In this paper, a recursive control theoretic smoothing spline approach is proposed for reconstructing a closed contour. Periodic splines are generated by minimizing a cost function subject to constraints imposed by a linear control system. The optimal control problem is shown to be proper, and sufficient optimality conditions are derived for a special case of the problem using Hamilton-Jacobi-Bellman theory.

The filtering effect of the smoothing splines allows for usage of noisy sensor data. An important feature of the method is that several data sets for the same closed contour can be processed recursively so that the accuracy is improved stepwise as new data becomes available.

This paper is the first in a two-part study of a variational Bayesian method and its application to a problem of reliable communication. The variational method expresses a Bayesian posterior distribution as the unique minimizer of a quantity dubbed apparent information. This has the same nature as the free energy of statistical mechanics. The minimum apparent information coincides with the full information of the observation.

Reliable communication over an error prone channel can be achieved by the use of random block coding, as originally proposed by Shannon. The primary Bayesian problem in this context, is that of estimating the transmitted block from observations of the output of the channel. Scaling limits for the various information quantities are derived for this problem; these show that the primary problem undergoes a second-order phase transition, in a very precise sense, at the channel capacity; the code rate is shown to play the role of absolute temperature.

Shannon’s reliability function is recovered as the scaling limit of the full information of a secondary Bayesian problem, in which the channel noise and random code are estimated from the observation of a block decoding error. This secondary problem undergoes a third-order phase transition at a second critical code rate.

The aim of this paper is deriving the exact analytical expressions of entropy for the Pareto-types and related distributions. Entropy for $i^{th}$ order statistics of these distributions corresponding to the random sample size $n$ is introduced. We have shown that all the expressions related to finding corresponding entropy for these families and their order statistics distributions are obtained through some particular techniques via integration. Indeed, these techniques for that improper integrals has its own importance also.