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

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, 2 nd 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 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://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. 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? Parameter estimation continued A B x 0 See p 35-60 in Sivia & Skilling

6. Data are e.g. photon counts in a particular channel, so expect count in k th chanel N k to be where S, B are signal and background Assume peak is Gaussian (for now), width w, centered on x o so ideal datum D k then given by Where n 0 is constant (  integration time). Unlike N k, D k not a whole no., so actual datum some integer close to D k Poisson distribution is pdf which represents this property i.e. Parameter estimation continued

7. 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 Aside: Poisson distribution 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. Data are e.g. photon counts in a particular channel, so expect count in k th chanel N k to be where S, B are signal and background Assume peak is Gaussian (for now), width w, centered on x o so ideal datum D k then given by Where n 0 is constant (  integration time). Unlike N k, D k not a whole no., so actual datum some integer close to D k Poisson distribution is pdf which represents this property i.e. Parameter estimation continued

9. 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 Common errors: reversed conditional Stewart, I. (1996) The Interrogator’s Fallacy, Sci. Am., 275(3), 172-175.