The minimum description length (MDL) principle is a formalization of Occam's razor in which the best hypothesis for a given set of data is the one that leads to the best compression of the data. MDL was introduced by Jorma Rissanen in 1978. It is an important concept in information theory and computational learning theory.
Contents
Overview
Any set of data can be represented by a string of symbols from a finite (say, binary) alphabet.
[The MDL Principle] is based on the following insight: any regularity in a given set of data can be used to compress the data, i.e. to describe it using fewer symbols than needed to describe the data literally." (Grünwald, 1998)
To select the hypothesis that captures the most regularity in the data, scientists look for the hypothesis with which the best compression can be achieved. In order to do this, a code is fixed to compress the data, most generally with a (Turing-complete) computer language. A program to output the data is written in that language; thus the program effectively represents the data. The length of the shortest program that outputs the data is called the Kolmogorov complexity of the data. This is the central idea of Ray Solomonoff's idealized theory of inductive inference.
Inference
However, this mathematical theory does not provide a practical way of reaching an inference. The most important reasons for this are:
MDL attempts to remedy these, by:
Rather than "programs", in MDL theory one usually speaks of candidate hypotheses, models or codes. The set of allowed codes is then called the model class. (Some authors refer to the model class as the model.) The code is then selected for which the sum of the description of the code and the description of the data using the code is minimal.
One of the important properties of MDL methods is that they provide a natural safeguard against overfitting, because they implement a tradeoff between the complexity of the hypothesis (model class) and the complexity of the data given the hypothesis. An illustration is given in the following example.
Example of MDL
A coin is flipped 1,000 times and the numbers of heads and tails are recorded. Consider two model classes:
For this reason a naive statistical method might choose the second model as a better explanation for the data. However, an MDL approach would construct a single code based on the hypothesis, instead of just using the best one. To do this, it is simplest to use a two-part code in which the element of the model class with the best performance is specified. Then the data is specified using that code. A lot of bits are needed to specify which code to use; thus the total codelength based on the second model class could be larger than 1,000 bits. Therefore, the conclusion when following an MDL approach is inevitably that there is not enough evidence to support the hypothesis of the biased coin, even though the best element of the second model class provides better fit to the data.
MDL notation
Central to MDL theory is the one-to-one correspondence between code length functions and probability distributions. (This follows from the Kraft–McMillan inequality.) For any probability distribution
Related concepts
MDL is very strongly connected to probability theory and statistics through the correspondence between codes and probability distributions mentioned above. This has led some researchers to view MDL as equivalent to Bayesian inference: code length of model and data together in MDL correspond to prior probability and marginal likelihood, respectively, in the Bayesian framework.
While Bayesian machinery is often useful in constructing efficient MDL codes, the MDL framework also accommodates other codes that are not Bayesian. An example is the Shtarkov normalized maximum likelihood code, which plays a central role in current MDL theory, but has no equivalent in Bayesian inference. Furthermore, Rissanen stresses that we should make no assumptions about the true data generating process: in practice, a model class is typically a simplification of reality and thus does not contain any code or probability distribution that is true in any objective sense. In the last mentioned reference Rissanen bases the mathematical underpinning of MDL on the Kolmogorov structure function.
According to the MDL philosophy, Bayesian methods should be dismissed if they are based on unsafe priors that would lead to poor results. The priors that are acceptable from an MDL point of view also tend to be favored in so-called objective Bayesian analysis; there, however, the motivation is usually different.
Other systems
MDL was not the first information-theoretic approach to learning; as early as 1968 Wallace and Boulton pioneered a related concept called Minimum Message Length (MML). The difference between MDL and MML is a source of ongoing confusion. Superficially, the methods appear mostly equivalent, but there are some significant differences, especially in interpretation: