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Concept
Minimum message length
Summary
Minimum message length (MML) is a Bayesian information-theoretic method for statistical model comparison and selection. It provides a formal information theory restatement of Occam's Razor: even when models are equal in their measure of fit-accuracy to the observed data, the one generating the most concise explanation of data is more likely to be correct (where the explanation consists of the statement of the model, followed by the lossless encoding of the data using the stated model). MML was invented by Chris Wallace, first appearing in the seminal paper "An information measure for classification". MML is intended not just as a theoretical construct, but as a technique that may be deployed in practice. It differs from the related concept of Kolmogorov complexity in that it does not require use of a Turing-complete language to model data.
Shannon's A Mathematical Theory of Communication (1948) states that in an optimal code, the message length (in binary) of an event , , where has probability , is given by .
Bayes's theorem states that the probability of a (variable) hypothesis given fixed evidence is proportional to , which, by the definition of conditional probability, is equal to . We want the model (hypothesis) with the highest such posterior probability. Suppose we encode a message which represents (describes) both model and data jointly. Since , the most probable model will have the shortest such message. The message breaks into two parts: . The first part encodes the model itself. The second part contains information (e.g., values of parameters, or initial conditions, etc.) that, when processed by the model, outputs the observed data.
MML naturally and precisely trades model complexity for goodness of fit. A more complicated model takes longer to state (longer first part) but probably fits the data better (shorter second part). So, an MML metric won't choose a complicated model unless that model pays for itself.
One reason why a model might be longer would be simply because its various parameters are stated to greater precision, thus requiring transmission of more digits.
Official source
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