Astrophysics and Space Sciences Transactions www.astrophys-space-sci-trans.net 1810-6528 1810-6536 3 1 2007 10.5194/astra-3-1-2007 http://www.astrophys-space-sci-trans.net/3/1/2007/ http://www.astrophys-space-sci-trans.net/3/1/2007/astra-3-1-2007.html http://www.astrophys-space-sci-trans.net/3/1/2007/astra-3-1-2007.pdf 1 13 2007-07-30 Predicting the future from observable sequences of events in astrophysics I. Maximum Likelihood Estimation for a fixed number of parameters C. Röken cr@tp4.rub.de I. Lerche Institut für Theoretische Physik, Lehrstuhl IV: Weltraum und Astrophysik, Ruhr-Universität-Bochum, 44780 Bochum, Germany Using analytical and numerical methods, estimates are given of future predictions in astrophysics that can be gathered from a sequence of observed events, for example for γ-ray bursts. Some general probability considerations are provided and then a maximum likelihood estimation, together with an approximation for the large number limit of possible events. Illustrations are given of the numerical maximum likelihood estimation programs in the situations of both a large number and a finite number of events. The effects of data uncertainty are also considered. Numerical calculations and comparisons with theoretical expectations are presented too. Aldrich,~J.: R A Fisher and the making of maximum likelihood 1912-1922, Stat. Science, 12, 162–167, 1997. Arps,~J J., Smith,~M B., and Mortada,~J.: Relationship between proved rese and exploratory effort, J. Pet. Technol., 23, 671–675 1971. Cronquist,~C.: Reserves and probabilities?synergism or anachronism?, J. Pet. Technol., 43, 1258–1264, 1991. Dingus,~B L. and Catelli,~J R.: EGRET Detections of the Highest Energy Emission from Gamma-Ray Bursts in: Abstracts of the 19th Texas Symposium on Relativistic Astrophysics and Cosmology, 63, 1998. Feller,~J.: Elements of probability theory, McGraw-Hill, Englewood Cliffs, 1–2, 1968. Fisher,~R A.: Theory of statistical estimation, Proc. Cambridge Philos. Soc., 22, 700–725, 1925. Fisher,~R A.: Two new properties of mathematical likelihood, Proc. Roy. Soc., 144, 285–307, 1934. Fishman,~G J. and Meegan,~C A.: Gamma-Ray Bursts, Annu. Rev. Astron. Astrophys., 33, 415–458, 1995. Frail,~D A., Kulkarni,~S R., Sari,~R., Djorgovski,~S G., Bloom,~J S., Galama,~T J., Reichart,~D E., Berger,~E., Harrison,~F A., Price,~P A., Yost,~S A., and Diercks,~A.: Beaming in Gamma-Ray Bursts: Evidence for a Standard Energy Reservoir, Astrophys J Lett., 562, L55, 2001. Harbaugh,~J W., Doveton,~J H., and Davis,~J C.: Probability methods in oil exploration, Wiley, New York, 1977. Hurley,~K., Sari,R., and Djorgovski,~S G.: Cosmic Gamma-Ray Bursts, Their Afterglows, and Their Host Galaxies, astro-ph/0211620, 2002. Jaynes,~E T.: Where do we stand on maximum entropy? in: The Maximum Entropy Formalism, MIT Press, Cambridge, MA, 15–118, 1978. Lerche,~I.: Geological risk and uncertainty in oil exploration, Academic Press, San Diego, 1997. Lumley,~J L.: Stochastic tools in turbulence, Academic Press, New York, 1970. McCray,~A W.: Evaluation of Exploratory Drilling Ventures By Statistical Decision Methods, J. Pet. Technol., 21, 1199–1209, 1969. Megill,~R E.: Exploration economics, Petroleum Publishing Co., Tulsa, 1971. Panaitescu,~A. and Kumar,~P.: Fundamental Physical Parameters of Collimated Gamma-Ray Burst Afterglows, Astrophys. J. Lett., 560, L49, 2001. Piran,~T., Kumar,~P., Panaitescu,~A., and Piro,~L.: The Energy of Long-Duration Gamma-Ray Bursts, Astrophys. J. Lett., 560, 167–169, 2001. Piran,~T.: Gamma-Ray Bursts and the Fireball Model, Phys. Rep., 314, 575–667, 1999. Piran,~T.: Gamma-ray Bursts - A Puzzle Being Resolved, Phys. Rep., 333, 529–553, 2000. Piran,~T.: The physics of gamma-ray bursts, Rev. of modern physics, 76, 1143–1211, 2004. van Paradijs,~J., Kouveliotou,~C., and Wijers,~R A M J.: Gamma-Ray Burst Afterglows, Annu. Rev. Astron. Astrophys., 38, 379–425, 2000. Rose,~P R.: Dealing with risk and uncertainty: how can we improve?, AAPG Bull., 71, 1–16, 1987.