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Emission tomography and Bayesian inverse problems Peter Green University of Bristol and UTS, Sydney Bernoulli Lecture 12 July 2012, Istanbul Peter Green (Bristol/UTS) Bayesian inverse problems Istanbul, July 2012 1 / 70 Outline The

  1. Inverse problems from a Bayesian perspective Basic formulation Why take a Bayesian view? The Bayesian formulation comes close to the way that most scientists intuitively regard the inferential task, and in principle allows the free use of subject knowledge in probabilistic model building. Mathematical analysis of inverse problems is usually focussed on how well the true solution can be recovered, in the presence of noise – in a Bayesian analysis we correspondingly focus on the behaviour of the posterior distribution. Peter Green (Bristol/UTS) Bayesian inverse problems Istanbul, July 2012 14 / 70

  2. Inverse problems from a Bayesian perspective Applications Cell microscopy & segmentation Al-Awadhi, Jennison and Hurn Peter Green (Bristol/UTS) Bayesian inverse problems Istanbul, July 2012 15 / 70

  3. Inverse problems from a Bayesian perspective Applications Electrical impedance tomography E Experimental set-up, i t l t MAP reconstruction, posterior mean and posterior mean and variance, under circular inclusions prior Colin Fox and Geoff Nicholls Peter Green (Bristol/UTS) Bayesian inverse problems Istanbul, July 2012 16 / 70

  4. Inverse problems from a Bayesian perspective Applications Deconvolution of Chandra X-ray images Overlay of the Hubble optical image first with the raw Chandra data and second with the posterior mean reconstruction, highlighting black holes. van Dyk Peter Green (Bristol/UTS) Bayesian inverse problems Istanbul, July 2012 17 / 70

  5. Inverse problems from a Bayesian perspective Applications Remote sensing Josiane Zerubia, Xavier Descombes, C. Lacoste, M. Ortner, R. Stoica Peter Green (Bristol/UTS) Bayesian inverse problems Istanbul, July 2012 18 / 70

  6. Inverse problems from a Bayesian perspective Applications Ocean circulation from tracer data Ocean circulation from tracer data data Posterior mean for advection field field superimposed on reconstructed oxygen and oxygen and salinity concentrations (left) compared (left) compared to interpolated data McKeague, Nicholls, Speer and Herbei, J. Marine Research, 2005 Peter Green (Bristol/UTS) Bayesian inverse problems Istanbul, July 2012 19 / 70

  7. Inverse problems from a Bayesian perspective Theory Some recent themes in inverse problems theory ℓ 1 penalisation for sparse high dimensional linear inverse problems, Candes & Tao (2007), Bickel, Ritov & Tsybakov (2009) Cavalier & Tsybakov (2002) – sharp adaptation Hofinger and Pikkarainen (2007, 2009) – convergence of posterior distribution in linear inverse problems, via Ky Fan metric Lasanen (2007, 2012) – Bayesian formalism for infinite dimensional inverse problems (more generally, on Banach spaces) Knapik, van der Vaart, van Zanten (2011) – Bayesian inverse problems with Gaussian priors; Recovery of initial conditions for the heat equation (Today at 3pm!) Pokern, Stuart, van Zanten (2012) – drift estimation for sde’s Agapiou, Larsson, Stuart (2012) – Bayesian nonparametric linear inverse problems in a separable Hilbert space Peter Green (Bristol/UTS) Bayesian inverse problems Istanbul, July 2012 20 / 70

  8. Single-photon emission computed tomography Principles SPECT Peter Green (Bristol/UTS) Bayesian inverse problems Istanbul, July 2012 21 / 70

  9. Single-photon emission computed tomography Principles SPECT SPECT is a medical imaging technique for creating a 3-dimensional visualisation of the pattern of concentration of a tracer of interest within the human body. As such, it images ‘function’ not ‘form’. It is complementary to other techniques such as CT and MRI/fMRI; useful for specific studies, and comparatively very cheap. The patient is injected with/ingests/inhales a tracer whose pattern of uptake within body tissue has a known relationship to physiological function. The tracer has been radioactively labelled. Some of the photons subsequently emitted are detected in a gamma camera for subsequent processing and interpretation. Peter Green (Bristol/UTS) Bayesian inverse problems Istanbul, July 2012 22 / 70

  10. Single-photon emission computed tomography Principles SPECT SPECT is a medical imaging technique for creating a 3-dimensional visualisation of the pattern of concentration of a tracer of interest within the human body. As such, it images ‘function’ not ‘form’. It is complementary to other techniques such as CT and MRI/fMRI; useful for specific studies, and comparatively very cheap. The patient is injected with/ingests/inhales a tracer whose pattern of uptake within body tissue has a known relationship to physiological function. The tracer has been radioactively labelled. Some of the photons subsequently emitted are detected in a gamma camera for subsequent processing and interpretation. Peter Green (Bristol/UTS) Bayesian inverse problems Istanbul, July 2012 22 / 70

