some hypothesis fits the data well, the codelength is short. The design of such codes is called universal coding. There are various types of universal codes one could use, often giving similar lengths for long data sequences but differing for short ones. The 'best' (in the sense that it has a minimax optimality property) are the normalized maximum likelihood (NML) or Shtarkov codes. A quite useful class of codes are the Bayesian marginal likelihood codes. For exponential families of distributions, when Jeffreys prior is used and the parameter space is suitably restricted, these asymptotically coincide with the NML codes; this brings MDL theory in close contact with objective Bayes model selection, in which one also sometimes adopts Jeffreys' prior, albeit for different reasons. The MDL approach to model selection "gives a selection criterion formally identical to the BIC approach" for large number of samples. Example of Statistical MDL Learning A coin is flipped 1000 times, and the numbers of heads and tails are recorded. Consider two model classes: The first is a code that represents outcomes with a 0 for heads or a 1 for tails. This code represents the hypothesis that the coin is fair. The code length according to this code is always exactly 1000 bits. The second consists of all codes that are efficient for a coin with some specific bias, representing the hypothesis that the coin is not fair. Say that we observe 510 heads and 490 tails. Then the code length according to the best code in the second model class is shorter than 1000 bits. 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. This code could be the normalized maximum likelihood code or a Bayesian code. If such a code is used, then the total codelength based on the second model class would be larger than 1000 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. Statistical 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