Minimum Description Length Model Selection, Problems and Extensions
Steven de Rooij
Abstract:
Model selection is a strange and wonderful topic in learning theory
and statistics. At first glance the question seems very clear-cut:
how should we decide which set of probability distributions matches
the observations at hand best. This question comes up time and again
in many different contexts, ranging from testing scientific hypotheses
in general (which among these psychological models describes best how
people behave?) to more concrete applications (what order polynomial
should we use to fit the data in this regression problem? What lossy
representation of this image best captures the structural properties
of the original?). Thus, model selection is ubiquitous, and the
one-size-fits-all criteria based on the Minimum Description Length
(MDL) principle and the closely related Bayesian statistics are
appreciated by many.
Upon closer inspection, many applications of model selection are not
as similar as they may first appear. They can be distinguished by
technical properties (are the models nested? Parametric? Countable?),
but also by a priori assumptions (is the process generating the data
believed to be an element of any of the considered models?), as well
as the motivation for performing model selection in the first place
(do we want to identify which model contains the data generating
process, or do we want to identify which model we may expect to
predict future data best?). The best choice of methodology in any
situation often depends on such particulars, and is further determined
by practical considerations such as whether or not the relevant
quantities can be evaluated analytically, and whether efficient
algorithms exist for their calculation. MDL/Bayesian model selection
has been shown to perform quite well in many different contexts and
applications; in this thesis we treat some of the puzzling problems and
limitations that have also become apparent over time. We also extend
the idea by linking it to other topics in machine learning and
statistical inference.
To apply MDL, universal codes or distributions have to be associated
with each of the considered models. The preferred code is the
Normalised Maximum Likelihood (NML) or Shtarkov code. However, this
code yields infinite code word lengths for many models. This first
issue with MDL model selection is investigated in Chapter 2, in which
we perform computer experiments to test the performance of some of the
available alternatives. One result is that the model selection
criterion based on the so-called prequential plug-in code displays
inferior performance. This observation seems important because the
prequential plug-in code is often thought of as a convenient
alternative to other universal codes such as the NML code, as it is
much easier to calculate. It was thought to result in code lengths
similar to those obtained for other universal codes (such as NML,
2-part codes or Bayesian mixtures), but we discovered that this is
only the case if the data generating process is in the model. We show
in Chapter 3 that the redundancy of the prequential plug-in code is
fundamentally different from the standard set by other universal codes
if the data generating process is not an element of the model, so that
caution should be exercised when it is applied to model selection.
The third problem treated in this thesis is that MDL/Bayesian model
selection normally does not take into account that, even in the ideal
case where one of the considered models is ``true'' (contains the data
generating process), and even if the data generating process is
stationary ergodic, then still the index of the model whose associated
universal code issues the best predictions of future data often
changes with the sample size. Roughly put, at small sample sizes
simple models often issue better predictions of future data than the
more complex ``true'' model, i.e. the smallest model that contains the
data generating distribution. When from a certain sample size onward
the true model predicts best, the simpler model has already built up a
lot of evidence in its favour, and a lot of additional data have to be
gathered before the true model ``catches up'' and is finally
identified by Bayesian/MDL model selection. This phenomenon is
described in Chapter 5, in which we also introduce a novel model
selection procedure that selects the true model almost as soon as
enough data have been gathered for it to be able to issue the best
predictions. The criterion is consistent: under mild conditions, the
true model is selected with probability one for sufficiently large
sample sizes. We also show that a prediction strategy based on this
model selection criterion achieves an optimal rate of convergence: its
cumulative KL-risk is as low as that of any other model selection
criterion. The method is based on the ``switch distribution'',
which can be evaluated using an efficient expert tracking algorithm.
More properties of this switch distribution are treated in Chapter 4,
which also contains a survey of this and other expert tracking
algorithms and shows how such algorithms can be formulated in terms of
Hidden Markov Models.
Finally, in Chapter 6 we evaluate the new theory of algorithmic
rate-distortion experimentally. This theory was recently proposed by
Vitanyi and Vereshchagin as an alternative to classical
rate-distortion theory. It allows analysis of the structural
properties of individual objects and does not require the
specification of a probability distribution on source objects. Instead
it is defined in terms of Kolmogorov complexity, which is
uncomputable. To be able to test this theory in practice we have
approximated the Kolmogorov complexity by the compressed size of a
general purpose data compression algorithm. This practical framework
is in fact a generalisation of MDL model selection.
The perspectives offered in this thesis on many aspects of MDL/Bayesian
model selection, contribute to a better understanding of the
relationships between model selection and such diverse topics as
universal learning, prediction with expert advice, rate distortion
theory and Kolmogorov complexity.