### abstract ###
We study the a priori semimeasure of sets of  SYMBOL -random  infinite sequences, where  SYMBOL  is a family of probability distributions depending on a real parameter  SYMBOL
In the case when for a computable probability distribution  SYMBOL  an effectively strictly consistent estimator exists, we show that the Levin's a priory semimeasure of the set of all  SYMBOL -random sequences is positive if and only if the parameter  SYMBOL  is a computable real number
For the Bernoulli family  SYMBOL , we show that the a priory semimeasure of the set  SYMBOL , where  SYMBOL  is the set of all  SYMBOL -random sequences and the union is taken over all non-random  SYMBOL , is positive
### introduction ###
We use algorithmic randomness theory to analyze ``the size'' of sets  of infinite sequences random with respect to parametric families of probability distributions
Let a parametric family of probability distributions  SYMBOL , where  SYMBOL  is a real number, be given such that an effectively strictly consistent estimator exists for this family
The Bernoulli family with a real parameter  SYMBOL  is an example of such family
Theorem~ shows that the Levin's a priory semimeasure of the set of all  SYMBOL -random sequences is positive if and only if the parameter value  SYMBOL  is a computable real number
We say that a property of infinite sequences has no ``empirical meaning'' if the Levin's a priory semimeasure of the set of all sequences possessing this property is  SYMBOL
In this respect, the model of the biased coin with ``a prespecified'' probability  SYMBOL  of head is meaningless when  SYMBOL  is a noncomputable real number; noncomputable parameters  SYMBOL  can have empirical meaning only in their totality, i e , as elements of some uncountable sets
For example,  SYMBOL -random sequences with noncomputable  SYMBOL  can be generated by a Bayesian mixture of these  SYMBOL  using a computable prior
In this case, evidently, the semicomputable semimeasure of the set of all sequences random with respect to this mixture is positive
We give in Appendix~ the simple proof of our previous result (formulated in Theorem~) which says that the Levin's a priory semimeasure of the set of all infinite binary sequences non-equivalent by Turing to Martin-L\"of random sequences is positive
In particular,  these sequences are non-random with respect to each computable probability  distribution
We use this result to prove Theorem~
This theorem  shows that a probabilistic machine can be constructed, which with probability close to  SYMBOL  outputs a random  SYMBOL -Bernoulli sequence such that the parameter  SYMBOL  is not random with respect to each computable probability distribution
This result can be interpreted such that the Bayesian statistical approach is insufficient to cover all possible ``meaningful'' cases for  SYMBOL -random sequences
