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 liminf (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.
- K. B. Athreya, J. M. Hitchcock, J. H. Lutz, and E. Mayordomo. Effective strong dimension in algorithmic information and computational complexity. SIAM Journal on Computing, 37(3):671--705, 2007. Google ScholarDigital Library
- P. Billingsley. Ergodic Theory and Information. John Wiley and Sons, 1965.Google Scholar
- C.-L. Chang and Y.-D. Lyuu. Efficient testing of forecasts. International Journal of Foundations of Computer Science, 21(1):61--72, 2010.Google ScholarCross Ref
- A. Dawid. The well-calibrated Bayesian. Journal of the American Statistical Association, 77(379):605--610, 1982.Google ScholarCross Ref
- H. Eggleston. The fractional dimension of a set defined by decimal properties. Quarterly Journal of Mathematics, 20:31--36, 1949.Google ScholarCross Ref
- L. Fortnow and R. V. Vohra. The complexity of forecast testing. Econometrica, 77:93--105, 2009.Google ScholarCross Ref
- D. P. Foster and R. V. Vohra. Asymptotic calibration. Biometrika, 85(2):379--390, 1998.Google ScholarCross Ref
- L. A. Hemaspaandra. Sigact news complexity theory column 48. SIGACT News, 36(3):24--38, 2005. Guest Column: The Fractal Geometry of Complexity Classes, by J. M. Hitchcock, J. H. Lutz, and E. Mayordomo. Google ScholarDigital Library
- J. H. Lutz. Dimension in complexity classes. SIAM Journal on Computing, 32(5):1236--1259, 2003. Google ScholarDigital Library
- J. H. Lutz. The dimensions of individual strings and sequences. Information and Computation, 187(1):49--79, 2003. Google ScholarDigital Library
- N. Merhav and M. Feder. Universal prediction. IEEE Transactions on Information Theory, 44(6):2124--2147, 1998. Google ScholarDigital Library
- A. Sandroni. The reproducible properties of correct forecasts. International Journal of Game Theory, 32(1):151--159, December 2003.Google ScholarCross Ref
Index Terms
- High-confidence predictions under adversarial uncertainty
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