Inconsistency of Bayesian Inference for Misspecified Linear Models, and a Proposal for Repairing It

We empirically show that Bayesian inference can be inconsistent under misspecification in simple linear regression problems, both in a model averaging/selection and in a Bayesian ridge regression setting. We use the standard linear model, which assumes homoskedasticity, whereas the data are heteroskedastic, and observe that the posterior puts its mass on ever more high-dimensional models as the sample size increases. To remedy the problem, we equip the likelihood in Bayes' theorem with an exponent called the learning rate, and we propose the Safe Bayesian method to learn the learning rate from the data. SafeBayes tends to select small learning rates as soon the standard posterior is not `cumulatively concentrated', and its results on our data are quite encouraging.Comment: 70 pages, 20 figure

MDL Convergence Speed for Bernoulli Sequences

The Minimum Description Length principle for online sequence estimation/prediction in a proper learning setup is studied. If the underlying model class is discrete, then the total expected square loss is a particularly interesting performance measure: (a) this quantity is finitely bounded, implying convergence with probability one, and (b) it additionally specifies the convergence speed. For MDL, in general one can only have loss bounds which are finite but exponentially larger than those for Bayes mixtures. 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. We discuss the application to Machine Learning tasks such as classification and hypothesis testing, and generalization to countable classes of i.i.d. models.Comment: 28 page

Inconsistency of Bayesian inference for misspecified linear models, and a proposal for repairing it

We empirically show that Bayesian inference can be inconsistent under misspecification in simple linear regression problems, both in a model averaging/selection and in a Bayesian ridge regression setting. We use the standard linear model, which assumes homoskedasticity, whereas the data are heteroskedastic (though, significantly, there are no outliers). As sample size increases, the posterior puts its mass on worse and worse models of ever higher dimension. This is caused by hypercompression, the phenomenon that the posterior puts its mass on distributions that have much larger KL divergence from the ground truth than their average, i.e. the Bayes predictive distribution. To remedy the problem, we equip the likelihood in Bayes' theorem with an exponent called the learning rate, and we propose the SafeBayesian method to learn the learning rate from the data. SafeBayes tends to select small learning rates, and regularizes more, as soon as hypercompression takes place. Its results on our data are quite encouraging

Fast rates in statistical and online learning

The speed with which a learning algorithm converges as it is presented with more data is a central problem in machine learning --- a fast rate of convergence means less data is needed for the same level of performance. The pursuit of fast rates in online and statistical learning has led to the discovery of many conditions in learning theory under which fast learning is possible. We show that most of these conditions are special cases of a single, unifying condition, that comes in two forms: the central condition for 'proper' learning algorithms that always output a hypothesis in the given model, and stochastic mixability for online algorithms that may make predictions outside of the model. We show that under surprisingly weak assumptions both conditions are, in a certain sense, equivalent. The central condition has a re-interpretation in terms of convexity of a set of pseudoprobabilities, linking it to density estimation under misspecification. For bounded losses, we show how the central condition enables a direct proof of fast rates and we prove its equivalence to the Bernstein condition, itself a generalization of the Tsybakov margin condition, both of which have played a central role in obtaining fast rates in statistical learning. Yet, while the Bernstein condition is two-sided, the central condition is one-sided, making it more suitable to deal with unbounded losses. In its stochastic mixability form, our condition generalizes both a stochastic exp-concavity condition identified by Juditsky, Rigollet and Tsybakov and Vovk's notion of mixability. Our unifying conditions thus provide a substantial step towards a characterization of fast rates in statistical learning, similar to how classical mixability characterizes constant regret in the sequential prediction with expert advice setting.Comment: 69 pages, 3 figure

Asymptotics of Discrete MDL for Online Prediction

Minimum Description Length (MDL) is an important principle for induction and prediction, with strong relations to optimal Bayesian learning. This paper deals with learning non-i.i.d. processes by means of two-part MDL, where the underlying model class is countable. We consider the online learning framework, i.e. observations come in one by one, and the predictor is allowed to update his state of mind after each time step. We identify two ways of predicting by MDL for this setup, namely a static} and a dynamic one. (A third variant, hybrid MDL, will turn out inferior.) We will prove that under the only assumption that the data is generated by a distribution contained in the model class, the MDL predictions converge to the true values almost surely. This is accomplished by proving finite bounds on the quadratic, the Hellinger, and the Kullback-Leibler loss of the MDL learner, which are however exponentially worse than for Bayesian prediction. We demonstrate that these bounds are sharp, even for model classes containing only Bernoulli distributions. We show how these bounds imply regret bounds for arbitrary loss functions. Our results apply to a wide range of setups, namely sequence prediction, pattern classification, regression, and universal induction in the sense of Algorithmic Information Theory among others.Comment: 34 page