The Impact of Overcon dence Bias on Practical Accuracy of Bayesian Network Models: An Empirical Study

Abstract

In this paper, we examine the in uence of overcon dence in parameter speci cation on the performance of a Bayesian network model in the context of Hepar II, a sizeable Bayesian network model for diagnosis of liver disorders. We enter noise in the parameters in such a way that the resulting distributions become biased toward extreme probabilities. We believe that this o ers a systematic way of modeling expert overcon dence in probability estimates. It appears that the diagnostic accuracy of Hepar II is less sensitive to overcon dence in probabilities than it is to undercon dence and to random noise, especially when noise is very large.

INTRODUCTION

Decision-analytic methods provide an orderly and coherent framework for modeling and solving decision problems in decision support systems [ 5 ]. A popular modeling tool for complex uncertain domains is a Bayesian network [ 13 ], an acyclic directed graph quanti ed by numerical parameters and modeling the structure of a domain and the joint probability distribution over its variables. There exist algorithms for reasoning in Bayesian networks that typically compute the posterior probability distribution over some variables of interest given a set of observations. As these algorithms are mathematically correct, the ultimate quality of reasoning depends directly on the quality of the underlying models and their parameters. These parameters are rarely precise, as they are often based on subjective estimates. Even when they are based on data, they may not be directly applicable to the decision model at hand and be fully trustworthy. Search for those parameters whose values are critical for the overall quality of decisions is known as sensitivity analysis. Sensitivity analysis studies how much a model output changes as various model parameters vary through the range of their plausible values. It allows to get insight into the nature of the problem and its formalization, helps in re ning the model so that it is simple and elegant (containing only those factors that matter), and checks the need for precision in re ning the numbers [ 8 ]. It is theoretically possible that small variations in a numerical parameter cause large variations in the posterior probability of interest. Van der Gaag and Renooij [ 17 ] found that practical networks may indeed contain such parameters. Because practical networks are often constructed with only rough estimates of probabilities, a question of practical importance is whether overall imprecision in network parameters is important. If not, the e ort that goes into polishing network parameters might not be justi ed, unless it focuses on their small subset that is shown to be critical.

There is a popular belief, supported by some anecdotal evidence, that Bayesian network models are overall quite tolerant to imprecision in their numerical parameters. Pradhan et al. [ 14 ] tested this on a large medical diagnostic model, the CPCS network [ 7, 16 ]. Their key experiment focused on systematic introduction of noise in the original parameters (assumed to be the gold standard) and measuring the in uence of the magnitude of this noise on the average posterior probability of the true diagnosis. They observed that this average was fairly insensitive to even very large noise. This experiment, while ingenious and thought provoking, had two weaknesses. The rst of these, pointed out by Coupe and van der Gaag [ 3 ], is that the experiment focused on the average posterior rather than individual posterior in each diagnostic case and how it varies with noise, which is of most interest. The second weakness is that the posterior of the correct diagnosis is by itself not a su cient measure of model robustness. The weaknesses of this experiment were also discussed in [ 6 ] and [ 9 ]. In our earlier work [ 9 ], we replicated the experiment of Pradhan et al. using Hepar II, a sizeable Bayesian network model for diagnosis of liver disorders. We systematically introduced noise in Hepar II's probabilities and tested the diagnostic accuracy of the resulting model. Similarly to Pradhan et al., we assumed that the original set of parameters and the model's performance are ideal. Noise in the original parameters led to deterioration in performance. The main result of our analysis was that noise in numerical parameters started taking its toll almost from the very beginning and not, as suggested by Pradhan et al., only when it was very large. The region of tolerance to noise, while noticeable, was rather small. That study suggested that Bayesian networks may be more sensitive to the quality of their numerical parameters than popularly believed. Another study that we conducted more recently [ 4 ] focused on the in uence of progressive rounding of probabilities on model accuracy. Here also, rounding had an effect on the performance of Hepar II, although the main source of performance loss were zero probabilities. When zeros introduced by rounding are replaced by very small non-zero values, imprecision resulting from rounding has minimal impact on Hepar II's performance.