  11. Single-photon emission computed tomography Principles SPECT 50 Journal of the American Statistical Association, March 1997 and Shepp and Vardi (1982) proposed using the EM al- x,y,z gorithm for obtaining its maximum likelihood estima- module position/pulse-height tor (MLE) solution. However, MLE solutions have the of being very noisy in appearance unattractive feature be- photomultiplier tubes cause the reconstruction problem is ill-posed. To overcome scintillation crystal this, Vardi, Shepp, and Kaufman (1985) suggested using a r ,JI'1111111111EIII. collimato slightly smoothed version of the MLE or running a limited a uniform number of EM iterations from start. With the lat- ter approach, the progression from smooth to noisy can be stopped at a point that yields a satisfactory reconstruction. Bayesian reconstruction from SPECT data was proposed by Geman and McClure (1985), following the general patient's head 4 3 paradigm of Bayesian image analysis introduced by Be- sag (1986) and Geman and Geman (1984). This involves 56 a probability on the space of true constructing distribution and patterns of isotope concentration. Geman McClure pro- Figure 1. Gamma Camera and Photon Interactions. (1) A photon posed a pairwise Gibbs prior with a potential difference Peter Green (Bristol/UTS) Bayesian inverse problems Istanbul, July 2012 23 / 70 rejected by the collimator; (2) a direct count; (3) a scattered undetected function that a smoothing scale incorporates and parameter. count; (4) an absorbed photon; (5) a scattered detected count; (6) an Reconstruction can then be accomplished by considering undetected count. the posterior distribution that follows from Bayes's theo- {ats} depend on various factors: the geometry of the detec- rem on the two model components. Various reconstruction tion system, the strength of the isotope and exposure time, have methods been used. Geman and McClure [1985, 1987, and the extent of attenuation and scattering between source 1991 (with Manbeck)] used iterated conditional modes and detector. Weir and Green (1994) discussed and com- (ICM) and Markov chain Monte Carlo (MCMC) methods, pared two methods of constructing these weights; the first and Green (1990) proposed the one-step-late (OSL) algo- method uses simple geometric and physical arguments, and rithm as an adaptation of the EM algorithm, which is im- the second is an extension of the empirical model of Geman, practical for posterior maximization and MCMC methods Manbeck, and McClure (1991). The first method is found to (1996). Regardless of the method used for reconstruction, be preferable and is used here. The weight modeling aims a Bayesian method could be fatally misleading with a bad to be sufficiently accurate for reasonable reconstruction of choice of prior. A simple approach to choosing appropriate x, without the need for auxiliary transmission experiments. prior parameter values is by experimentation using train- The values involve only known physical constants and eas- ing data sets. In earlier work (Weir 1993), I used simulated ily measured dimensions, and they do not depend on the training sets from a suitably designed phantom and found data obtained during the scanning of the patient. It is as- the transition from simulated to real data effortless. A case sumed that the rate of elimination of the radiopharmaceu- can be made for using fixed parameters in clinical prac- tical from the organ is negligible within the duration of the data acquisition and that scattering can be neglected. Each ats term is assumed to be the product of three factors: Camera bins 1. the proportion of radioactivity that has not decayed by the time at which photons are collected in detector t 2. the proportion of emissions that survive attenuation 3. the solid angle of view of the center of pixel s into detector t, which is treated as a cylindrical tube of known length and radius. Attenuation is assumed to be at a constant rate per unit Gamma camera distance within an idealised elliptical boundary represent- T ing the patient's body. The dimensions of the ellipse can be easily measured from the patient, or estimated from a trial reconstruction from the data, presuming that the body outline can be readily distinguished. In my experience, five EM iterations are sufficient to determine the approximate center and length of axes of the ellipse. Ange_ Caer is Roae Thog The joint probability function of the data given the iso- tope concentration is given by p(ylx) = I exp(- Zs atsxs)(Z8 (1 atsxs)Yt

  12. Single-photon emission computed tomography Principles Cartoon of SPECT data acquisition Sinogram: raw data from a single slice, SPECT scan of pelvis Peter Green (Bristol/UTS) Bayesian inverse problems Istanbul, July 2012 24 / 70

  13. Single-photon emission computed tomography Principles Cartoon of SPECT data acquisition Sinogram: raw data from a single slice, SPECT scan of pelvis Peter Green (Bristol/UTS) Bayesian inverse problems Istanbul, July 2012 24 / 70

  14. Single-photon emission computed tomography Principles Cartoon of SPECT data acquisition Sinogram: raw data from a single slice, SPECT scan of pelvis Peter Green (Bristol/UTS) Bayesian inverse problems Istanbul, July 2012 24 / 70

  15. Single-photon emission computed tomography Principles Cartoon of SPECT data acquisition Sinogram: raw data from a single slice, SPECT scan of pelvis Peter Green (Bristol/UTS) Bayesian inverse problems Istanbul, July 2012 24 / 70

  16. Single-photon emission computed tomography Modelling Statistical modelling of SPECT The Poisson linear model y ∼ Poisson ( Ax ) (componentwise, independently) is close to reality (there are some dead-time effects and other artifacts in recording). x represents the spatial distribution of the isotope in question, typically discretised on a square/cubic grid, x = { x j } , y the array of detected photons, also discretised y = { y i } by the recording process, the array A = ( a ij ) - discrete Radon transform - quantifies the emission, transmission, attenuation, decay and recording process; a ij is the mean number of photons recorded at i per unit concentration at pixel/voxel j . Peter Green (Bristol/UTS) Bayesian inverse problems Istanbul, July 2012 25 / 70

  17. Single-photon emission computed tomography Modelling Bayesian analysis of SPECT The reconstruction problem is to infer the pattern of concentration x for the array of detected photon counts y ; a statistical inverse problem, linear, but with (approximately) Poisson variation. Earlier work: Geman & McClure ( Proc ASA , 1985); PJG ( JRSSB , 1990; IEEE-TMI , 1990); PJG & Weir ( J. Appl. Stat. , 1994); Weir ( JASA , 1997) discussed: modelling of SPECT – particularly building the matrix A from physical, geometric and data-analytic considerations Bayesian methods of reconstruction, using both EM- and sampling-based computational methods assessment on simulated data and real gamma camera images of both physical phantoms and actual patients. Peter Green (Bristol/UTS) Bayesian inverse problems Istanbul, July 2012 26 / 70

  18. Single-photon emission computed tomography Modelling Bayesian analysis of SPECT The reconstruction problem is to infer the pattern of concentration x for the array of detected photon counts y ; a statistical inverse problem, linear, but with (approximately) Poisson variation. Earlier work: Geman & McClure ( Proc ASA , 1985); PJG ( JRSSB , 1990; IEEE-TMI , 1990); PJG & Weir ( J. Appl. Stat. , 1994); Weir ( JASA , 1997) discussed: modelling of SPECT – particularly building the matrix A from physical, geometric and data-analytic considerations Bayesian methods of reconstruction, using both EM- and sampling-based computational methods assessment on simulated data and real gamma camera images of both physical phantoms and actual patients. Peter Green (Bristol/UTS) Bayesian inverse problems Istanbul, July 2012 26 / 70

