[Paper Review] On Finding Predictors for Arbitrary Families of Processes

Daniil Ryabko|arXiv (Cornell University)|Dec 24, 2009

Computability, Logic, AI Algorithms16 references3 citations

TL;DR

This paper establishes that for any family of stochastic processes 𝒞, if a universal predictor exists that converges to the true measure μ in total variation, then a Bayesian predictor with a discrete prior also exists and achieves the same performance. The key contribution is a characterization of when such predictors exist, showing that existence reduces to topological and local behavior conditions on 𝒞, and that predictors can be constructed via channel capacity analysis or NML-based methods when capacity grows sublinearly.

ABSTRACT

The problem is sequence prediction in the following setting. A sequence $x_1,...,x_n,...$ of discrete-valued observations is generated according to some unknown probabilistic law (measure) $μ$. After observing each outcome, it is required to give the conditional probabilities of the next observation. The measure $μ$ belongs to an arbitrary but known class $C$ of stochastic process measures. We are interested in predictors $ρ$ whose conditional probabilities converge (in some sense) to the "true" $μ$-conditional probabilities if any $μ\in C$ is chosen to generate the sequence. The contribution of this work is in characterizing the families $C$ for which such predictors exist, and in providing a specific and simple form in which to look for a solution. We show that if any predictor works, then there exists a Bayesian predictor, whose prior is discrete, and which works too. We also find several sufficient and necessary conditions for the existence of a predictor, in terms of topological characterizations of the family $C$, as well as in terms of local behaviour of the measures in $C$, which in some cases lead to procedures for constructing such predictors. It should be emphasized that the framework is completely general: the stochastic processes considered are not required to be i.i.d., stationary, or to belong to any parametric or countable family.

Motivation & Objective

• To determine under what conditions a universal predictor exists for arbitrary families of stochastic processes.
• To characterize families 𝒞 of measures for which prediction in total variation is possible.
• To show that if any predictor works, then a Bayesian predictor with discrete prior also works.
• To provide constructive sufficient conditions for predictor existence based on local behavior and channel capacity.
• To extend the applicability of Bayesian prediction beyond countable or parametric families.

Proposed method

• Prove that if a predictor exists for a family 𝒞 of measures, then a Bayesian predictor with a discrete prior also exists and predicts every μ ∈ 𝒞 in total variation.
• Use topological separability of 𝒞 as a sufficient condition for predictor existence, though not necessary.
• Analyze the local behavior of measures in 𝒞 truncated to finite horizons (n-length sequences), deriving sufficient conditions for predictor construction.
• Apply information-theoretic tools, particularly channel capacity C(𝒞ⁿ), to determine predictor feasibility: sublinear growth implies constructibility.
• Construct predictors via NML (normalized maximum likelihood) estimates when channel capacity is sublinear.
• Leverage known results on weak merging and total variation convergence (e.g., Blackwell-Dubins, Kalai-Lehrer) to derive conditions for convergence.

Experimental results

Research questions

• RQ1 Under what conditions on a family 𝒞 of stochastic processes does a universal predictor exist that converges to the true measure in total variation?
• RQ2 Can the existence of a universal predictor for 𝒞 be reduced to the existence of a Bayesian predictor with a discrete prior?
• RQ3 What topological or local structural properties of 𝒞 ensure the existence of a predictor?
• RQ4 How does channel capacity C(𝒞ⁿ) relate to the feasibility of constructing a predictor for 𝒞?
• RQ5 Can predictor construction be reduced to computing NML estimates or similar information-theoretic quantities?

Key findings

• If any predictor works for a family 𝒞, then a Bayesian predictor with a discrete prior also works, thus justifying the Bayesian approach in general settings.
• Topological separability of 𝒞 is a sufficient but not necessary condition for predictor existence.
• A predictor can be constructed if the channel capacity C(𝒞ⁿ) grows sublinearly with n, as in the case of all i.i.d. Bernoulli processes.
• For the class of all stationary processes, C(𝒞ⁿ) grows linearly, so the sublinear condition fails, and no such predictor can be constructed via this method.
• Sufficient conditions for predictor existence can be derived from the local behavior of measures in 𝒞, particularly through truncated distributions on finite sequences.
• The result extends to prediction in other metrics (e.g., KL divergence), though the paper focuses on total variation and provides a foundation for future work on alternative performance measures.