## 8 Bayes' Approach in the Interpretation of Test Results from the Proposed Formula
Next, we continue the transformation process for formula \((\frac{\gamma}{\gamma_{w}},k_{1}\times log\left[64\times\left[\frac{p_{0}-u_{0}} {p_{1}}\right]\right]-k_{2}\times log\left(\frac{p_{1}}{\sigma_{atm}}\right)-k_ {3}\)_and_\(\varepsilon\)). The following modification was introduced for: \(p_{0}\rightarrow\log\) (p\({}_{0}\)), \(p_{1}\rightarrow\log\) (p\({}_{1}\)), \(u_{0}\rightarrow\log\)(u\({}_{0}\)), \(\sigma_{atm}\rightarrow\log\)(\(\sigma_{\text{atm}}\)) parameters. A separate analysis was carried out for each soil peat, gyttja (Gy), organic mud, mud, clayey sand (Sicl), for boulder clay (Cl), \(u_{0}\) = 0 kPa, sand (Sa, CSa, MSa, FSa)). Then, the normality of the Shapiro-Wilk method was checked for each type of soil. A normal distribution was obtained after the transformation. The results of the analysis are presented in Figure 7 and Table 8 below. Figure 6: Matrix of extended charts for selected variables (soil unit weight). Previous statistical methods using mostly the least squares estimation took data from field and laboratory studies as a closed number of random variables to determine the expected values, variances, and correlations for a given expression as a function of a random variable. The main disadvantage of the least squares method used to estimate the parameters is the difficulty in the representation of knowledge about the expected values of parameters, which should result from the estimation process. A significant progress in parameter estimation, as shown by increasingly numerous examples of applications in various fields, is the Bayesian analysis, in which a set of test results may be increased \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & \(\frac{\gamma}{\gamma_{w}}\) & \(=\)k\(\times log\Big{[}64\times\Big{[}\frac{\mathbb{P}_{w}-u_{0}}{\mathbb{P}_{1}} \Big{]}\Big{]}\) & \(+k_{2}\) & \(\text{for}\ log\frac{\gamma}{\gamma_{w}}\), \(p_{0}\to\log(p_{0})\), \(p_{1}\to\log(p_{1})\), \(u_{0}\to\log(u_{0})\), \\ \cline{2-7}
**Soil Type** & & \(\times log\Big{[}\frac{\mathbb{P}_{w}}{\text{data}}\Big{]}\) & \(+k_{3}+\epsilon\) & \(\sigma_{\text{data}}\to\log(\sigma_{\text{data}})\) \\ \cline{2-7}  & **MRD (\%)** & **MRSD (\%)** & **Rest \(\epsilon\)** & **MRD (\%)** & **MRSD (\%)** & **Rest \(\epsilon\)** \\ \hline \hline \multirow{2}{*}{\begin{tabular}{c} peat and gyt\# (Gy) \\ \end{tabular} } & 25.0 & 7.4 & \(-5.53\times 1.29\) & 15.0 & 3.0 & \(-0.05\times 0.17\) \\ \cline{2-7}  & 8.8 & 2.4 & \(-0.93\times 1.28\) & 6.3 & 4.4 & \(-0.01\times 0.07\) \\ \hline \multirow{2}{*}{\
\begin{tabular}{c} clayey sand (Sicl) \\ \end{tabular} } & 18.0 & 8.5 & \(-1.03\times 3.78\) & 5.0 & 2.7 & \(-0.03\times 0.06\) \\ \cline{2-7}  & 16.5 & 4.0 & \(-3.3\times 1.68\) & \(\ast\) & \(\ast\) & \(\ast\) \\ \hline \hline \multirow{2}{*}{
\begin{tabular}{c} sand (Sa, CSa, MSa, FSa) \\ \end{tabular} } & 22.0 & 6.3 & \(-3.9\times 2.72\) & \(\ast\) & \(\ast\) & \(\ast\) \\ \cline{2-7}  & 22.2 & **10.2** & \(-5.53\times 3.78\) & **0.50** & **0.34** & \(-0.19\times 0.09\) \\ \hline \hline \end{tabular}  
* calculation was not possible because u\({}_{0}\) = 0 kPa for this subsoil; MRD—maximum relative deviation, MRSD—mean square relative deviation.  
\end{table} 
Table 8: Comparison of MRD, MRSD, and the rest of the obtained values before and after transformation for Equation (13). by new data, and on this basis, the probability of occurrence is determined. In order to determine the values of characteristic geotechnical parameters (\(X_{k}\)), [PERSON] (1997) [48] proposed a formula based on comparative calculations, proposing to use the following formula to select the characteristic value of geotechnical parameters (Table 9): \(\gamma_{k}=\gamma_{m}-0.5\times S_{d}\), where: \(X_{m}\)--mean value; \(S_{d}\)--standard deviation. Unit weight parameters were calculated for the soil found in each site. The calculation was based on the use of [PERSON]'s patterns existing in the literature and on [PERSON]'s nomogram, which has not changed since 1980.