36,493 research outputs found
On the Convergence Speed of MDL Predictions for Bernoulli Sequences
We consider the Minimum Description Length principle for online sequence
prediction. If the underlying model class is discrete, then the total expected
square loss is a particularly interesting performance measure: (a) this
quantity is bounded, implying convergence with probability one, and (b) it
additionally specifies a `rate of convergence'. Generally, for MDL only
exponential loss bounds hold, as opposed to the linear bounds for a Bayes
mixture. We show that this is even the case if the model class contains only
Bernoulli distributions. We derive a new upper bound on the prediction error
for countable Bernoulli classes. This implies a small bound (comparable to the
one for Bayes mixtures) for certain important model classes. The results apply
to many Machine Learning tasks including classification and hypothesis testing.
We provide arguments that our theorems generalize to countable classes of
i.i.d. models.Comment: 17 page
Strong Asymptotic Assertions for Discrete MDL in Regression and Classification
We study the properties of the MDL (or maximum penalized complexity)
estimator for Regression and Classification, where the underlying model class
is countable. We show in particular a finite bound on the Hellinger losses
under the only assumption that there is a "true" model contained in the class.
This implies almost sure convergence of the predictive distribution to the true
one at a fast rate. It corresponds to Solomonoff's central theorem of universal
induction, however with a bound that is exponentially larger.Comment: 6 two-column page
The Allotropic Transformation of Hafnium
The existence of an allotropic transformation in hafnium, suggested by Zwikker in 1926, has been confirmed. The transformation temperature is 1310±10°C. The high temperature beta-form is probably body-centered cubic
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