Nowadays, constructing effective statistical estimates with a limited amount of statistical information constitutes a significant practical problem. The article is devoted to applying the Bayesian scientific approach to the construction of statistical estimates of the parameters of the laws of distribution of random variables. Five distribution laws are considered: The Poisson law, the exponential law, the uniform law, the Pareto law, and the ordinary law. The concept of distribution laws that conjugate with the observed population was introduced and used. It is shown that for considered distribution laws, the parameters of the laws themselves are random variables and obey the typical law, gamma law, gamma - normal law, and Pareto law. Recalculation formulas are obtained to refine the parameters of these laws, taking into account posterior information. If we apply the recalculation formulas several times in a row, we will get some convergent process. Based on a converging process, it is possible to design a process for self-learning a system or self-tuning a system. The developed scientific approach was applied to solve the measuring problems for the testing measuring devices and technical systems. The results of constructing point estimates and constructing interval estimates for these laws' parameters are given. The results of comparison with the corresponding statistical estimates constructed by the classical maximum likelihood method are presented.