  19. Single-photon emission computed tomography Modelling Pelvis data 50 40 Sinogram: raw data 30 detector from a single slice, SPECT scan of 20 pelvis 10 10 20 30 40 50 60 angle Peter Green (Bristol/UTS) Bayesian inverse problems Istanbul, July 2012 27 / 70

  20. Single-photon emission computed tomography Modelling Ill-posed character of SPECT matrix A: Single slice, SPECT scan of pelvis Peter Green (Bristol/UTS) Bayesian inverse problems Istanbul, July 2012 28 / 70

  21. Single-photon emission computed tomography Modelling Eigenvalues ● ● ● ● ● ● ● ● ● ● ● 1 ● ● ● ● ● ● ● ● ● ● ● ● ● ● Single slice, SPECT ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● scan of pelvis. ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● Small scale log10(eigenvalues) ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● problem: 24 × 24 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● −1 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● body space, 26 × 32 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● detector space, ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● −2 ● Gaussian ● ● ● ● ● ● ● ● ● ● ● ● approximation ● ● ● ● −3 ● 0 100 200 300 400 500 Peter Green (Bristol/UTS) Bayesian inverse problems Istanbul, July 2012 29 / 70

  22. Single-photon emission computed tomography Reconstruction Bayesian SPECT reconstruction In practical work, we have typically used non-Gaussian pairwise-interaction Markov random field priors: � � � p ( x ) ∝ exp − βδ ( 1 + δ ) log cosh (( x s − x s ′ ) /δ ) s ∼ s ′ This has attractive properties log-convex penalises less reconstructions x with physically-realistic abrupt boundaries (e.g. between tissue types), which are smoothed over with Gaussian priors bridges Gaussian ( δ → ∞ ) and Laplace ( δ → 0) Mrf’s and β and δ can also be inferred. Peter Green (Bristol/UTS) Bayesian inverse problems Istanbul, July 2012 30 / 70

  23. Single-photon emission computed tomography Reconstruction Bayesian SPECT reconstruction approx mle Some demonstration reconstructions... approx mle Peter Green (Bristol/UTS) Bayesian inverse problems Istanbul, July 2012 31 / 70

  24. Single-photon emission computed tomography Reconstruction Bayesian SPECT reconstruction osl iteration 30 Some demonstration reconstructions... approx MAP using log cosh prior Peter Green (Bristol/UTS) Bayesian inverse problems Istanbul, July 2012 32 / 70

  25. Single-photon emission computed tomography Reconstruction Bayesian SPECT reconstruction osl iteration 30 Some demonstration reconstructions... approx MAP using log cosh prior, finer body-space resolution Peter Green (Bristol/UTS) Bayesian inverse problems Istanbul, July 2012 33 / 70

  26. Single-photon emission computed tomography Current practice Current practice Standard clinical gamma-camera systems still supplied with (non-probabilistic) filtered back-projection algorithms Smoothed/penalised EM methods also now supplied as standard, increasingly used given attenuation information (e.g. from CT) In PET, EM (in Hudson’s ‘ordered subsets’ variant) is standard, due to improved noise characteristics in low count regions Interest in 4D problems that incorporate motion or tracer kinetics Modelling approaches have particular advantage over FBP with low counts and/or irregular acquisition protocols In all these areas, algorithms rely on OSL to implement penalised/Bayesian methods. See Qi & Leahy, 2006; Hutton, 2011. Peter Green (Bristol/UTS) Bayesian inverse problems Istanbul, July 2012 34 / 70

  27. Single-photon emission computed tomography Current practice Current practice Standard clinical gamma-camera systems still supplied with (non-probabilistic) filtered back-projection algorithms Smoothed/penalised EM methods also now supplied as standard, increasingly used given attenuation information (e.g. from CT) In PET, EM (in Hudson’s ‘ordered subsets’ variant) is standard, due to improved noise characteristics in low count regions Interest in 4D problems that incorporate motion or tracer kinetics Modelling approaches have particular advantage over FBP with low counts and/or irregular acquisition protocols In all these areas, algorithms rely on OSL to implement penalised/Bayesian methods. See Qi & Leahy, 2006; Hutton, 2011. Peter Green (Bristol/UTS) Bayesian inverse problems Istanbul, July 2012 34 / 70

  28. Single-photon emission computed tomography Current practice Current practice Standard clinical gamma-camera systems still supplied with (non-probabilistic) filtered back-projection algorithms Smoothed/penalised EM methods also now supplied as standard, increasingly used given attenuation information (e.g. from CT) In PET, EM (in Hudson’s ‘ordered subsets’ variant) is standard, due to improved noise characteristics in low count regions Interest in 4D problems that incorporate motion or tracer kinetics Modelling approaches have particular advantage over FBP with low counts and/or irregular acquisition protocols In all these areas, algorithms rely on OSL to implement penalised/Bayesian methods. See Qi & Leahy, 2006; Hutton, 2011. Peter Green (Bristol/UTS) Bayesian inverse problems Istanbul, July 2012 34 / 70

  29. Single-photon emission computed tomography Current practice SPECT-CT Buck et al, 2008. Peter Green (Bristol/UTS) Bayesian inverse problems Istanbul, July 2012 35 / 70

  30. Single-photon emission computed tomography Current practice SPECT-MRI phantom study Hutton, 2011. Peter Green (Bristol/UTS) Bayesian inverse problems Istanbul, July 2012 36 / 70

  31. Concentration and approximation of posterior Framework Theoretical analysis of SPECT This section is joint work with Natalia Bochkina (Edinburgh) There is plenty of empirical evidence of the value of taking a Bayesian approach to inverse problems in emission tomography – what mathematical statements can we make about the results? Our framework includes: studying frequentist properties of a Bayesian method the likelihood part of model assumed to be true (cf nonparametric analysis of van der Vaart etc.) treating the prior as the construct of the analyst, not a statement about nature a treatment that addresses non-gaussianity, non-regularity and constraints Peter Green (Bristol/UTS) Bayesian inverse problems Istanbul, July 2012 37 / 70

