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Bayesian Methods in Parameter Estimation and Uncertainty Analysis

This lecture provides an overview of Bayesian methods for parameter estimation, uncertainty analysis, and linear models. Topics covered include two-parameter estimation, basic Bayesian estimation, and the use of Poisson distribution in data analysis. Reading materials include scientific method texts and computational methods resources. The lecture also delves into error analysis and common errors in Bayesian reasoning. Relevant examples and practical applications are discussed to aid in understanding complex mathematical concepts.

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Bayesian Methods in Parameter Estimation and Uncertainty Analysis

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  1. MSc Methods part XX: YY Dr. Mathias (Mat) Disney UCL Geography Office: 113, Pearson Building Tel: 7670 0592 Email: mdisney@ucl.geog.ac.uk www.geog.ucl.ac.uk/~mdisney

  2. Lecture outline • Two parameter estimation • Some stuff • Uncertainty & linear approximations • parameter estimation, uncertainty • Practical – basic Bayesian estimation • Linear Models • parameter estimation, uncertainty • Practical – basic Bayesian estimation

  3. Reading and browsing Bayesian methods, data analysis • Gauch, H., 2002, Scientific Method in Practice, CUP. • Sivia, D. S., with Skilling, J. (2008) Data Analysis, 2nd ed., OUP, Oxford. Computational • Numerical Methods in C (XXXX) • Flake, W. G. (2000) Computational Beauty of Nature, MIT Press. • Gershenfeld, N. (2002) The Nature of Mathematical Modelling,, CUP. • Wainwright, J. and Mulligan, M. (2004) (eds) Environmental Modelling: Finding Simplicity in Complexity, John Wiley and Sons. Mathematical texts, inverse methods • Tarantola (XXXX) Kalman filters • Welch and Bishop • Maybeck

  4. Reading and browsing Papers, articles, links P-values • Siegfried, T. (2010) “Odds are it’s wrong”, Science News, 107(7), http://www.sciencenews.org/view/feature/id/57091/title/Odds_Are,_Its_Wrong • Ioannidis, J. P. A. (2005) Why most published research findings are false, PLoS Medicine, 0101-0106. Bayes • Hill, R. (2004) Multiple sudden infant deaths – coincidence or beyond coincidence, Pediatric and Perinatal Epidemiology, 18, 320-326 (http://www.cse.salford.ac.uk/staff/RHill/ppe_5601.pdf) • http://betterexplained.com/articles/an-intuitive-and-short-explanation-of-bayes-theorem/ • http://yudkowsky.net/rational/bayes Error analysis • http://level1.physics.dur.ac.uk/skills/erroranalysis.php • http://instructor.physics.lsa.umich.edu/int-labs/Statistics.pdf

  5. Parameter estimation continued • Example: signal in the presence of background noise • Very common problem: e.g. peak of lidar return from forest canopy? Presence of a star against a background? Transitioning planet? A B 0 x See p 35-60 in Sivia & Skilling

  6. Parameter estimation continued • Data are e.g. photon counts in a particular channel, so expect count in kthchanelNk to be where S, B are signal and background • Assume peak is Gaussian (for now), width w, centered on xo so ideal datum Dk then given by • Where n0 is constant (integration time). Unlike Nk, Dk not a whole no., so actual datum some integer close to Dk • Poisson distribution is pdf which represents this property i.e.

  7. Aside: Poisson distribution • Poisson: describes the p of N events occurring over some fixed time interval if events occur at a known rate and independently of the time of the previous event • If expected number over a given interval is D, then prob. of exactly N events • Widely used in discrete counting experiments • Particularly cases where large number of outcomes, each of which is rare (law of rare events) e.g. • Nuclear decay • No. of calls arriving at a call centre per minute – large number arriving BUT rare from POV of general population…. http://en.wikipedia.org/wiki/File:Poisson_pmf.svg

  8. Parameter estimation continued • Data are e.g. photon counts in a particular channel, so expect count in kthchanelNk to be where S, B are signal and background • Assume peak is Gaussian (for now), width w, centered on xo so ideal datum Dk then given by • Where n0 is constant (integration time). Unlike Nk, Dk not a whole no., so actual datum some integer close to Dk • Poisson distribution is pdf which represents this property i.e.

  9. Common errors: reversed conditional • If P(innocent|match) ~ 1:1000000 then P(match|innocent) ~ 1:1000 • Other priors? Strong local ethnic identity? Many common ancestors within 1-200 yrs(isolated rural areas maybe)? • P(match|innocent) >> 1:1000, maybe 1:100 • Says nothing about innocence, but a jury must consider whether the DNA evidence establishes guilt beyond reasonable doubt Stewart, I. (1996) The Interrogator’s Fallacy, Sci. Am., 275(3), 172-175.

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