Abstract
We study the setting in which the bits of an unknown infinite binary sequence x are revealed sequentially to an observer. We show that very limited assumptions about x allow one to make successful predictions about unseen bits of x. First, we study the problem of successfully predicting a single 0 from among the bits of x. In our model, we have only one chance to make a prediction, but may do so at a time of our choosing. This model is applicable to a variety of situations in which we want to perform an action of fixed duration, and need to predict a “safe” time-interval to perform it.
Letting Nt denote the number of 1s among the first t bits of x, we say that x is “ε-weakly sparse” if lim inf (Nt/t) ≤ ε. Our main result is a randomized algorithm that, given any ε-weakly sparse sequence x, predicts a 0 of x with success probability as close as desired to 1 - ε. Thus, we can perform this task with essentially the same success probability as under the much stronger assumption that each bit of x takes the value 1 independently with probability ε.
We apply this result to show how to successfully predict a bit (0 or 1) under a broad class of possible assumptions on the sequence x. The assumptions are stated in terms of the behavior of a finite automaton M reading the bits of x. We also propose and solve a variant of the well-studied “ignorant forecasting” problem. For every ε>0, we give a randomized forecasting algorithm Sε that, given sequential access to a binary sequence x, makes a prediction of the form: “A p fraction of the next N bits will be 1s.” (The algorithm gets to choose p, N, and the time of the prediction.) For any fixed sequence x, the forecast fraction p is accurate to within ±ε with probability 1 - ε.
- Athreya, K. B., Hitchcock, J. M., Lutz, J. H., and Mayordomo, E. 2007. Effective strong dimension in algorithmic information and computational complexity. SIAM J. Comput. 37, 3, 671--705. DOI:http://dx.doi.org/10.1137/S0097539703446912. Google Scholar
Digital Library
- Billingsley, P. 1965. Ergodic Theory and Information. John Wiley and Sons.Google Scholar
- Dawid, A. 1982. The well-calibrated Bayesian. J. Amer. Statist. Assoc. 77, 379, 605--610.Google Scholar
- Eggleston, H. 1949. The fractional dimension of a set defined by decimal properties. Quart. J. Math. 20, 31--36.Google Scholar
Cross Ref
- Fortnow, L. and Vohra, R. V. 2009. The complexity of forecast testing. Econometrica 77, 93--105.Google Scholar
Cross Ref
- Foster, D. P. and Vohra, R. V. 1998. Asymptotic calibration. Biometrika 85, 2, 379--390. http://www.jstor.org/stable/2337364.Google Scholar
Cross Ref
- Hemaspaandra, L. A. 2005. SIGACT news complexity theory column 48. SIGACT News 36, 3, 24--38. (Guest Column: The fractal geometry of complexity classes, by J. M. Hitchcock, J. H. Lutz, and E. Mayordomo.) Google Scholar
Digital Library
- Kapralov, M. and Panigrahy, R. 2011. Prediction strategies without loss. In Proceedings of the 25th Annual Conference on Neural Information Processing Systems. 828--836.Google Scholar
- Lutz, J. H. 2003a. Dimension in complexity classes. SIAM J. Comput. 32, 5, 1236--1259. DOI:http://dx.doi.org/10.1137/S0097539701417723. Google Scholar
Digital Library
- Lutz, J. H. 2003b. The dimensions of individual strings and sequences. Inf. Comput. 187, 1, 49--79. Google Scholar
Digital Library
- Merhav, N. and Feder, M. 1998. Universal prediction. IEEE Trans. Inf. Theory 44, 6, 2124--2147. Google Scholar
Digital Library
- Sandroni, A. 2003. The reproducible properties of correct forecasts. Int. J. Game Theory 32, 1, 151--159.Google Scholar
Digital Library
Index Terms
High-confidence predictions under adversarial uncertainty
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