A problem (and solution) with estimating rare event frequencies

Industry uses many numbers in process safety associated with predicting the likelihood of rare, catastrophic events (e.g., failure rates, demand rates, incident rates, probability of failure, probability of ignition, etc.). Yet have you given serious thought to the accuracy and trustworthiness of those numbers? For example, layer of protection analysis (LOPA) often uses target numbers in the range of 10-4 per year or lower. How can you determine or prove whether you’re actually meeting these numbers? How can you use traditional frequentist-based statistics to make inferences about rare events that haven’t happened yet in your plant? After all, it’s neither practical nor ethical to determine rare event frequencies of catastrophic accidents by experiment.

Instead, your achieving such rare event frequencies must be inferred. And there are many inferences that must be made when arriving at a calculated LOPA number. For example, the assumed probability of failure of each safety layer is an inference. And the data you may be using for the inferences may not even be from your plant. Yet you’re interested in your own situation, not everyone else’s. What to do?

Enter Bayes. Bayes’ theorem (a.k.a. Bayes’ rule) provides the likelihood of occurrence of one-off events. Bayes’ probability is not defined by long-run, frequentist-based averages. In Bayes’ rule, qualitative knowledge of a process can be used to quantify the uncertainty of assumptions about the process. It can help you quantify the odds of the proverbial safety dice before you throw them.

A Bayesian engine in simplified form.

Bayes’ rule works with sparse data. It’s a simple formula for updating current beliefs based on new evidence (data that is both quantitative and qualitative) as it trickles in, as shown in the accompanying figure. It treats parameters as variables, not fixed-point values. It provides a way to update a parameter as new evidence (data) is gathered, as opposed to waiting potentially decades to pool enough data to make a valid frequentist inference. Bayes’ rule is also able to account for information that may not be showing up in your data.

As new data and evidence trickles in, use of Bayes’ rule can provide the most consistent and rational method to update your current beliefs about safe operation. It’s the best we can do in a complex changing operating environment where we can’t afford to wait decades to gather enough data to use frequentist-based methods.

To learn more about how Bayes’ rule can be applied in process safety, read the full paper “Reverend Bayes, meet Process Safety” .

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