Universal models are an outgrowth of the ideas rooted in universal coding, stochastic complexity, and modeling by the shortest code length or the MDL principle, which themselves are modifications of algorithmic complexity, as introduced by Solomonoff. Much like algorithmic complexity, which permits the definition of a universal probability model, albeit a noncomputable one, stochastic complexity can be used to define a computable model, which is universal for a class of probability measures. Although it can fully learn to imitate these models from their samples only asymptotically, we can prove for many classes of models that asymptotically no "estimate" of the data-generating machinery can perform better than the universal model, which then may be regarded as the ultimate "estimator".
In reality, such a foundation for statistics does not require the data, let alone the parameters, to be a sample from any population. The selected probability models are just a convenient way via the Kraft inequality to describe good codes without need to resort to the (false) claim that any of them has generated the data. The central issue then becomes model fitting rather than estimation, in which the elusive model complexity plays a natural and essential part. This brings advantages well beyond the reach of the traditional techniques, especially in complex modeling problems.
This talk will outline algorithms for universal density and
regression models as well as a new way to do universal coding.
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