  32. Concentration and approximation of posterior Framework Theoretical analysis of SPECT This section is joint work with Natalia Bochkina (Edinburgh) There is plenty of empirical evidence of the value of taking a Bayesian approach to inverse problems in emission tomography – what mathematical statements can we make about the results? Our framework includes: studying frequentist properties of a Bayesian method the likelihood part of model assumed to be true (cf nonparametric analysis of van der Vaart etc.) treating the prior as the construct of the analyst, not a statement about nature a treatment that addresses non-gaussianity, non-regularity and constraints Peter Green (Bristol/UTS) Bayesian inverse problems Istanbul, July 2012 37 / 70

  33. Concentration and approximation of posterior Framework Theoretical analysis of SPECT This section is joint work with Natalia Bochkina (Edinburgh) There is plenty of empirical evidence of the value of taking a Bayesian approach to inverse problems in emission tomography – what mathematical statements can we make about the results? Our framework includes: studying frequentist properties of a Bayesian method the likelihood part of model assumed to be true (cf nonparametric analysis of van der Vaart etc.) treating the prior as the construct of the analyst, not a statement about nature a treatment that addresses non-gaussianity, non-regularity and constraints Peter Green (Bristol/UTS) Bayesian inverse problems Istanbul, July 2012 37 / 70

  34. Concentration and approximation of posterior Framework Theoretical analysis of SPECT Using SPECT reconstruction as an example for an asymptotic study, there are three directions in which it is very natural in practical terms to go to a limit: exposure time becomes large resolution of data becomes finer resolution of reconstruction becomes finer We will concentrate on the first of these – thus we hold the dimensions of data and reconstruction fixed, but allow relative noise levels to decrease towards 0. If the exposure time is extended by a factor T , the model becomes T y ∼ Poisson ( T Ax ) , T → ∞ . Peter Green (Bristol/UTS) Bayesian inverse problems Istanbul, July 2012 38 / 70

  35. Concentration and approximation of posterior Framework Theoretical analysis of SPECT Using SPECT reconstruction as an example for an asymptotic study, there are three directions in which it is very natural in practical terms to go to a limit: exposure time becomes large resolution of data becomes finer resolution of reconstruction becomes finer We will concentrate on the first of these – thus we hold the dimensions of data and reconstruction fixed, but allow relative noise levels to decrease towards 0. If the exposure time is extended by a factor T , the model becomes T y ∼ Poisson ( T Ax ) , T → ∞ . Peter Green (Bristol/UTS) Bayesian inverse problems Istanbul, July 2012 38 / 70

  36. Concentration and approximation of posterior Framework Theoretical analysis of SPECT Using SPECT reconstruction as an example for an asymptotic study, there are three directions in which it is very natural in practical terms to go to a limit: exposure time becomes large resolution of data becomes finer resolution of reconstruction becomes finer We will concentrate on the first of these – thus we hold the dimensions of data and reconstruction fixed, but allow relative noise levels to decrease towards 0. If the exposure time is extended by a factor T , the model becomes T y ∼ Poisson ( T Ax ) , T → ∞ . Peter Green (Bristol/UTS) Bayesian inverse problems Istanbul, July 2012 38 / 70

  37. Concentration and approximation of posterior Framework Small-variance SPECT Our model for SPECT is an example of an ill-posed generalised linear inverse problem – p ( y | x ) depends on x only through the linear predictor Ax where A has (numerically) linearly dependent columns. We study inference for x given observed y , in the limit as a noise parameter σ 2 (here, 1 / T ) goes to 0. We assume an ‘identity link function’, so that y becomes concentrated on Ax as σ 2 → 0. Because of the ill-posed/ill-conditioned character of the problem, we cannot expect consistency in inference about x based on the likelihood alone. Even as σ 2 → 0, so that y converges to ‘exact data’ y exact = Ax true , we will not be able to distinguish between { x : Ax = Ax true } . Peter Green (Bristol/UTS) Bayesian inverse problems Istanbul, July 2012 39 / 70

  38. Concentration and approximation of posterior Framework Small-variance SPECT Our model for SPECT is an example of an ill-posed generalised linear inverse problem – p ( y | x ) depends on x only through the linear predictor Ax where A has (numerically) linearly dependent columns. We study inference for x given observed y , in the limit as a noise parameter σ 2 (here, 1 / T ) goes to 0. We assume an ‘identity link function’, so that y becomes concentrated on Ax as σ 2 → 0. Because of the ill-posed/ill-conditioned character of the problem, we cannot expect consistency in inference about x based on the likelihood alone. Even as σ 2 → 0, so that y converges to ‘exact data’ y exact = Ax true , we will not be able to distinguish between { x : Ax = Ax true } . Peter Green (Bristol/UTS) Bayesian inverse problems Istanbul, July 2012 39 / 70

  39. Concentration and approximation of posterior Framework Small-variance SPECT Our model for SPECT is an example of an ill-posed generalised linear inverse problem – p ( y | x ) depends on x only through the linear predictor Ax where A has (numerically) linearly dependent columns. We study inference for x given observed y , in the limit as a noise parameter σ 2 (here, 1 / T ) goes to 0. We assume an ‘identity link function’, so that y becomes concentrated on Ax as σ 2 → 0. Because of the ill-posed/ill-conditioned character of the problem, we cannot expect consistency in inference about x based on the likelihood alone. Even as σ 2 → 0, so that y converges to ‘exact data’ y exact = Ax true , we will not be able to distinguish between { x : Ax = Ax true } . Peter Green (Bristol/UTS) Bayesian inverse problems Istanbul, July 2012 39 / 70

