![]() (Compare page 1021 on coarse-graining in thermodynamics.) Starting in the late 1940s the development of information theory began to suggest connections between randomness and inability to compress data, but emphasis on p Log measures of information content (see page 1071) reinforced the idea that block frequencies are the only real criterion for randomness. To disallow procedures say specially set up to pick out all the infinite number of 1's in a sequence Alonzo Church in 1940 suggested that only procedures corresponding to finite computations be considered. And in the 1920s Richard von Mises-attempting to capture the observed lack of systematically successful gambling schemes-suggested that randomness for individual infinite sequences could be defined in general by requiring that "collectives" consisting of elements appearing at positions specified by any procedure should show equal frequencies. In 1909 Emile Borel had formulated the notion of normal numbers (see page 912) whose infinite digit sequences contain all blocks with equal frequency. In many fields outside of statistics, however, the idea persisted even to the 1990s that block frequencies (or flat frequency spectra) were somehow the only ultimate tests for randomness. Sometimes these were claimed to have some kind of general significance, but mostly they were just viewed as simple practical methods. With the development of statistical hypothesis testing in the early 1900s various tests for randomness were proposed (see page 1084). By the 1800s there was extensive debate about this, but in the early 1900s with the advent of statistical mechanics and measure theory the use of ensembles (see page 1020) turned discussions of probability away from issues of randomness in individual sequences. ![]() ![]() When probability theory emerged in the mid-1600s it implicitly assumed sequences random in the sense of having limiting frequencies following its predictions. THEORETICAL NOTION DEFINITION FREERandomness and unpredictability were discussed as general notions in antiquity in connection both with questions of free will (see page 1135) and games of chance. ![]()
0 Comments
Leave a Reply. |
AuthorWrite something about yourself. No need to be fancy, just an overview. ArchivesCategories |