36,169 research outputs found
Optimality of Universal Bayesian Sequence Prediction for General Loss and Alphabet
Various optimality properties of universal sequence predictors based on
Bayes-mixtures in general, and Solomonoff's prediction scheme in particular,
will be studied. The probability of observing at time , given past
observations can be computed with the chain rule if the true
generating distribution of the sequences is known. If
is unknown, but known to belong to a countable or continuous class \M
one can base ones prediction on the Bayes-mixture defined as a
-weighted sum or integral of distributions \nu\in\M. The cumulative
expected loss of the Bayes-optimal universal prediction scheme based on
is shown to be close to the loss of the Bayes-optimal, but infeasible
prediction scheme based on . We show that the bounds are tight and that no
other predictor can lead to significantly smaller bounds. Furthermore, for
various performance measures, we show Pareto-optimality of and give an
Occam's razor argument that the choice for the weights
is optimal, where is the length of the shortest program describing
. The results are applied to games of chance, defined as a sequence of
bets, observations, and rewards. The prediction schemes (and bounds) are
compared to the popular predictors based on expert advice. Extensions to
infinite alphabets, partial, delayed and probabilistic prediction,
classification, and more active systems are briefly discussed.Comment: 34 page
On Universal Prediction and Bayesian Confirmation
The Bayesian framework is a well-studied and successful framework for
inductive reasoning, which includes hypothesis testing and confirmation,
parameter estimation, sequence prediction, classification, and regression. But
standard statistical guidelines for choosing the model class and prior are not
always available or fail, in particular in complex situations. Solomonoff
completed the Bayesian framework by providing a rigorous, unique, formal, and
universal choice for the model class and the prior. We discuss in breadth how
and in which sense universal (non-i.i.d.) sequence prediction solves various
(philosophical) problems of traditional Bayesian sequence prediction. We show
that Solomonoff's model possesses many desirable properties: Strong total and
weak instantaneous bounds, and in contrast to most classical continuous prior
densities has no zero p(oste)rior problem, i.e. can confirm universal
hypotheses, is reparametrization and regrouping invariant, and avoids the
old-evidence and updating problem. It even performs well (actually better) in
non-computable environments.Comment: 24 page
Sequential Predictions based on Algorithmic Complexity
This paper studies sequence prediction based on the monotone Kolmogorov
complexity Km=-log m, i.e. based on universal deterministic/one-part MDL. m is
extremely close to Solomonoff's universal prior M, the latter being an
excellent predictor in deterministic as well as probabilistic environments,
where performance is measured in terms of convergence of posteriors or losses.
Despite this closeness to M, it is difficult to assess the prediction quality
of m, since little is known about the closeness of their posteriors, which are
the important quantities for prediction. We show that for deterministic
computable environments, the "posterior" and losses of m converge, but rapid
convergence could only be shown on-sequence; the off-sequence convergence can
be slow. In probabilistic environments, neither the posterior nor the losses
converge, in general.Comment: 26 pages, LaTe
Ultimate Intelligence Part I: Physical Completeness and Objectivity of Induction
We propose that Solomonoff induction is complete in the physical sense via
several strong physical arguments. We also argue that Solomonoff induction is
fully applicable to quantum mechanics. We show how to choose an objective
reference machine for universal induction by defining a physical message
complexity and physical message probability, and argue that this choice
dissolves some well-known objections to universal induction. We also introduce
many more variants of physical message complexity based on energy and action,
and discuss the ramifications of our proposals.Comment: Under review at AGI-2015 conference. An early draft was submitted to
ALT-2014. This paper is now being split into two papers, one philosophical,
and one more technical. We intend that all installments of the paper series
will be on the arxi
Kolmogorov's Structure Functions and Model Selection
In 1974 Kolmogorov proposed a non-probabilistic approach to statistics and
model selection. Let data be finite binary strings and models be finite sets of
binary strings. Consider model classes consisting of models of given maximal
(Kolmogorov) complexity. The ``structure function'' of the given data expresses
the relation between the complexity level constraint on a model class and the
least log-cardinality of a model in the class containing the data. We show that