  40. Concentration and approximation of posterior Framework Model formulation: GLIP Rather than restricting to the SPECT model, our theoretical analysis is set in a framework of generalised linear inverse problems: specifically, p ( y | x ) = c y ( σ 2 ) exp ( − � f y ( Ax ) /σ 2 ) for a given n × p matrix A and appropriate functions f y and c y . There is a continuous bijective link function G , and we write G ( y exact ) = Ax true . We suppose the data are generated from this distribution, with x = x true , and attempt to recover x true as the dispersion parameter σ 2 → 0. We assume that for all µ such that G ( µ ) ∈ A X , p if y has the above distribution with Ax = G ( µ ) , then y → µ as σ → 0 � f µ ( η ) has a unique minimum over η ∈ A X at η = G ( µ ) Peter Green (Bristol/UTS) Bayesian inverse problems Istanbul, July 2012 40 / 70

  41. Concentration and approximation of posterior Framework Model formulation: GLIP Rather than restricting to the SPECT model, our theoretical analysis is set in a framework of generalised linear inverse problems: specifically, p ( y | x ) = c y ( σ 2 ) exp ( − � f y ( Ax ) /σ 2 ) for a given n × p matrix A and appropriate functions f y and c y . There is a continuous bijective link function G , and we write G ( y exact ) = Ax true . We suppose the data are generated from this distribution, with x = x true , and attempt to recover x true as the dispersion parameter σ 2 → 0. We assume that for all µ such that G ( µ ) ∈ A X , p if y has the above distribution with Ax = G ( µ ) , then y → µ as σ → 0 � f µ ( η ) has a unique minimum over η ∈ A X at η = G ( µ ) Peter Green (Bristol/UTS) Bayesian inverse problems Istanbul, July 2012 40 / 70

  42. Concentration and approximation of posterior Framework Prior modelling One of the roles of the prior in the Bayesian approach is to resolve this ambiguity (as well as generally improve reconstruction through ‘regularisation’, even without σ 2 → 0). Our results apply for a wider class of suitably smooth priors, but for this talk we simply assume p ( x ) ∝ exp ( − 1 / ( 2 γ 2 ) || x − x 0 || 2 B ) = exp ( − 1 / ( 2 γ 2 )( x − x 0 ) T B ( x − x 0 )) subject to x ∈ X = { x ∈ R p : x j ≥ 0 ∀ j } . Thus the posterior is proportional to p ( y | x ) × exp ( − 1 / ( 2 γ 2 ) || x − x 0 || 2 B ) subject to x ∈ X . Peter Green (Bristol/UTS) Bayesian inverse problems Istanbul, July 2012 41 / 70

  43. Concentration and approximation of posterior Framework Prior modelling One of the roles of the prior in the Bayesian approach is to resolve this ambiguity (as well as generally improve reconstruction through ‘regularisation’, even without σ 2 → 0). Our results apply for a wider class of suitably smooth priors, but for this talk we simply assume p ( x ) ∝ exp ( − 1 / ( 2 γ 2 ) || x − x 0 || 2 B ) = exp ( − 1 / ( 2 γ 2 )( x − x 0 ) T B ( x − x 0 )) subject to x ∈ X = { x ∈ R p : x j ≥ 0 ∀ j } . Thus the posterior is proportional to p ( y | x ) × exp ( − 1 / ( 2 γ 2 ) || x − x 0 || 2 B ) subject to x ∈ X . Peter Green (Bristol/UTS) Bayesian inverse problems Istanbul, July 2012 41 / 70

  44. Concentration and approximation of posterior Framework Prior modelling One of the roles of the prior in the Bayesian approach is to resolve this ambiguity (as well as generally improve reconstruction through ‘regularisation’, even without σ 2 → 0). Our results apply for a wider class of suitably smooth priors, but for this talk we simply assume p ( x ) ∝ exp ( − 1 / ( 2 γ 2 ) || x − x 0 || 2 B ) = exp ( − 1 / ( 2 γ 2 )( x − x 0 ) T B ( x − x 0 )) subject to x ∈ X = { x ∈ R p : x j ≥ 0 ∀ j } . Thus the posterior is proportional to p ( y | x ) × exp ( − 1 / ( 2 γ 2 ) || x − x 0 || 2 B ) subject to x ∈ X . Peter Green (Bristol/UTS) Bayesian inverse problems Istanbul, July 2012 41 / 70

  45. Concentration and approximation of posterior Framework Small-variance SPECT The posterior is proportional to p ( y | x ) × exp ( − 1 / ( 2 γ 2 ) || x − x 0 || 2 B ) subject to x ∈ X . We study this in the σ → 0 limit; to obtain convergence to a point, we will need γ → 0 as well (though, as we will see, at a slower rate than σ ). In particular, we will show that as σ → 0 and γ → 0 in such a way that σ/γ → 0, the posterior converges to the point x ⋆ = argmin x ∈X : Ax = y exact || x − x 0 || 2 B Peter Green (Bristol/UTS) Bayesian inverse problems Istanbul, July 2012 42 / 70

  46. Concentration and approximation of posterior Framework Small-variance SPECT The posterior is proportional to p ( y | x ) × exp ( − 1 / ( 2 γ 2 ) || x − x 0 || 2 B ) subject to x ∈ X . We study this in the σ → 0 limit; to obtain convergence to a point, we will need γ → 0 as well (though, as we will see, at a slower rate than σ ). In particular, we will show that as σ → 0 and γ → 0 in such a way that σ/γ → 0, the posterior converges to the point x ⋆ = argmin x ∈X : Ax = y exact || x − x 0 || 2 B Peter Green (Bristol/UTS) Bayesian inverse problems Istanbul, July 2012 42 / 70

  47. Concentration and approximation of posterior Framework Small-variance SPECT The posterior is proportional to p ( y | x ) × exp ( − 1 / ( 2 γ 2 ) || x − x 0 || 2 B ) subject to x ∈ X . We study this in the σ → 0 limit; to obtain convergence to a point, we will need γ → 0 as well (though, as we will see, at a slower rate than σ ). In particular, we will show that as σ → 0 and γ → 0 in such a way that σ/γ → 0, the posterior converges to the point x ⋆ = argmin x ∈X : Ax = y exact || x − x 0 || 2 B Peter Green (Bristol/UTS) Bayesian inverse problems Istanbul, July 2012 42 / 70

