Why bayesian networks

If we know something about an intermediate variable C , the model splits, and each side of the chain turns conditionally dependent given the observed variable—now A has an effect on B Fig. Bayesian networks are statistical tools to model the qualitative and quantitative aspects of complex multivariate problems and can be used for diagnostics, classification and prediction.

Time series and feedback loops, common in biological systems, can be modeled by using dynamic Bayesian networks, which allow cycles. One of the most interesting fields where Bayesian networks are used is the identification of 'latent' structures of relations in big databases 4. Learning a Bayesian network automatically by estimating the nodes, edges and associated probabilities from data is difficult, but it can help to discover unsuspected relations between, for example, genes and diseases.

Needham, C. PLoS Comput. Article Google Scholar. Beaumont, M. Puga, J. Methods 12 , — Nagarajan, R. Book Google Scholar. Download references. You can also search for this author in PubMed Google Scholar. Reprints and Permissions. Bayesian networks. Nat Methods 12, — Download citation. Published : 28 August Issue Date : September Anyone you share the following link with will be able to read this content:.

Sorry, a shareable link is not currently available for this article. Provided by the Springer Nature SharedIt content-sharing initiative. Nature Methods Advanced search. The collected data set was used to construct Bayesian networks for predicting post-stroke outcomes. We extracted a total of 76 random variables of each instance for patient data. A Bayesian network consists of a directed acyclic graph whose nodes represent random variables and links express dependences between nodes.

P , a joint probability over V , is described as. Training Bayesian network classifiers is the process of parameter learning to find optimal Bayesian structures estimating parameter set of P that best represents given data set with labeled instances Given a data set D with variable V i , the observed distribution P D is described as a joint probability distribution over D. The learning process now measures and compares the quality of Bayesian networks to evaluate how well the represented distribution explains the given data set.

The log-likelihood is the basic common value used for measuring the quality of a Bayesian network as follows:. Diverse quality measurement methods have been investigated The algorithm searched the best Bayesian network based on the Bayesian information criterion 32 , Bayesian Dirichlet equivalence score 19 , Akaike information criterion AIC 33 , and the maximum description length MDL scores 30 , In this study, we used the MDL score to evaluate the quality of a Bayesian network.

The MDL score is described as. The smaller the MDL score, the better the network. The search algorithm, greedy hill-climbing algorithm 35 in our study, selects the best Bayesian network by calculating MDL scores of candidate networks. For the type of Bayesian network structure, we constructed tree-augmented network TAN structures that restrict the number of parents to two nodes The entire process of a Bayesian network-based prediction system is shown in Figure 1.

A total of 76 features were extracted from the Yonsei Stroke Registry and data preparation process filtered records with missing outcome variables and exclusion criteria. For feasible prediction service in clinical environment, we performed two different feature selection methods. Feature selection or dimension reduction is the process of reducing the number of random variables under consideration by obtaining a set of principal variables 37 , Feature selection improves the overfitting problem caused by irrelevant or redundant variables that may strongly bias the performance of the classifier.

The definition of feature selection in formal expression is described in Drugan and Wiering 30 and Hruschka et al. In many studies, feature selection methods are categorized into filters, wrappers, or embedded methods that are applied to the data set in advance of the training learning algorithm, or to embed feature selection in the learning process 37 , Filter methods select features based on a performance measure regardless of the employed data modeling algorithm.

The filter approach selects random variables based on information gain score, ReliefF, or correlation-based method by ranking variables or searching subset of variables. Information gain measures the amount of entropy as a measure of uncertainty reduced by knowing a feature 41 — 43 ; ReliefF evaluates the worth of an attribute by repeatedly sampling an instance and considering the value of the given attribute for the nearest instance of the same and the different class 44 , 45 ; and correlation evaluates the worth of a subset of attributes by considering the individual predictive ability of each feature along with the degree of redundancy between them 46 , Unlike the filter approach, wrapper methods measure the usefulness of a subset of features by actually training a model on it.