the structure function determines all stochastic properties of the data: for
every constrained model class it determines the individual best-fitting model
in the class irrespective of whether the ``true'' model is in the model class
considered or not. In this setting, this happens {\em with certainty}, rather
than with high probability as is in the classical case. We precisely quantify
the goodness-of-fit of an individual model with respect to individual data. We
show that--within the obvious constraints--every graph is realized by the
structure function of some data. We determine the (un)computability properties
of the various functions contemplated and of the ``algorithmic minimal
sufficient statistic.''Comment: 25 pages LaTeX, 5 figures. In part in Proc 47th IEEE FOCS; this final
version (more explanations, cosmetic modifications) to appear in IEEE Trans
Inform T
New error bounds for Solomonoff prediction
Solomonoff sequence prediction is a scheme to predict digits of binary strings without knowing the underlying probability distribution. We call a prediction scheme informed when it knows the true probability distribution of the sequence. Several new relations between universal Solomonoff sequence prediction and informed prediction and general probabilistic prediction schemes will be proved. Among others, they show that the number of errors in Solomonoff prediction is finite for computable distributions, if finite in the informed case. Deterministic variants will also be studied. The most interesting result is that the deterministic variant of Solomonoff prediction is optimal compared to any other probabilistic or deterministic prediction scheme apart from additive square root corrections only. This makes it well suited even for difficult prediction problems, where it does not suffice when the number of errors is minimal to within some factor greater than one. Solomonoff's original bound and the ones presented here complement each other in a useful way
Self-Optimizing and Pareto-Optimal Policies in General Environments based on Bayes-Mixtures
The problem of making sequential decisions in unknown probabilistic
environments is studied. In cycle action results in perception
and reward , where all quantities in general may depend on the complete
history. The perception and reward are sampled from the (reactive)
environmental probability distribution . This very general setting
includes, but is not limited to, (partial observable, k-th order) Markov
decision processes. Sequential decision theory tells us how to act in order to
maximize the total expected reward, called value, if is known.
Reinforcement learning is usually used if is unknown. In the Bayesian
approach one defines a mixture distribution as a weighted sum of
distributions \nu\in\M, where \M is any class of distributions including
the true environment . We show that the Bayes-optimal policy based
on the mixture is self-optimizing in the sense that the average value
converges asymptotically for all \mu\in\M to the optimal value achieved by
the (infeasible) Bayes-optimal policy which knows in advance. We
show that the necessary condition that \M admits self-optimizing policies at
all, is also sufficient. No other structural assumptions are made on \M. As
an example application, we discuss ergodic Markov decision processes, which
allow for self-optimizing policies. Furthermore, we show that is
Pareto-optimal in the sense that there is no other policy yielding higher or
equal value in {\em all} environments \nu\in\M and a strictly higher value in
at least one.Comment: 15 page
Gaussian process domain experts for model adaptation in facial behavior analysis
We present a novel approach for supervised domain adaptation that is based upon the probabilistic framework of Gaussian processes (GPs). Specifically, we introduce domain-specific GPs as local experts for facial expression classification from face images. The adaptation of the classifier is facilitated in probabilistic fashion by conditioning the target expert on multiple source experts. Furthermore, in contrast to existing adaptation approaches, we also learn a target expert from available target data solely. Then, a single and confident classifier is obtained by combining the predictions from multiple experts based on their confidence. Learning of the model is efficient and requires no retraining/reweighting of the source classifiers. We evaluate the proposed approach on two publicly available datasets for multi-class (MultiPIE) and multi-label (DISFA) facial expression classification. To this end, we perform adaptation of two contextual factors: where (view) and who (subject). We show in our experiments that the proposed approach consistently outperforms both source and target classifiers, while using as few as 30 target examples. It also outperforms the state-of-the-art approaches for supervised domain adaptation
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