  48. Concentration and approximation of posterior Geometry Basic geometry Suppose that A is a real n × p matrix, and B a real symmetric non-negative definite p × p matrix, both possibly of deficient rank. Assume that the p × ( n + p ) block matrix [ A T . . . B ] has full rank p . Then x ⋆ = argmin x ∈X : Ax = y exact || x − x 0 || 2 B is unique, and if σ → 0, γ → 0 and σ/γ → 0, then approaching the limit the posterior is a truncated Gaussian, with variance scaling differently in different directions. If q is the rank of A , then asymptotically the variance of the posterior distribution (before truncation) has q eigenvalues scaling like σ 2 and the remaining ( p − q ) like the (larger) γ 2 . Peter Green (Bristol/UTS) Bayesian inverse problems Istanbul, July 2012 43 / 70

  49. Concentration and approximation of posterior Geometry Basic geometry Suppose that A is a real n × p matrix, and B a real symmetric non-negative definite p × p matrix, both possibly of deficient rank. Assume that the p × ( n + p ) block matrix [ A T . . . B ] has full rank p . Then x ⋆ = argmin x ∈X : Ax = y exact || x − x 0 || 2 B is unique, and if σ → 0, γ → 0 and σ/γ → 0, then approaching the limit the posterior is a truncated Gaussian, with variance scaling differently in different directions. If q is the rank of A , then asymptotically the variance of the posterior distribution (before truncation) has q eigenvalues scaling like σ 2 and the remaining ( p − q ) like the (larger) γ 2 . Peter Green (Bristol/UTS) Bayesian inverse problems Istanbul, July 2012 43 / 70

  50. Concentration and approximation of posterior Geometry Geometry Visualisation of posterior when n = 1, p = 2, q = 1. Peter Green (Bristol/UTS) Bayesian inverse problems Istanbul, July 2012 44 / 70

  51. Concentration and approximation of posterior Geometry Geometry Visualisation of posterior when n = 1, p = 2, q = 1. Peter Green (Bristol/UTS) Bayesian inverse problems Istanbul, July 2012 44 / 70

  52. Concentration and approximation of posterior Geometry Geometry Visualisation of posterior when n = 1, p = 3, q = 1. In this case x ⋆ lies internal to X . Peter Green (Bristol/UTS) Bayesian inverse problems Istanbul, July 2012 45 / 70

  53. Concentration and approximation of posterior Geometry Geometry Visualisation of posterior when n = 1, p = 3, q = 1. In this case x ⋆ lies internal to X . Peter Green (Bristol/UTS) Bayesian inverse problems Istanbul, July 2012 45 / 70

  54. Concentration and approximation of posterior Geometry Geometry Visualisation of posterior when n = 1, p = 3, q = 1. In this case x ⋆ lies internal to X . σ and γ getting smaller. Peter Green (Bristol/UTS) Bayesian inverse problems Istanbul, July 2012 46 / 70

  55. Concentration and approximation of posterior Geometry Geometry Visualisation of posterior when n = 1, p = 3, q = 1. In this case x ⋆ lies on boundary of X . Peter Green (Bristol/UTS) Bayesian inverse problems Istanbul, July 2012 47 / 70

  56. Concentration and approximation of posterior Geometry Geometry Visualisation of posterior when n = 1, p = 3, q = 1. In this case x ⋆ lies on boundary of X . Peter Green (Bristol/UTS) Bayesian inverse problems Istanbul, July 2012 47 / 70

  57. Concentration and approximation of posterior Geometry Geometry Visualisation of posterior when n = 1, p = 3, q = 1. In this case x ⋆ lies on boundary of X . σ and γ getting smaller. Peter Green (Bristol/UTS) Bayesian inverse problems Istanbul, July 2012 48 / 70

  58. Concentration and approximation of posterior Geometry Karush–Kuhn–Tucker We are interested in the solution to the constrained minimisation problem x ⋆ = argmin x ∈X : Ax = y exact || x − x 0 || 2 B By the Karush–Kuhn–Tucker theory, for this particular problem, it is necessary and sufficient to find ( x ⋆ , ζ, λ ) ∈ R p × R p × R n such that B ( x ⋆ − x 0 ) − ζ + A T λ = 0 x ⋆ ≥ 0 Ax ⋆ = y exact ζ ≥ 0 for all j , ζ j = 0 or x ⋆ j = 0 Peter Green (Bristol/UTS) Bayesian inverse problems Istanbul, July 2012 49 / 70

  59. Concentration and approximation of posterior Geometry Karush–Kuhn–Tucker Geometrical interpretation of KKT conditions when n = 1, p = 2, q = 1, in case B = I . Peter Green (Bristol/UTS) Bayesian inverse problems Istanbul, July 2012 50 / 70

  60. Concentration and approximation of posterior Geometry Karush–Kuhn–Tucker Geometrical interpretation of KKT conditions when n = 1, p = 2, q = 1, in case B = I . Peter Green (Bristol/UTS) Bayesian inverse problems Istanbul, July 2012 50 / 70

  61. Concentration and approximation of posterior Concentration of posterior Metrics and convergence To study convergence of the posterior simultaneously over all y = y ( ω ) , we need a metric that metrises convergence in probability. The Ky Fan metric between two random variables ξ 1 and ξ 2 in a metric space ( Y , d Y ) is ρ K ( ξ 1 , ξ 2 ) = inf { ε > 0 : p { d Y ( ξ 1 ( ω ) , ξ 2 ( ω )) > ε } < ε } . Weak convergence of the posterior µ post = p ( x ∈ ·| y ( ω )) as a random variable to the point mass δ x ⋆ is equivalent to its convergence in the Ky Fan metric, where ( Y , d Y ) is a space of distributions with the Prohorov metric. Peter Green (Bristol/UTS) Bayesian inverse problems Istanbul, July 2012 51 / 70

