## 3 Background  
Deduction in classical mathematical statistics is based on random samples drawn from a given population. In an alternative approach, derived from the [PERSON] theorem [5, 6, 7, 8, 9], deduction can be based not only on a random sample, but also on a priori information. The _a posteriori_ information is generated from these two sets of data. A priori knowledge may refer to expert knowledge, deriving from earlier investigations whose results are known, but not for the entire dataset. The character of knowledge does not allow us to take advantage of it in the classical approach. Therefore, the Bayesian approach should be used. If the same data can be used in both cases, classical and Bayesian analysis would give similar conclusions. In its most basic form, the [PERSON] theorem presents the dependence of the conditional probability of event \(A\) on condition \(B\) and the conditional probabilities of event \(B\) on condition \(A\), provided that event \(A\) is completed and the probability of event \(A\) and its complement, which can be expressed in the following form: \(\mathrm{P}(A\mid B)=\frac{\mathrm{P}(B|A)\cdot\mathrm{P}(A)}{\mathrm{P}(B|A) \mathrm{P}(A)+\mathrm{P}(B|A)\mathrm{P}(A)}\). The formula above is generalized to a situation in which the occurrence of many mutually exclusive events is considered, and not as in the given relationship between only event \(A\) and its complement: \(\mathrm{P}(\theta\mid x)=\frac{\mathrm{P}(\mathrm{i}\theta)\cdot\mathrm{P}( \theta)}{\sum_{i}\mathrm{P}(\mathrm{i}\theta)\cdot\mathrm{P}(\theta)}\). Here, _f_(\(\theta\)) means a function of the _a priori_ probability density of parameter \(\theta\), while _f_(x\(|\theta\)) is a function of reliability, i.e., a function of the density of the conditional observation result at a given value of \(\theta\). The symbol \(\Omega\) used under the integral indicates a set of possible values of the estimated parameter \(\theta\). To the left of the formula is the density function of the _a posteriori_ probability of the parameter \(\theta\), after observing the result of the sample x. Thus, based on [PERSON] theorem, the function of the a priori probability density of parameter \(\theta\) is updated, using information from the sample. The population parameters to be estimated, e.g., parameter \(\theta\) (for example mean, standard deviation, specified type element fraction) are treated differently in the two cases. In the classical approach, parameter values are specific but remain unknown. In the Bayesian analysis, they are treated as random variables. For random variables with a continuous probability distribution, [PERSON] theorem may be presented as follows:  
\[\mathrm{f}(\theta\mid x)=\frac{f(\theta\mid x)\cdot f(\theta)}{\int_{\Omega} f(\theta\mid x)\cdot f(\theta)d\theta} \tag{1}\]  
f(\(\theta\))_--a posteriori_ density function of parameter \(\theta\), after sample result \(x\) has been observed  
f(\(\theta\))_--a priori_ distribution density function of parameter \(\theta\);  
\(\Omega\)--set of possible values of parameter \(\theta\). Therefore, based on [PERSON] theorem, _the a priori_ density function of parameter \(\theta\) is updated with the use of information from the sample. In the present paper, Bayesian analysis has been adopted for a case, in which the unknown distribution of parameter \(\theta\) is to be estimated, and \(\theta\) is the mean in a normal population. The population standard deviation \(\sigma_{0}\) is known. It is derived from a priori knowledge that the mean \(\theta\) is a normal random variable with parameters \(m_{1}\) and \(\sigma_{1}\).