  We study algorithmic randomness and monotone complexity on product of the set
of infinite binary sequences. We explore the following problems: monotone
complexity on product space, Lambalgen's theorem for correlated probability,
classification of random sets by likelihood ratio tests, decomposition of
complexity and independence, Bayesian statistics for individual random
sequences. Formerly Lambalgen's theorem for correlated probability is shown
under a uniform computability assumption in [H. Takahashi Inform. Comp. 2008].
In this paper we show the theorem without the assumption.