  62. Concentration and approximation of posterior Concentration of posterior Concentration We are able to give explicit upper bounds on ρ K ( µ post , δ x ⋆ ) , depending on the spectral properties of the asymptotic information matrix and prior precision at x ⋆ , on ρ K ( y , y exact ) , and on the likelihood and prior dispersion parameters σ 2 and γ 2 . For example, in the interior point case, under conditions, ρ K ( µ post , δ x ⋆ ) ≤ max { 2 ρ K ( y , y exact ) , c 1 ρ K ( y , y exact ) + c 2 ν � � � � κ P �� 1 / 2 τ τ + − ( 1 + ∆ ⋆, K ( δ )) } λ min ( H ν ) log C P λ min ( H ν ) Peter Green (Bristol/UTS) Bayesian inverse problems Istanbul, July 2012 52 / 70

  63. Concentration and approximation of posterior Concentration of posterior Concentration We are able to give explicit upper bounds on ρ K ( µ post , δ x ⋆ ) , depending on the spectral properties of the asymptotic information matrix and prior precision at x ⋆ , on ρ K ( y , y exact ) , and on the likelihood and prior dispersion parameters σ 2 and γ 2 . For example, in the interior point case, under conditions, ρ K ( µ post , δ x ⋆ ) ≤ max { 2 ρ K ( y , y exact ) , c 1 ρ K ( y , y exact ) + c 2 ν � � � � κ P �� 1 / 2 τ τ + − ( 1 + ∆ ⋆, K ( δ )) } λ min ( H ν ) log C P λ min ( H ν ) Peter Green (Bristol/UTS) Bayesian inverse problems Istanbul, July 2012 52 / 70

  64. Concentration and approximation of posterior Concentration of posterior Concentration Ky Fan metrics can be readily evaluated for specific distributions. If Y t ∼ σ 2 Poisson ( µ t /σ 2 ) , then � − σ 2 M log ( σ 2 M ) ρ K ( Y , µ ) ∼ as σ 2 → 0, where M = � t µ t . If Y ∼ N p ( µ, Σ) then there exist constants C p , θ p such that for any Σ with || Σ || < θ p , � −|| Σ || log { C p || Σ || max { 1 , p − 2 } } ρ K ( Y , µ ) ≤ If Y ∼ Exp ( λ/σ 2 ) then ρ K ( Y , 0 ) ∼ − ( σ 2 /λ ) log ( σ 2 /λ ) (and is always ≤ ) as σ → 0. Peter Green (Bristol/UTS) Bayesian inverse problems Istanbul, July 2012 53 / 70

  65. Concentration and approximation of posterior Concentration of posterior Concentration in the Bayesian SPECT model If x ⋆ is an interior point of X , then for small enough σ 2 and γ 2 , � � σ 2 − σ 2 log σ + C 2 ρ K ( µ post , δ x ⋆ ) ≤ C 1 γ 2 � � + C 3 σ 1 − α γ α − log ( σ 1 − α γ α ) ( 1 + o ( 1 )) where α = 0 is A T ∇ 2 ˜ f y exact ( Ax ⋆ ) A is of full rank, and α = 1 otherwise. � If α = 0 (well-posed case), the fastest rate is σ − log σ , with γ ≥ σ 1 / 2 [ − log σ ] − 1 / 4 If α = 1 (ill-posed case), the fastest rate is σ 2 / 3 � − log σ , with γ = σ 2 / 3 [ − log σ ] − 1 / 6 If x ⋆ is on the boundary of X , there are additional terms in ρ K ( µ post , δ x ⋆ ) . Peter Green (Bristol/UTS) Bayesian inverse problems Istanbul, July 2012 54 / 70

  66. Concentration and approximation of posterior Concentration of posterior Concentration in the Bayesian SPECT model If x ⋆ is an interior point of X , then for small enough σ 2 and γ 2 , � � σ 2 − σ 2 log σ + C 2 ρ K ( µ post , δ x ⋆ ) ≤ C 1 γ 2 � � + C 3 σ 1 − α γ α − log ( σ 1 − α γ α ) ( 1 + o ( 1 )) where α = 0 is A T ∇ 2 ˜ f y exact ( Ax ⋆ ) A is of full rank, and α = 1 otherwise. � If α = 0 (well-posed case), the fastest rate is σ − log σ , with γ ≥ σ 1 / 2 [ − log σ ] − 1 / 4 If α = 1 (ill-posed case), the fastest rate is σ 2 / 3 � − log σ , with γ = σ 2 / 3 [ − log σ ] − 1 / 6 If x ⋆ is on the boundary of X , there are additional terms in ρ K ( µ post , δ x ⋆ ) . Peter Green (Bristol/UTS) Bayesian inverse problems Istanbul, July 2012 54 / 70

  67. Concentration and approximation of posterior Concentration of posterior Concentration in the Bayesian SPECT model If x ⋆ is an interior point of X , then for small enough σ 2 and γ 2 , � � σ 2 − σ 2 log σ + C 2 ρ K ( µ post , δ x ⋆ ) ≤ C 1 γ 2 � � + C 3 σ 1 − α γ α − log ( σ 1 − α γ α ) ( 1 + o ( 1 )) where α = 0 is A T ∇ 2 ˜ f y exact ( Ax ⋆ ) A is of full rank, and α = 1 otherwise. � If α = 0 (well-posed case), the fastest rate is σ − log σ , with γ ≥ σ 1 / 2 [ − log σ ] − 1 / 4 If α = 1 (ill-posed case), the fastest rate is σ 2 / 3 � − log σ , with γ = σ 2 / 3 [ − log σ ] − 1 / 6 If x ⋆ is on the boundary of X , there are additional terms in ρ K ( µ post , δ x ⋆ ) . Peter Green (Bristol/UTS) Bayesian inverse problems Istanbul, July 2012 54 / 70

  68. Concentration and approximation of posterior Concentration of posterior Geometry Visualisation of posterior when n = 1, p = 3, q = 1. In this case x ⋆ lies internal to X . Peter Green (Bristol/UTS) Bayesian inverse problems Istanbul, July 2012 55 / 70

