Univ. Nice
Prof. D. Mary, M. Carbillet
Statistical Methods and Inverse Problems
(S2, elective, 3 ECTS)
Learning Outcomes:The aim of this elective course is to provide the students with a set of skills that are mandatory to any astronomer aiming at working with astrophysical data (time series, images or data cubes). The scope is on mathematical and statistical concepts about data analysis down to code implementation in Python and use of an IDL-based package.
Knowledge and Understanding:
The students will understand some of the main problems and solutions arising when estimating physical quantities from noisy data. They will learn how to design and implement tools to search for the desired information and how to establish confidence guarantees on their results. A special focus will be on the deconvolution of post adaptive optics images.
Applying Knowledge and Understanding:The students have theoretical exercises and numerical exercises with Jupyter notebooks, as well as post-AO deconvolution practice by means of the IDL-based package AIRY used within the CAOS problem-solving environment (lagrange.oca.eu/caos).
PrerequisitesIntroductory course in statistics (probability and distributions theory). Introductory courses to continuous and discrete Fourier transform. Introductory courses to astronomical imaging.
ProgramPart A (D. Mary)
1 Statistical estimation Definitions: estimators, deterministic vs random parameters, bias and variance of estimators, MSE, optimality, consistency, efficiency. Cramer-Rao bound and Fisher information Maximum likelihood estimation (MLE): definition, main characteristics, invariance, distribution and asymptotic properties d) Confidence intervals; practical approaches : bootstrap, jacknife Lab work : distribution and convergence of the MLE, confidence intervals.
2) Statistical detection Definition : Test statistic, risks, FAP, p-values, Neyman-Pearson and Fisher approaches Optimality of the Likelihood Ratio test (Neyman Pearson lemma) Generalized Likelihood Ratio test Tests frequently used in practice: Kolmogorov, chi2, correlations (Pearson, Spearman), Student,… Lab work : exoplanet detection by the photometric transit technique 3) Bayesian inference [if time permits]
a) History, Bayes’ rule, distribution a priori, a posteriori
b) Priors (Laplace, Jeffrey, improper), conjugate; case of the MLE
c) A posteriori exploitation: Max a posteriori, alpha-credible regions: definition, estimation in practice, law a posteriori
d) Detection: Bayes’ factor
e) Frequentism vs Bayes: pros and cons; empirical and objective Bayes.
Lab work : prior’s influence on posterior estimation.
4) Least squares and inverses problems
a) Least squares: definition, properties, orthogonality, link with MLE: weighted, constrained LS.
b) Uncertainty and confidence intervals
c) LS as an inverse problem: definition, ill-posedness, regularization.
d) Fourier view of convolution and deconvolution
e) Some famous methods: SVD regularization, Wiener, Richardson-Lucy, ISRA.
Lab work: Implementation of algorithms in image deconvolution. Part B (M. Carbillet): Practice of deconvolution of past-adaptive optics data by means of the package AIRY used through the CAOS problem-solving environment (lagrange.oca.eu/caos). “Standard” post-AO deconvolution (Lucy-Richardson, binary star). Computing super-resolution (if time permits) Strehl-constrained blind deconvolution.

Description of how the course is conductedPart A (D. Mary 18h) : 4 lectures of 4 hours + 2 hours.
Each lectures entails theoretical information and exercises, plus: 1) A list of caveats/frequent errors. 2) A quizz with quick questions, given before and after the lecture. 3) A numerical Lab work (python).
Part B (M. Carbillet, 6h): 2 computer-aided (IDL needed) lab works/lectures of 3h.
Description of the didactic methods
Description of the evaluation methodsPart A (D. Mary, 75%) One theoretical written exam at the end (50%). One report for each of the 4 lab works (12,5% each).
Part B (M. Carbillet, 25%): One report.
Adopted Textbooks
Recommended readingsPart A (D. Mary):
S. M Kay, Fundamentals of Statistical Signal Processing, Estimation Theory, Vol. I, Prentice Hall, Signal Processing Series, 2009 S. M Kay, Fundamentals of Statistical Signal Processing, Detection Theory, Vol. II, Prentice Hall, Signal Processing Series, 2009 Effron : Computer age statistical Inference 2010 Cambridge University Press; Large scale inference, Cambridge University Press, 2014 S. Mallat, A Wavelet Tour of Signal Processing The Sparse Way, 3rd ed., Burlington: Academic Press, 2009. Part B (M. Carbillet): Introduction to inverse problems in imaging, M. Bertero, P. Boccacci, C. De Mol, https://doi.org/10.1201/9781003032755 (2021) Restoration of interferometric images: I. The software package AIRY, S. Correia, M. Carbillet, P. Boccacci, M. Bertero, L. Fini, Astron. Astrophys. 387 (2), 733 (2002) Restoration of interferometric images: IV. An algorithm for super-resolution of binary systems, B. Anconelli, M. Bertero, P. Boccacci, M. Carbillet, Astron. Astrophys. 431, 747 (2005) Strehl-constrained iterative blind deconvolution for post-adaptive-optics data,G.Desiderà & M. Carbillet, Astron. Astrophys. 507 (3), 1759 (2009)