## 3 Background
If, in turn, the average of an \(n\)-element sample drawn from the population is equal to \(m_{2}\), the _a posteriori_ distribution of the random variable \(\theta\) is also normal, with the mean \(m\) and standard deviation \(\sigma\), calculated as follows:  
\[m=\frac{\left(1/\sigma_{1}^{2}\right)\times m_{1}+\left(n/\sigma_{0}^{2} \right)\times m_{2}}{\left(1/\sigma_{1}^{2}\right)+\left(n/\sigma_{0}^{2} \right)} \tag{2}\]  
\[\sigma=\frac{1}{\left(1/\sigma_{1}^{2}\right)+\left(n/\sigma_{0}^{2}\right)} \tag{3}\]  
The presented Bayes' theorem gives a valuable practical possibility of the successive inclusion of new information coming from consecutively drawn random samples. During a consecutive step, knowledge about the posterior distribution of parameter \(\theta\) is treated as a priori knowledge of this parameter. Including new portions of information does not affect the final result. When the probability distribution of the examined parameter is estimated, a credible set of the parameter can be constructed. The credible set (the highest posterior density set) is an interval, to which the value of the parameter drops with a defined probability. This is the analog of a confidence interval concept that is used in classical statistical analysis. The confidence interval predicts the unknown value of the parameter with an assumed probability. The selection of parameters requires careful and cautious use of statistical analysis, including the use of the Bayesian approach. In Bayesian statistical analysis, results of research from the soil of the analyzed sites were subsequently included. In the calculations, it was stated that the final result is independent of the order of the objects. In addition, the mean values of the unit weight of organic soils, Pliocene clays, and boulder clays are the same in the case of classical analysis and the Bayesian approach. Bayesian analysis (i.e., Equations (1)-(3)) is performed in order to provide an a posteriori probability distribution function for model parameters, the most likely thickness of layers of the soil, and its unit weight. Here, the asymptotic technique is used to estimate the posterior model parameters. The asymptotic technique includes an approximation of the _a posteriori_ probability function as a Gaussian probability distribution function (PDF), and the _a posteriori_ probability distribution function for model parameters is a combined Gaussian probability distribution function with a mean value equal to the most probable posterior probability distribution function [10]. The asymptotic technique was successfully used in previous studies to interpret the undrained shear strength of mineral and organic soils from DMT tests. In the Bayesian approach, some preliminary knowledge about the distribution of parameter values is modified after confronting the data. Using the _a priori_ distribution and knowledge about the sample taken, a new parameter distribution is determined, which takes into account both the original _a priori_ beliefs and the empirical data obtained. An important characteristic of the Bayesian approach is that the sequential modification of knowledge about the distribution of the tested parameter gives the same result as in the case, when all doses of information are included in the conclusion at once--that is, if successively taken samples are treated as one larger sample. It also implies that the order of attaching new portions of information is arbitrary. Therefore, a question should be answered: When is the Bayesian approach worth applying in practice--that is, when will the classical approach not give better results? The classical approach will not give better results when the a priori information is only in the form of results of the analyses, but the tests on the basis of which these analyses were made are no longer available (therefore, it is impossible to extend the data sample based on which the inference is made in the classic way).