MSc Methods part XX: YY

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.