Empirical studies conducted so far that focused on the impact of noise in probabilities on Bayesian network results disagree in their conclusions. Also, the noise introduced in parameters was usually assumed to be random, which may not be a reasonable assumption. Human experts, for example, often tend to be overcon dent [ 8 ]. This paper describes a follow-up study that probes the issue of sensitivity of model accuracy to noise in probabilities further. We examine whether a bias in the noise that is introduced into the network makes a di erence. We enter noise in the parameters in such a way that the resulting distributions become biased toward extreme probabilities. We believe that this o ers a systematic way of modeling expert overcon dence in probability estimates. Our results show again that the diagnostic accuracy of Hepar II is sensitive to imprecision in probabilities. It appears, however, that the diagnostic accuracy of Hepar II is less sensitive to overcon dence in probabilities than it is to random noise. We also test the sensitivity of Hepar II to undercon dence in parameters and show that undercon dence in paramaters leads to more error than random noise.

The remainder of this paper is structured as follows. Section 2 introduces the Hepar II model. Section 3 describes how we introduced noise into our probabilities. Section 4 describes the results of our experiments. Finally, Section 5 discusses our results in light of previous work.

THE Hepar II MODEL

Our experiments are based on Hepar II [ 10, 11 ], a Bayesian network model consisting of over 70 variables modeling the problem of diagnosis of liver disorders. The model covers 11 di erent liver diseases and 61 medical ndings, such as patient self-reported data, signs, symptoms, and laboratory tests results. The structure of the model, (i.e., the nodes of the graph along with arcs among them) was built based on medical literature and conversations with domain experts and it consists of 121 arcs. Hepar II is a real model and it consists of nodes that are a mixture of propositional, graded, and general variables. There are on the average 1.73 parents per node and 2.24 states per variable. The numerical parameters of the model (there are 2,139 of these in the most recent version), i.e., the prior and conditional probability distributions, were learned from a database of 699 real patient cases. Readers interested in the Hepar II model can download it from Decision Systems Laboratory's model repository at http://genie.sis.pitt.edu/. As our experiments study the in uence of precision of Hepar II's numerical parameters on its accuracy, we owe the reader an explanation of the metric that we used to test the latter. We focused on diagnostic accuracy, which we de ned in our earlier publications as the percentage of correct diagnoses on real patient cases. When testing the diagnostic accuracy of Hepar II, we were interested in both (1) whether the most probable diagnosis indicated by the model is indeed the correct diagnosis, and (2) whether the set of w most probable diagnoses contains the correct diagnosis for small values of w (we chose a \window" of w=1, 2, 3, and 4). The latter focus is of interest in diagnostic settings, where a decision support system only suggest possible diagnoses to a physician. The physician, who is the ultimate decision maker, may want to see several alternative diagnoses before focusing on one. With diagnostic accuracy de ned as above, the most recent version of the Hepar II model reached the diagnostic accuracy of 57%, 69%, 75%, and 79% for window sizes of 1, 2, 3, and 4 respectively [ 12 ]. 3

INTRODUCTION OF NOISE

INTO Hepar II PARAMETERS

When introducing noise into parameters, we used essentially the same approach as Pradhan et al. [ 14 ], which is transforming each original probability into log-odds function, adding noise parametrized by a parameter (as we will show, even though is proportional to the amount of noise, in our case it cannot be directly interpreted as standard deviation), and transNow, we designed the Noise() function as follows. Given a discrete probability distribution Pr, we identify the smallest probability pS . We transform this smallest probability pS into p0S by making it even smaller, according to the following formula: p0S = Lo 1[Lo(pS ) jNormal(0; )j] : We make the largest probability in the probability distribution Pr, pL larger by precisely the amount by which we decreased pS , i.e., p0L = pL + pS p0S :

Figure 1 shows the e ect of introducing the noise. As we can see, the transformation is such that small probabilities are likely to become smaller and large probabilities are likely to become larger. Please note that distributions have become more biased towards the extreme probabilities. It is straightforward to prove that the entropy of Pr0 is smaller than the entropy of Pr. The transformed probability distributions re ect overcon dence bias, common among human experts. An alternative way of introducing biased noise, suggested by one of the reviewers, is by means of building a logistic regression/IRT model (e.g., [ 1, 2, 15 ])for each conditional probability table and, subsequently, manipulating the slope parameter. 3.2