We evaluated the performance of Bayesian networks with a reduced variable set selected by information gain and Bayesian network algorithms that are popular in filter and wrapper methods 42 , 48 , First, we tested the Bayesian network classifier with features chosen by information gain based on entropy of each feature. The other feature selection method, considering the characteristics of Bayesian network classifiers, reduces the variable set by evaluating the performance of the Bayesian network classifier in cross-validation in which a search algorithm extracts a subset of attributes to maximize AUC in prediction Figure 1.

The optimization for AUC is to solve the imbalance between the number of survival and mortal subjects. Using the reduced variables by feature selection, the system constructed a Bayesian network prediction model to search optimal Bayesian network structures and parameters. We evaluated the performance of prediction algorithms using 1 a basic tree-augmented Bayesian network, 2 a tree-augmented Bayesian network with features filtered by information gain, and 3 a tree-augmented Bayesian network with features filtered by the wrapper of a Bayesian network.

The performances of all Bayesian networks and predictive models were evaluated based on the AUC, specificity, and sensitivity of fold cross-validations We also implemented an online prediction system for post-stroke outcomes embedding the trained classifiers.

In the validation process, we bound the minimum sensitivity as 0. During the study period, 4, consecutive patients with acute ischemic stroke or transient ischemic attack were registered to the Yonsei Stroke Registry. After exclusion, a total of 3, patients were finally enrolled for this study. The mean age was A comparison of demographic characteristics between the outcome at 3 months and death within 1 year is shown at Table 1.

Patients with poor outcome were older, more likely to be women, not a current smoker, frequently had old stroke, hypertension, atrial fibrillation, congestive heart failure, peripheral artery obstructive disease, or anemia. Thrombolysis or endovascular mechanical thrombectomy, symptomatic intracranial hemorrhage, and herniation are frequent in patients with poor outcome. Laboratory data showed that patients with poor outcome showed lower hemoglobin, hematocrit, albumin, prealbumin, body weight and higher ESR, fibrinogen, hsCRP, and D-dimer level.

The differences of demographics of patients between survival and death within 1 year were similar with functional outcome at 3 months. D-dimer levels were significantly higher in patients who died within 1 year compared with survivors Table 1.

Demographic characteristics and comparison of outcome at 3 months and death within 1 year. As we described in Figure 1 , two different feature selection techniques were performed in our experiment: variables selected by information gain with ranking or variables selected by a wrapper embedding Bayesian network with greedy stepwise subset selection in cross-validation. The top-ranked variables in the filter by information gain and the wrapper of the Bayesian network in forecasting functional independence at 3 months are shown in Figures 2A,B , and variables for predicting 1-year mortality are shown in Figures 2C,D.

The most affective factor for functional recovery prediction was Initial NIHSS, while D-dimer ranked top in 1-year mortality prediction. However, the subset-searching algorithm selects a method differently from the ranking method that evaluates the individual variables separately; thus, certain variables were excluded from the selected subset even though their ranks are high in individual evaluation.

Figure 2. Top 15 variables in dimension reduction for post-stroke outcome prediction: A variables filtered by ranks of information gain for predicting functional independence at 3 months, B variables selected by the wrapper of the Bayesian network classifier with greedy subset selection for predicting functional independence at 3 months, C variables filtered by ranks of information gain for predicting 1-year mortality, and D variables selected by the wrapper of the Bayesian network classifier with greedy subset selection for predicting 1-year mortality.

Using the result of feature selection, we trained three tree-augmented Bayesian network classifiers; 1 Tree-augmented Bayesian network with the entire dataset, 2 tree-augmented Bayesian network with features filtered by ranking of information gain, and 3 tree-augmented Bayesian network with features filtered by the wrapper of the Bayesian network classifier see Figure 3.

The predictive performance for 3-month outcomes is shown in Figure 3A. The classifier trained with features chosen by the Bayesian network's subset evaluation performs in prediction of 3-month functional recovery with the specificity of 0. The tree-augmented Bayesian network without feature selection achieved the AUC of 0.