  69. Concentration and approximation of posterior Concentration of posterior (Lack of) consistency In summary, the posterior concentrates at the point x ⋆ = argmin x ≥ 0 : Ax = y exact || x − x 0 || 2 B – but this is usually not equal to the truth x true . We do have Ax ⋆ = Ax true but ( I − P A T ) x ⋆ is determined solely by the prior and the constraints. Peter Green (Bristol/UTS) Bayesian inverse problems Istanbul, July 2012 56 / 70

  70. Concentration and approximation of posterior Approximation of posterior Bernstein–von Mises Can we quantify probabilistically the variation of p ( x | y ) about the concentration point x ⋆ ? More concretely, can x − x ⋆ be (transformed and) scaled so that it has a non-trivial limit posterior law? Such questions are traditionally the subject of the Bernstein–von Mises (BvM) theorem, in various versions. Here we need a version of the BvM theorem that deals, as well as singularity of the prior, with the non-regular aspects of our set-up, namely positivity constraints ill-posed-ness of A (so x not identifiable) likelihood non-regularity, e.g. from the possibility of Poisson counts of 0 Peter Green (Bristol/UTS) Bayesian inverse problems Istanbul, July 2012 57 / 70

  71. Concentration and approximation of posterior Approximation of posterior Bernstein–von Mises Can we quantify probabilistically the variation of p ( x | y ) about the concentration point x ⋆ ? More concretely, can x − x ⋆ be (transformed and) scaled so that it has a non-trivial limit posterior law? Such questions are traditionally the subject of the Bernstein–von Mises (BvM) theorem, in various versions. Here we need a version of the BvM theorem that deals, as well as singularity of the prior, with the non-regular aspects of our set-up, namely positivity constraints ill-posed-ness of A (so x not identifiable) likelihood non-regularity, e.g. from the possibility of Poisson counts of 0 Peter Green (Bristol/UTS) Bayesian inverse problems Istanbul, July 2012 57 / 70

  72. Concentration and approximation of posterior Approximation of posterior BvM: transformation of coordinates Recall that the point of concentration of the posterior is x ⋆ = argmin x ≥ 0 : Ax = y exact || x − x 0 || 2 B j = 0 } . When we scale x − x ⋆ to get a non-degenerate Let J = { j : x ⋆ limit as σ, γ → 0, the support of the scaled posterior is X J = { x : x J ≥ 0 } . We can decompose X J = W 0 ⊕ W 1 ⊕ W + 2 ⊕ W + 3 to focus on the different characters of the limit in different directions: different scalings and different shapes Peter Green (Bristol/UTS) Bayesian inverse problems Istanbul, July 2012 58 / 70

  73. Concentration and approximation of posterior Approximation of posterior BvM: transformation of coordinates We can decompose X J = W 0 ⊕ W 1 ⊕ W + 2 ⊕ W + 3 The four subsets can be characterised as follows: identi- projection dim fiable? on W k scaling subspace W 0 p 0 yes interior σ subspace W 1 p 1 no interior γ W + σ 2 polyhedral cone p 2 yes boundary 2 W + γ 2 polyhedral cone p 3 no boundary 3 Any of { p k } can be 0, depending on the details of the problem, but p 0 + p 1 + p 2 + p 3 = p . Peter Green (Bristol/UTS) Bayesian inverse problems Istanbul, July 2012 59 / 70

  74. Concentration and approximation of posterior Approximation of posterior BvM: transformation of coordinates W 0 is the whole space when the problem is regular W 1 appears in an ill-posed linear Gaussian problem 3 arises from the positivity constraints on x ⋆ being active W + W + 2 arises with an irregular likelihood (zero Poisson counts) Peter Green (Bristol/UTS) Bayesian inverse problems Istanbul, July 2012 60 / 70

  75. Concentration and approximation of posterior Approximation of posterior BvM theorem for non-regular GLIP x 3 Visualisation of a case with n = 1, p = 3, x 2 x 1 Peter Green (Bristol/UTS) Bayesian inverse problems Istanbul, July 2012 61 / 70

  76. Concentration and approximation of posterior Approximation of posterior BvM theorem for non-regular GLIP x 3 Visualisation of a case with n = 1, p = 3, in which x ⋆ 1 , x ⋆ 2 > 0 , x ⋆ 3 = 0. x 2 X * x 1 Peter Green (Bristol/UTS) Bayesian inverse problems Istanbul, July 2012 61 / 70

  77. Concentration and approximation of posterior Approximation of posterior BvM theorem for non-regular GLIP x 3 Visualisation of a case with n = 1, p = 3, in which x ⋆ 1 , x ⋆ 2 > 0 , x ⋆ 3 = 0. x 2 x 1 Peter Green (Bristol/UTS) Bayesian inverse problems Istanbul, July 2012 61 / 70

  78. Concentration and approximation of posterior Approximation of posterior BvM theorem for non-regular GLIP x 3 Visualisation of a case with n = 1, p = 3, in which x ⋆ 1 , x ⋆ 2 > 0 , x ⋆ 3 = 0. x 2 p 0 = p 1 = p 3 = 1, W 3 + p 2 = 0. W 1 W 0 x 1 Peter Green (Bristol/UTS) Bayesian inverse problems Istanbul, July 2012 61 / 70

  79. Concentration and approximation of posterior Approximation of posterior BvM theorem for non-regular GLIP Suppose we fill the p × p k matrix V k with column vectors forming a basis for W k ( W + k ) for each k , . . . . . . we set V = [ V 0 . V 1 . V 2 . V 3 ] , we assume σ → 0, γ → 0, σ = o ( γ ) and σ = O ( γ 2 ) , � � − 1 , σ 2 V 2 γ 2 V 3 we write S = σ V 0 γ V 1 then S ( x − x ⋆ ) | y → µ ⋆ = N ( a 0 , Ω − 1 00 ) × N ( 0 , B − 1 11 ) × Exp ( a 2 ) × Exp ( a 3 ) in total variation norm. Peter Green (Bristol/UTS) Bayesian inverse problems Istanbul, July 2012 62 / 70

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