V'yugin, V. V. (1999) Algorithmic complexity and stochastic properties of finite binary sequences. Comput. J., this issue.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....our examples this somewhat vague assumption will be obviously satisfied) The function # is assumed to be computable. Within our general framework of games of prediction, Kolmogorov complexity (more accurately, its predictive variant, known as the minus log of Levin s apriorisemi measure; see [8, 9, 10]) describes complexity with respect to a particular game, the so called log loss game. There are, however, many other interesting games; e.g. Example 1 involves the so called square loss game. A data sequence is defined to be a finite sequence z # (X Y ) # of signal outcome pairs. But ....

....to be #. Typically we do not specify the signal space, though we still use the definite article in expressions like the log loss game . The predictive complexity for this game coincides with the variant KM of Kolmogorov complexity (the minus logarithm of Levin s apriorisemi measure; see [8, 9, 10]) The base of the logarithm is usually taken to be 2 in algorithmic information theory and exponential e in machine learning. An important (especially in machine learning and statistics) game is the square loss game, in which Y = Y = a, b] #(y, y) y y) 2 , where a and b are two ....

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V'yugin, V. V. (1999) Algorithmic complexity and stochastic properties of finite binary sequences. Comput. J., this issue.

.... in [11, 12] Despite its evident epistimological prominence in the theory of hypothesis selection and prediction, only some scattered aspects of the subject have been studied before, for example as related to the Kolmogorov structure function [14, 2] and absolutely non stochastic objects [14, 17, 15, 18], notions also defined or suggested by Kolmogorov at the mentioned meeting. For the relation with inductive reasoning according to minimum description length principle see [16] The entire approach is based on Kolmogorov complexity [8] also known as algorithmic information theory) For a general ....

V.V. V'yugin, Algorithmic complexity and stochastic properties of finite binary sequences, The Computer Journal, 42:4(1999), 294--317.

....n is the length of the finite binary sequence x. In the following the length of x will be denoted by l(x) An analogous function (for the logarithmic loss function) as a measure of stochasticity was introduced by Kolmogorov in the seventies [1] 4] 3] and studied by Levin and V yugin [9] [10]. Evidently, the snooping function is non negative and non decreasing. For any sequence x we have by (2) L x (ff c) l(x) for each non negative ff, c is a constant. A more refined estimate is L x (ff c) n Gamma ff for all positive ff n, where n = l(x) is the length of x and c is a ....

V'yugin, V.V. (1999) Algorithmic complexity and stochastic properties of finite binary sequences, The Computer Journal, 42:4, 294--317.

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