استنتاج در شبکه های بیزی هیبریدی

Since the 1980s, Bayesian networks (BNs) have become increasingly popular for building statistical models of complex systems. This is particularly true for boolean systems, where BNs often prove to be a more efficient modelling framework than traditional reliability techniques (like fault trees and reliability block diagrams). However, limitations in the BNs’ calculation engine have prevented BNs from becoming equally popular for domains containing mixtures of both discrete and continuous variables (the so-called hybrid domains). In this paper we focus on these difficulties, and summarize some of the last decade's research on inference in hybrid Bayesian networks. The discussions are linked to an example model for estimating human reliability.

A reliability analyst will often find himself making decisions based on uncertain information. Examples of decisions he may need to take include defining a maintenance strategy or choosing between different system configurations for a safety system. These decisions are typically based on only limited knowledge about the failure mechanisms that are in play and the environment the system will be deployed in. This uncertainty, which can be both aleatory and epistemic, requires the analyst to use a statistical model representing the system in question. This model must be mathematically sound, and at the same time easy to understand for the reliability analyst and his team. To build the models, the analyst can employ different sources of information, e.g., historical data or expert judgement. Since both of these sources of information can have low quality, as well as come with a cost, one would like the modelling framework to use the available information as efficiently as possible. Finally, the model must be encoded such that the quantities we are interested in (e.g., the availability of a system) can be calculated efficiently. All of these requirements have led to a shift in focus, from traditional frameworks, like fault trees, to more flexible modelling frameworks. One such framework for building statistical models for complex systems is the Bayesian network (BN) framework [1], [2] and [3]. BNs have gained popularity over the last decade [4], partly because a number of comparisons between BNs and the classical reliability formalisms have shown that BNs have significant advantages [5], [6], [7], [8], [9] and [10]. BNs consist of a qualitative part, an acyclic directed graph, where the nodes mirror stochastic variables and a quantitative part, a set of conditional probability functions (CPFs). An example of the qualitative part of a BN is shown in Fig. 1. This BN models the risk of an explosion in a process system. An explosion (Explosion?) might occur if there is a leak (Leak?) of chemical substance that is not detected by the gas detection (GD) system. The GD system detects all leaks unless it is in its failed state (GD Failed?). The environment (Environment?) will influence the probability of a leak as well as the probability of a failure in the GD system. Finally, an explosion may lead to a number of casualties (Casualties?). Full-size image (11 K) Fig. 1. An example BN describing a gas leak scenario. Only the qualitative part of the BN is shown. Figure options The graphical structure has an intuitive interpretation as a model of causal influences. Although this interpretation is not necessarily entirely correct, it is helpful when the BN structure is to be elicited from experts. Furthermore, it can also be defended if some additional assumptions are made [11]. BNs originated as a robust and efficient framework for reasoning with uncertain knowledge. The history of BNs in reliability can (at least) be traced back to [12] and [13]; the first real attempt to use BNs in reliability analysis is probably the work of Almond [13], where he used the GRAPHICAL-BELIEF tool to calculate reliability measures concerning a low pressure coolant injection system for a nuclear reactor (a problem originally addressed by Martz [14]). BNs constitute a modelling framework which is particularly easy to use in interaction with domain experts, also in the reliability field [15]. BNs have found applications in, e.g., fault detection and identification, monitoring, software reliability, troubleshooting systems, and maintenance optimization. Common to these models are that all variables are discrete. As we shall see in Section 3, there is a purely technical reason why most BN models fall in this class. However, in Section 4 we introduce a model for human reliability analysis, where both discrete and continuous variables are in the same model. Attempts to handle such models are considered in Section 5 and we conclude in Section 6.

In this paper we have explored four approaches to inference in hybrid BNs: discretization, mixtures of truncated exponentials (MTEs), variational methods, and Markov chain Monte Carlo (MCMC). Each of them have their pros and cons, which we will briefly summarize here. We note that this paper is about inference, hence the specification of the models (either manually or by learning from data) is outside the scope of this discussion. Furthermore, we have considered the inference problem in the context of reliability analysis. This means that we are interested in obtaining good approximations for low probability events, and will therefore give the tails of the approximations some attention in the following. The simplest approach to inference in hybrid domains is to use discretization. Discretization entails only a simple transformation of the continuous variables, it is implemented in almost all commercial BN tools, and the user only has to decide upon one parameter, namely m, the number of intervals the continuous variables are discretized into. Choosing a “good” value for m can be a bit of a problem, though, since a too high value leads to complexity problems and a too low value leads to poor approximations. We note that practitioners in reliability who use discretization without investigating this effect further are in danger of under-estimating the probability of unwanted events considerably. MTEs are generalizations of standard discretization, with the aim of avoiding the complexity problems discretization are hampered by. MTEs benefit from the BNs’ efficient inference engine. Furthermore, MTEs define a rather general framework, which can approximate any distribution accurately. In particular, MTEs are better at approximating the tail of the distribution of our running example than discretization (compare Figs. 5 and 6). MTEs are currently receiving a lot of attention from the research community; both refining the inference and exploring new applications are hot research topics. On the downside, the MTE framework is still in its infancy, and in particular methods for learning MTEs from data must be further explored. The variational approximations provide satisfactory answers to the kind of queries associated with inference in hybrid BNs. However, the variational approximations are still rather ad hoc, and the formulae have to be rewritten depending on the underlying distribution used. It is also difficult to have a well-founded understanding of the error the variational approximation makes, and as we saw in the example, the error can be substantial. MCMC is a very general inference technique, and it can take advantage of a BN's structure to speed up the simulation process. Together with standard discretization, MCMC is currently the most popular technique for inference in hybrid BNs. This is partly due to a strong mathematical foundation and well-known statistical properties of the generated estimates. From a practitioners point of view, one should, however, be vigilant when using MCMC to estimate the probability of rare events. If the probability of a gas leak, say, is p=10-4p=10-4 one would on average need to generate 1/p=1041/p=104 samples after burn-in to obtain a single sample of the event. It is also particularly important to consider the auto-correlation in the samples before conclusions regarding rare events are drawn. It is our experience that practitioners are not always aware of these facts, and sometimes abuse the methodology by underestimating the demands to obtain representative samples. In our experience, the discretization method (with moderate number of regions) is the fastest technique, outperforming the MTE method (with the same number of regions) by a factor of about 2. On the other hand, MTEs are about four times faster than the variational approximation. This is not surprising, as the variational approximation requires a number of iterations to converge (refer to Algorithm 1). MCMC is comparably much slower than MTEs (a factor of about 103103 to obtain results of comparable quality). Among the four explored approaches, the MTE framework appears to be the one best suited for reliability applications: it balances the need for good approximations in the tail of the distributions with not-too-high computational complexity. MTEs are flexible from the modelling point of view, and there exist efficient methods for inference building on the classical BN inference scheme.