The Bayesian network classifier with feature selection achieved best performance in most metrics except sensitivity, although it reduced the variable set from 76 variables to 19 variables, resulting in a great reduction in model construction time.

Figure 3. Performance evaluation of Bayesian network-based classifiers: A performance of classifiers forecasting day functional independence and B performance of classifiers for 1-year mortality prediction. In prediction of 1-year mortality, AUCs of three algorithms were not significantly different 0. All algorithms achieved higher specificities in predicting 1-year mortality than those for the prediction of functional independence 0.

The Bayesian network algorithm with feature selection for 1-year mortality cuts out the entire variable set to 24 variables that curtail network construction time. The final Bayesian networks predicting functional recovery and 1-year mortality are shown in Figures 4 , 5 , respectively.

Figure 4. Bayesian network for predicting functional independence at 3 months. The tree-augmented Bayesian network used 19 variables selected by the wrapper of the Bayesian network for prediction.

Figure 5. Bayesian network for predicting 1-year mortality. The tree-augmented Bayesian network used 24 variables selected by the wrapper of the Bayesian network for prediction. Figure 6. Interpretability is a core requirement for machine learning models in medicine, because both patients and physicians need to understand the reason behind a prediction This study presents an evaluation of Bayesian networks in providing post-stroke outcomes estimates based on the collected demographic data, lab result, and initial neurological assessment.

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Rajapakse, J. Learning effective brain connectivity with dynamic Bayesian networks. Neuroimage 37, — The problem is that joint probability distribution, particularly over discrete variables, can be very large. Consider a network with 30 binary discrete variables. Binary simply means a variable has 2 states e. The joint probability would require 2 30 entries which is a very large number. This would not only require large amounts of memory but also queries would be slow.

Bayesian networks are a factorized representation of the full joint. This just means that many of the values in the full joint can be computed from smaller distributions.

This property used in conjunction with the distributive law enable Bayesian networks to query networks with thousands of nodes. The Distributive law simply means that if we want to marginalize out the variable A we can perform the calculations on the subset of distributions that contain A.

The distributive law has far reaching implications for the efficient querying of Bayesian networks, and underpins much of their power. Bayes theorem allows us to update our belief in a distribution Q over one or more variables , in the light of new evidence e.

The term P e is the Probability of Evidence , and is simply a normalization factor so that the resulting probability sums to 1. This is because, given that we know e , P e Q is a measure of how likely it is that Q caused the evidence.

Yes and no. They do make use of Bayes Theorem during inference, and typically use priors during batch parameter learning. However they do not typically use a full Bayesian treatment in the Bayesian statistical sense i. The matter is further confused, as Bayesian networks typically DO use a full Bayesian approach for Online learning. Inference is the process of calculating a probability distribution of interest e. The terms inference and queries are used interchangeably.

The following terms are all forms of inference will slightly difference semantics. Importantly Bayesian networks handle missing data during inference and also learning , in a sound probabilistic manner. Exact inference is the term used when inference is performed exactly subject to standard numerical rounding errors.

Is is often possible to refactor a Bayesian network before resorting to approximate inference, or use a hybrid approach. There are a large number of exact and approximate inference algorithms for Bayesian networks.

Bayes Server supports both exact and approximate inference with Bayesian networks, Dynamic Bayesian networks and Decision Graphs. Bayes Server algorithms. Bayes Server also includes a number of analysis techniques that make use of the powerful inference engines, in order to extract automated insight, perform diagnostics, and to analyze and tune the parameters of the Bayesian network.

Dynamic Bayesian networks DBNs are used for modeling times series and sequences. They extend the concept of standard Bayesian networks with time. For more information please see the Dynamic Bayesian network help topic.

Once you have built a model, often the next step is to make a decision based on that model. In order to make those decisions, costs are often involved. The problem with doing this manually is that there can be many different decisions to make, costs to trade off against each other, and all this in an uncertain environment i. Decision Graphs are an extension to Bayesian networks which handle decision making under uncertainty. For more information please see the Decision Graphs help topic.

Bayesian networks - an introduction This article provides a general introduction to Bayesian networks.