We review the recent progress on the definition of randomness with respect to conditional probabilities and a generalization of van Lambalgen theorem (Takahashi 2006, 2008, 2009, 2011). In addition we generalize Kjos Hanssen theorem (2010) when the consistency of the posterior distributions holds. Finally we propose a definition of random sequences with respect to conditional probabilities as the section of the Martin-Lof random set at the random parameters and argue the validity of the definition from the Bayesian statistical point of view.

A new negative result for nonparametric distribution estimation of binary ergodic processes is shown. The problem of estimation of distribution with any degree of accuracy is studied. Then it is shown that for any countable class of estimators there is a zero-entropy binary ergodic process that is inconsistent with the class of estimators. Our result is different from other negative results for universal forecasting scheme of ergodic processes. We also introduce a related result by B. Weiss.

A new negative result for nonparametric distribution estimation of binary ergodic processes is shown. The problem of estimation of distribution with any degree of accuracy is studied. Then it is shown that for any countable class of estimators there is a zero-entropy binary ergodic process that is inconsistent with the class of estimators. Our result is different from other negative results for universal forecasting scheme of ergodic processes.

We introduce a universal Bayes test, which is a Bayesian version of Martin-Lof test. Then we define random sequences with respect to parametric models based on our universal Bayes test. We state some theorems related to Bayesian statistical inference in terms of random sequence

We introduce a universal Bayes test, which is a Bayesian version of Martin-Lof test. Then we define random sequences with respect to parametric models based on our universal Bayes test. We study relations between random samples and random parameters, and apply our results to a parameter estimation problem.

We study asymptotic code lengths of universal codes for parametric models. We show a universal code whose code length is asymptotically less than or equal to that of the minimum description length (MDL) code. Especially when some of the parameters of a source are not random reals, the coefficient of the logarithm in the formula of our universal code is less than that of the MDL code. We describe the redundancy in terms of Kolmogorov complexity and Hausdorff dimension. We show that our universal code is asymptotically optimal in the sense that the coefficient of the logarithm in the formula of the code length is minimal. Our universal code can be considered to be a natural extension of the Shannon code and the MDL code.