Introduction to bayesian inference and its application in medical biology

In statistics the data become available via a random number generator mechanism, known as distribution, that depends on some unknown parameter(s). Two schools of thought exist in statistics that handle the unknown parameter(s) with a different philosophy: the Frequentist and the Bayesian. In the frequentist-based approach the unknown parameter is treated as a fixed unknown constant. The probability assessment is then based on the long-term frequency of an infinite number of replications of the experiment. On the contrary, the bayesian approach uses probability theory to quantify the unknown parameter. Precisely, adopting a subjective bayesian approach, we can incorporate into the prior distribution for the unknown parameter(s) any relevant information that we have (from our experience, from expert’s opinion etc.). In absence of prior knowledge, a non-informative (objective) prior could be employed. Once the data will arrive, the Bayes theorem will update the prior to posterior distribution and provide via decision theory a more user-friendly environment for statistical inference. Furthermore, one can derive the predictive distribution where we can talk about the uncertainty of future observable(s), given the available data only, i.e. the unknown parameter has been integrated out of the problem. Apart from better interpretability, the bayesian methods are also sequentially updated, offering the option to build methods that are up and running with even very few data (i.e. they are considered as self-starting methods). Furthermore, as more data become available the effect of the prior distribution decreases. In the field of Statistical Process Control and Monitoring the bayesian perspective provides unique tools that allow to perform (internal or external) quality control in an online fashion. The bayesian tools are capable of being self-starting, that is they break free from the necessity of a startup (calibration) phase, which is needed from the basic frequentist-based control charts. Within the bayesian framework two major tools for quality monitoring that are based on the predictive distribution have been developed: PCC: Predictive Control Charts [1], for detecting transient shifts of large magnitude (outliers) and the PRC: Predictive Ratio Cusum [2, 3]), for detection of persistent shifts of medium/small size (Predictive Residual Cusum is the special case of PRC, when we handle normal data). Jointly PCC and PRC provide a framework that is capable to provide efficient quality monitoring, outperforming standard frequentist-based alternatives.