Statistical Image Modelling and Particle Physics Comments on talk by D.M. Titterington Glen Cowan RHUL Physics PHYSTAT05 Glen Cowan Royal Holloway, University of London PHYSTAT05 Oxford, 15 September 2005

Unfolding for particle physicists Glen Cowan RHUL Physics PHYSTAT05 V. Blobel, Unfolding Methods in High Energy Physics, DESY ; also CERN G. Zech, Comparing statistical data to Monte Carlo simulation-- parameter fitting and unfolding, DESY (1995). R. Barlow, SLUO Lectures on Numerical Methods in HEP (Lecture 9: Unfolding), G. Cowan, Statistical Data Analysis (Ch. 11), OUP, N. Bissantz, Regularized inversion methods and error bounds for general statistical inverse problems, PHYSTAT05. N. Gagunashvili, Unfolding with system identification, PHYSTAT05. + now DMT’s PHYSTAT paper and its 2 pages of refs.

Connections with and differences from HEP Glen Cowan RHUL Physics PHYSTAT05 Image reco: many pixels, Gaussian noise HEP: (often) data are numbers of events, ~10 2 bins true scene noisy image measured histogram ‘true histogram’ expectation value of data response matrix

Connections with and differences from HEP Glen Cowan RHUL Physics PHYSTAT05 In image reco, point spread function (response matrix) is essentially a property of the measuring device: In HEP this is often almost true, but resolution in a variable of interest x can depend on the value of some other variable y. E.g. energy resolution might be better if particles often have large angles w.r.t. beam → H has model dependence. This model dependence is one source of systematic uncertainty in the unfolded distribution; also modeling of measurement device itself difficult (probably worse problem...). Also technical difficulties in determining H: need lots of MC, smoothing(?)

Connections with and differences from HEP Glen Cowan RHUL Physics PHYSTAT05 Image reco: if new feature visible, it worked. HEP: need not only unfolded distribution but full covariance matrix and estimate of systematic biases. Image reco: Bayesian method using priors motivated by statistical physics (Ising model). HEP: could we write down an appropriate prior?

Bias vs. variance Glen Cowan RHUL Physics PHYSTAT05 Unregularized solution unbiased, huge variance. But (at least in Poisson case), the variance is at as small as you can get for zero bias (Minimum Variance Bound). → to reduce catastrophic variance you must put up with some bias. Bias vs. variance trade-off determined by regularization parameter. DMT mentions several recipes not yet explored in HEP, e.g., (generalized) cross-validation, Bayesian prescription

Iterative methods Glen Cowan RHUL Physics PHYSTAT05 In HEP iterative technology used e.g. G. D’Agostini NIM A362 (1995) 487. DMT describes many variations on this theme -- we need to read these references!