# Metrics Related to Confusion Matrix as Tools for Conformity Assessment Decisions
## 1 Introduction
Figure 2: A different relationship between acceptance interval and tolerance interval: (**a**) Model of the producer’s risk minimization; (**b**) Shared risk; (**c**) Model of the consumer’s risk minimization. The symbols \(A_{L}\) and \(A_{U}\) in figures are marks for the lower and upper limits of the acceptance interval, respectively. global producer's risk. In contrast to the specific risk related to a particular tested item, the global risk of producers and consumers refers to the risks calculated for a sample drawn from some statistical population of such items obtained in some production process [1]. Foremost, the calculation of risk, specific and global, is carried out due to the improvement of the production process, the reduction of scrap, i.e., the number of products that do not meet the given standards, and the reduction of the producer's costs. Also, consumer risk assessment provides the probability of purchasing a defective product. The global consumers' and producers' risks are calculated using the Bayesian approach [11; 12]. The Bayesian approach combines two sources of information. The first source of information refers to prior beliefs of a measurer about the possible distribution of parameters that describes the measurement data. In addition, such a source of information can be the results of previous measurements, data from manuals, historical data, etc. [13]. These data are treated as a random variable Y, and they are expressed by the probability density function (PDF), which is usually called a prior, denoted by \(g_{0}(\eta)\). The values that take by the prior are marked with \(\eta\). According to the principle of maximum entropy (PME), when the first two moments of the distribution are known, in this case, that is the best estimate of a measurand \(\overline{y}\), and the standard measurement uncertainty \(u_{0}\), the prior is usually a two-parameter distribution [14]. Considering that these data are an outcome of a measurement, the prior is usually a normal distribution. Depending on the nature of the measurand and the number of known moments, other two-parameter distributions can be used, as well as one-parameter or non-parametric distributions [9; 15]. The second source of information is data assigned to the measurements. These data are also regarded as a random variable \(Y_{m}\), and they are expressed via the likelihood function for normal distribution denoted by \(h(\eta_{m}|\eta)\). Likelihood function formula includes standard measurement uncertainty \(u_{m}\). Values of the likelihood function are denoted by \(\eta_{m}\). It is important to notice that according to [PERSON] rule, the prior distribution is independent of the likelihood function, so data for one and the other function must come from different sources. Thus, data for the likelihood function are usually obtained from subsequent production process quality control [1] (pp. 27-32). Determining the global risk of producers and consumers according to the Bayesian approach requires the numerical solution of the double integral. This procedure involves knowledge of programming that can be difficult for users to apply, especially if Monte Carlo methods are required for the posterior distribution simulation [6; 16; 17]. Therefore, risk calculation is often avoided in the quality control of the production process. However, such an attitude is wrong. Since the risks of the consumer and producer are related, the producer can reduce his costs by assessing the risk and, at the same time, guaranteeing the consumer a certain level of product quality. The conformity assessment procedure can be applied in a wide range of disciplines. For example: in the assessment of the efficiency of liquid chromatography analytical procedures [18], as a tool in the pharmaceutical industry for drug quality control [19], for the evaluation of the risk of a false decision on compliance of multicomponent materials [20], for quality evaluation of automotive fuels [21], or in a food control [22].