Undercon dence bias

Now, we designed the Noise() function as follows. Given a discrete probability distribution Pr, we identify the highest probability pS . We transform this largest probability pL into p0L by making it smaller, according to the following formula: p0L = Lo 1[Lo(pL) jNormal(0; )j] : We make the smallest probability in the probability distribution Pr, pS larger by precisely the amount by which we decreased pL, i.e., p0S = pS + pL p0L : improved upon. In the experiment, we studied how this baseline performance degrades under the condition of noise, as described in Section 3. For illustration purpose, Figure 3 shows the transformation applied in our previous study [ 9 ]. For > 1 the amount of noise becomes so large that any value of probability can be transformed in any other value. This suggests strongly that > 1 is not really a region that is of interest in practice. The main reason why we look at such high values is that this was the range used in Pradhan et al. paper. 4

EXPERIMENTAL RESULTS

We have performed an experiment investigating the in uence of biased noise in Hepar II's probabilities on its diagnostic performance. For the purpose of our experiment, we assumed that the model parameters were perfectly accurate and, e ectively, the diagnostic performance achieved was the best possible. Of course, in reality the parameters of the model may not be accurate and the performance of the model can be We tested 30 versions of the network (each for a different standard deviation of the noise 2< 0:0; 3:0 > with 0.1 increments) on all records of the Hepar data set and computed Hepar II's diagnostic accuracy. We plotted this accuracy in Figures 4 and 5 as a function of for di erent values of window size w.

It is clear that Hepar II's diagnostic performance deteriorates with noise. In order to facilitate comparison between biased and unbiased noise and, by this, judgment of the in uence of overcon dence bias on the results, we reproduce the experimental result of [ 9 ] in Figure 6. The results are qualitatively similar, although it can be seen that performance under overcon dence bias degrades more slowly with the amount of noise than performance under random noise. Performance under undercon dence bias degrades the fastest of the three. Figure 7 shows the accuracy of Hepar II (w = 1) for biased and unbiased noise on the same plot, where this e ect is easier to see.

It is interesting to note that for small values of , such as < 0:2, there is only a minimal e ect of noise on performance. This observation may o er some justication to the belief that Bayesian networks are not too sensitive to imprecision of their probability parameters. 5

SUMMARY

This paper has studied the in uence of bias in parameters on model performance in the context of a practical medical diagnostic model, Hepar II. We believe that the study was realistic in the sense of focusing on a real, context-dependent performance measure. Our study has shown that the performance of Hepar II is sensitive to noise in numerical parameters, i.e., the diagnostic accuracy of the model decreases after introducing noise into numerical parameters of the model. While our result is merely a single data point that sheds light on the hypothesis in question, it looks like overcon dence bias has a smaller negative e ect on model performance than random noise. Undercon dence bias leads to most serious deterioration of performance. While it is only a wild speculation that begs for further investigation, one might see our results as an explanation of the fact that humans tend to be overcon dent rather than undercon dent in their probability estimates.

Acknowledgments

This work was supported by the Air Force O ce of Scienti c Research grant FA9550-06-1-0243, by Intel Research, and by the MNiI (Ministerstwo Nauki i Informatyzacji) grant 3T10C03529. We thank Linda van der Gaag for suggesting extending our earlier work on sensitivity of Bayesian networks to precision of their numerical parameters by introducing bias in the noise. Reviewers for The Sixth Bayesian Modelling Applications Workshop provided several useful suggestions that have improved the readability and extended the scope of the paper.

The Hepar II model was created and tested using SMILE, an inference engine, and GeNIe, a development environment for reasoning in graphical probabilistic models, both developed at the Decision Systems Laboratory, University of Pittsburgh, and available at http://genie.sis.pitt.edu/. We used SMILE in our experiments and the data pre-processing module of GeNIe for plotting scatter plot graphs in Figure 1.

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