general phenomenon of partition dependence. We propose an alternative mechanism: a judge begins with equal probabilities for all events to be evaluated and then adjusts this uniform distribution based on his or her beliefs about how the likelihoods of the events differ. Bias arises because the adjustment is typically insufficient. Although current best practices in subjective probability elicitation are designed to guard against availability and the other major causes of pruning bias that have been previously advanced in the literature, such best practices provide inadequate protection against a more pervasive tendency to anchor on equal probabilities. Understanding the nature and causes of partition dependence can help analysts identify conditions under which this bias may arise, predict conditions that may exacerbate or mitigate the effect, and develop more effective debiasing techniques.

[FIGURE 1 OMITTED]

In [section]2, we review literature on pruning bias and partition dependence. In [section]3, we describe a series of studies that document the robustness of partition dependence across a variety of contexts beyond fault trees, provide support for our interpretation of this phenomenon, and cast doubt on the necessity of alternative accounts that have been proposed to explain pruning bias. We close with a discussion of the interpretation and robustness of partition dependence, other manifestations of this phenomenon, and prescriptive implications of our results.

2. Literature Review

Fischhoff et al. (1978) presented professional automobile mechanics and laypeople with trees that identified several categories of reasons why a car might fail to start, as well as a residual category of reasons labeled "all other problems." Participants were asked to estimate the number of times out of 1,000 that a car would fail to start for each of the categories of causes specified. When the experimenters removed (pruned) specific categories of causes from the tree (e.g., battery charge insufficient) and relegated them to the residual category as in Figure 1, the judged probability of the residual category, as assessed by a new a group of participants, did not increase by a corresponding amount. Instead, the probability from the pruned categories tended to be distributed across all of the remaining categories. Because the probability assigned to the residual category in the pruned tree was lower than the sum of probabilities of corresponding events in the unpruned tree, the pattern has subsequently come to be known as the pruning bias (e.g., Russo and Kolzow 1994).

Since the publication of Fischhoff et al. (1978), numerous authors have replicated and extended the basic result and proposed three major explanations for pruning bias: availability, ambiguity, and credibility. Below we review each of these accounts.

Availability. In explaining pruning bias, Fischhoff et al. (1978) invoked the availability heuristic (Tversky and Kahneman 1973): judged probabilities depend on the ease with which instances can be recalled or scenarios constructed. In the case of fault trees, explicitly mentioning a cause or category of causes will make that cause or category more salient, easing retrieval of related instances or construction of relevant scenarios, and hence leading to an increase in the corresponding judged probability. Support for such a mechanism has been provided by a number of researchers since Fischhoff et al. (1978), notably van der Pligt et al. (1987), Dube-Rioux and Russo (1988), Russo and Kolzow (1994), and Ofir (2000). (1)

Ambiguity. Hirt and Castellan (1988) argued that some categories of problems in Fischhoff et al. (1978) are ambiguous. For example, suppose that the branch labeled "battery charge insufficient" were removed from the tree. Specific causes that might fit into that category, such as "faulty ground connection" or "loose connection to alternator," could just as well be assigned to a remaining branch labeled "ignition system defective" as to the residual "all other causes" category. Such ambiguous mapping of specific causes to categories could give rise to the observed pattern in which probabilities of pruned branches are distributed across remaining branches.

Credibility. A third explanation of the pruning bias is that people assume that a credible real-world fault tree would list enough possible causes so that the catchall category would be relatively unlikely, and each explicitly listed cause should have a nontrivial probability (Dube-Rioux and Russo 1988, Fischhoff et al. 1978). This argument suggests that the pruning bias represents a demand effect (Clark 1985, Grice 1975, Orne 1962), whereby a participant considers the assessment as an implicit conversation with the experimenter in which the experimenter is expected to adhere to accepted conversational norms, including the expectation that any contribution should be relevant to the aims of the conversation. In the case of fault trees, the probability assessor may presume that any branch (other than the catchall) for which a probability is solicited must have a nontrivial probability; otherwise the probability of that item would be irrelevant, and therefore the query would violate conversational norms.

Although each of the three foregoing accounts (availability, ambiguity, and credibility) may contribute to some instances of pruning bias, previous studies suggest that the availability mechanism is most robust, contributing to pruning bias even in situations where the other mechanisms can be ruled out (Fischhoff et al. 1978, Russo and Kolzow 1994). (2) We assert, however, that even availability does not provide an adequate explanation of pruning bias. In particular, the availability account predicts that there should be little or no effect of pruning causes from a full tree if these causes are explicitly mentioned as part of the catchall category (so that the pruned causes are no longer out of sight even though their probabilities are not assessed separately). However, when Fischhoff et al. (1978) did this (Study 5), they nevertheless observed a strong pruning bias--a result that has received surprisingly little subsequent attention in the literature and that begs for a new interpretation of the phenomenon.

Anchoring and Insufficient Adjustment. We propose a fourth mechanism driving pruning bias: people anchor on a uniform distribution of probability across all branches of the fault tree and adjust according to features that distinguish each branch. Because such adjustment is usually insufficient (Tversky and Kahneman 1974, Epley and Gilovich 2001), assessors are biased toward probabilities of 1/n for each of n branches in the tree. To illustrate, consider a fault tree consisting of seven branches plus a residual category. According to the anchoring account, the assessed probability of the residual will be biased toward 1/8 because it is one branch of eight. Now imagine pruning this tree so that three branches remain, plus a residual category. Although the residual subsumes five of the original eight branches, it now represents a single branch of four. The anchoring account predicts that the assessed probability of the residual in this pruned tree will be biased toward 1/4 rather than 5/8 and that the remaining branches will be biased toward 1/4 rather than 1/8.

Starting with equal probabilities for all branches can be interpreted as an intuitive application of the so-called principle of insufficient reason that has been attributed to Leibniz and Laplace (Hacking 1975). We say that a probability assessor adopts an ignorance prior, by which we mean a default judgment that branch probabilities are equal. Taking equal probabilities as a starting point, a probability assessor then adjusts (usually insufficiently) to account for his or her beliefs about how the likelihood of the events differ. Although we interpret this phenomenon in terms of anchoring and insufficient adjustment, a bias toward the ignorance prior may also be driven in some cases by enhanced accessibility of information that is consistent with an equal distribution of probability (Chapman and Johnson 2002) or the intrusion of error variance into the processing of frequency information (Fiedler and Armbruster 1994).

The anchoring hypothesis has not been extensively investigated, and the existing empirical evidence for it is rather indirect. Van Schie and van der Pligt (1990) asked undergraduates to estimate the proportion of acid rain that could be attributed to various causes and found that the cause "traffic" received a median rating of 14% in a (full) eight-branch tree and a median rating of 24% in a (pruned) four-branch tree, very close to the corresponding ignorance prior probabilities of 1/8 and 1/4, respectively. Johnson et al. (1991) asked undergraduates to judge the relative frequency of possible outcomes when a baseball player is at bat (e.g., single, double, out), the true values of which were known to the experimenters. Participants tended to underestimate relative frequencies when the corresponding ignorance prior was below the true value and overestimate relative frequencies when the corresponding ignorance prior was above the true value. Harries and Harvey (2000, pp. 441-442) obtained a similar result using a causes of death probability estimation task. Russo and Kolzow (1994, p. 26, footnote 13) asked participants "what should be" the probability of a residual category for a typical tree with different numbers (n) of labeled branches …