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As classical definitions of correlation give rise to counterintuitive statements when extreme probability events are involved, we introduce enhanced notions of positive and negative correlation in the general framework of coherent conditional probability. These notions allow to handle extreme probability events in a principled way by accommodating the different levels of strength of the zero probabilities involved (namely, zero layers). Since the detection of correlations by means of zero layers is computationally challenging, we provide a full characterization relying on only conditional probability values. [ABSTRACT FROM AUTHOR]

Automatic milking systems (AMS) generate alert lists reporting cows likely to have clinical mastitis (CM). Dutch farmers indicated that they use non-AMS cow information or the detailed alert information from the AMS to decide whether to check an alerted cow for CM. However, it is not yet known to what extent such information can be used to discriminate between true-positive and false-positive alerts. The overall objective was to investigate whether selection of the alerted cows that need further investigation for CM can be made. For this purpose, non-AMS cow information and detailed alert information were used. During a 2-yr study period, 11,156 alerts for CM, including 159 true-positive alerts, were collected at one farm in the Netherlands. Non-AMS cow information on parity, days in milk, season of the year, somatic cell count history, and CM history was added to each alert. In addition, 6 alert information variables were defined. These were the height of electrical conductivity, the alert origin (electrical conductivity, color, or both), whether or not a color alert for mastitic milk was given, whether or not a color alert for abnormal milk was given, deviation from the expected milk yield, and the number of alerts of the cow in the preceding 12 to 96 h. Subsequently, naive Bayesian networks (NBN) were constructed to compute the posterior probability of an alert being truly positive based only on non-AMS cow information, based on only alert information, or based on both types of information. The NBN including both types of information had the highest area under the receiver operating characteristic curve (AUC; 0.78), followed by the NBN including only alert information (AUC = 0.75) and the NBN including only non-AMS cow information (AUC = 0.62). By combining the 2 types of information and by setting a threshold on the computed probabilities, the number of false-positive alerts on a mastitis alert list was reduced by 35%, and 10% of the true-positive alerts would not be identified. To detect CM cases at a farm with an AMS, checking all alerts is still the best option but would result in a high workload. Checking alerts based on a single alert information variable would result in missing too many true-positive cases. Using a combination of alert information variables, however, is the best way to select cows that need further investigation. The effect of adding non-AMS cow information on making a distinction between true-positive and false-positive alerts would be minor. [ABSTRACT FROM AUTHOR]

Clinical mastitis (CM) can be caused by a wide variety of pathogens and farmers must start treatment before the actual causal pathogen is known. By providing a probability distribution for the causal pathogen, naive Bayesian networks (NBN) can serve as a management tool for farmers to decide which treatment to use. The advantage of providing a probability distribution for the causal pathogen, rather than only providing the most likely causal pathogen, is that the uncertainty involved is visible and a more informed treatment decision can be made. The objective of this study was to illustrate provision of probability distributions for the gram status and for the causal pathogen for CM cases. For constructing the NBN, data were used from 274 Dutch dairy herds in which the occurrence of CM was recorded over an 18-mo period. The data set contained information on 3,833 CM cases. Two-thirds of the data set was used for the construction process and one-third was retained for validation. One NBN was constructed with the CM cases classified according to their gram status, and another was built with the CM cases classified into streptococci, Staphylococcus aureus, or Escherichia coli. Information usually available at a dairy farm was included in both NBN (parity, month in lactation, season of the year, quarter position, SCC and CM history, being sick or not, and color and texture of the milk). Accuracy was calculated to obtain insight in the quality of the constructed NBN. The accuracy of classifying CM cases into gram-positive or gram-negative pathogens was 73%, while the accuracy of classifying CM cases into streptococci, Staph. aureus, or E. coli was 52%. Because only CM cases with a high probability for a single causal pathogen will be considered for pathogen-specific treatment, accuracies based on only classifying CM cases above a particular probability threshold were determined. For instance, for CM cases in which either gram-negative or gram-positive had a probability >0.90, classification according to the gram status reached an accuracy of 97%. We found that the greater the probability for a particular pathogen was for a CM case, the more accurate was the classification of this case as being caused by this pathogen. The probability distributions provided by the NBN and the associated accuracies for varying classification thresholds provide the farmer with considerable insight about the most likely causal pathogen for a CM case and the uncertainty involved. [ABSTRACT FROM AUTHOR]

In artificial intelligence research, the belief network framework for automated reasoning with uncertainty is rapidly gaining in popularity. The framework provides a powerful formalism for representing a joint probability distribution on a set of statistaical variables. In addition, it offers algorithms for efficient probabilistic inference. At present, more and moe knowledge-based systems employing the framework are being developed for various domains of application ranging from probabilistic information retrieval to medical diagnosis. This paper provides a tutorial introduction to the belief network framework and highlights some issues of ongoing research in applying the framework for real-life problem solving. [ABSTRACT FROM AUTHOR]