Manufacturing quality depends on keeping process outputs within tight specifications. Traditional statistical process control (SPC) relies on Shewhart charts, CUSUM procedures, and hypothesis tests rooted in frequentist statistics. Bayesian quality control reframes these problems as inference about process parameters — means, variances, defect rates — given both observed data and prior knowledge from engineering specifications, previous production runs, and physical understanding of the manufacturing process.
Bayesian Process Monitoring
A Bayesian control chart monitors a process by computing the posterior probability that the process has shifted away from its target. Rather than flagging a point as "out of control" when it exceeds a fixed control limit, the Bayesian approach maintains a running posterior probability of a shift, which accumulates evidence over successive observations. This naturally handles the tradeoff between detecting small shifts quickly (sensitivity) and avoiding false alarms (specificity).
An alarm is triggered when P(shift | data) exceeds a decision threshold,
integrating evidence across multiple observations.
The Bayesian approach unifies many classical SPC methods. The CUSUM chart, for example, is the sequential probability ratio test — which has a direct Bayesian interpretation as computing the posterior odds of a shift. Bayesian methods extend naturally to multivariate process monitoring, where correlations among quality characteristics must be considered.
Acceptance Sampling
Acceptance sampling decides whether to accept or reject a batch of products based on inspecting a sample. Bayesian acceptance sampling computes the posterior distribution of the batch defect rate given the sample data and a prior informed by the supplier's quality history. The decision to accept or reject minimizes expected loss — balancing the cost of accepting a bad batch against the cost of rejecting a good one. This is more economically rational than the traditional approach of fixing producer's and consumer's risk levels independently.
Six Sigma's DMAIC framework (Define, Measure, Analyze, Improve, Control) can be enhanced at every stage by Bayesian methods. In the Measure phase, Bayesian gauge R&R studies estimate measurement system variation with proper uncertainty. In the Analyze phase, Bayesian regression and ANOVA identify significant factors while incorporating prior knowledge about effect directions and magnitudes. In the Control phase, Bayesian control charts provide probabilistic monitoring. The key advantage is that Bayesian methods produce direct probability statements about process parameters, rather than requiring the indirect reasoning of p-values and confidence intervals.
Capability Analysis
Process capability indices like Cp and Cpk measure how well a process meets specifications. Bayesian capability analysis provides posterior distributions for these indices, giving probability statements like "there is a 95% probability that Cpk exceeds 1.33." This is far more useful for decision-making than a point estimate with a confidence interval, particularly for small sample sizes common in short production runs or expensive testing.
Designed Experiments in Manufacturing
Bayesian design of experiments (DOE) in manufacturing settings uses prior knowledge about factor effects to design more efficient experiments. When screening many factors, Bayesian variable selection identifies active factors while naturally accounting for the multiple comparison problem. Bayesian response surface methods optimize process settings while properly quantifying uncertainty in the predicted response at untested conditions.
"Quality improvement is fundamentally about learning — and Bayesian statistics is the mathematics of learning from data combined with prior knowledge." — George E. P. Box, pioneer of statistical quality control and Bayesian thinking
Current Frontiers
Industry 4.0 and smart manufacturing generate high-frequency sensor data suited to online Bayesian monitoring. Bayesian machine learning models predict product quality from in-process measurements, enabling real-time adjustment. And Bayesian methods for functional data — monitoring entire profiles, curves, or images rather than single measurements — address the increasingly complex quality characteristics of advanced manufacturing.