doi: 10.3934/ipi.2021030

Where Bayes tweaks Gauss: Conditionally Gaussian priors for stable multi-dipole estimation

1. 

Dipartimento di Matematica, Via Dodecaneso 35, 16146 Genova, Italy

2. 

BESA GmbH, Freihamer Str. 18, 82166 Gräfelfing, Germany

* Corresponding author: Alberto Sorrentino

Received  June 2020 Revised  December 2020 Published  May 2021

We present a very simple yet powerful generalization of a previously described model and algorithm for estimation of multiple dipoles from magneto/electro-encephalographic data. Specifically, the generalization consists in the introduction of a log-uniform hyperprior on the standard deviation of a set of conditionally linear/Gaussian variables. We use numerical simulations and an experimental dataset to show that the approximation to the posterior distribution remains extremely stable under a wide range of values of the hyperparameter, virtually removing the dependence on the hyperparameter.

Citation: Alessandro Viani, Gianvittorio Luria, Alberto Sorrentino, Harald Bornfleth. Where Bayes tweaks Gauss: Conditionally Gaussian priors for stable multi-dipole estimation. Inverse Problems & Imaging, doi: 10.3934/ipi.2021030
References:
[1]

Z. A. Acar and S. Makeig, Neuroelectromagnetic forward head modeling toolbox, Journal of Neuroscience Methods, 190 (2010), 258-270.  doi: 10.1016/j.jneumeth.2010.04.031.  Google Scholar

[2]

C. AguerrebereA. AlmansaJ. DelonY. Gousseau and P. Musé, A bayesian hyperprior approach for joint image denoising and interpolation, with an application to hdr imaging, IEEE Transactions on Computational Imaging, 3 (2017), 633-646.  doi: 10.1109/TCI.2017.2704439.  Google Scholar

[3]

A. F. Ansari and H. Soh, Hyperprior induced unsupervised disentanglement of latent representations, Proceedings of the AAAI Conference on Artificial Intelligence, 33 (2019), 3175-3182.  doi: 10.1609/aaai.v33i01.33013175.  Google Scholar

[4]

J. Ballé, D. Minnen, S. Singh, S. J. Hwang and N. Johnston, Variational image compression with a scale hyperprior, preprint, (2018), arXiv: 1802.01436. Google Scholar

[5]

D. CalvettiH. HakulaS. Pursiainen and E. Somersalo, Conditionally gaussian hypermodels for cerebral source localization, SIAM Journal on Imaging Sciences, 2 (2009), 879-909.  doi: 10.1137/080723995.  Google Scholar

[6]

D. CalvettiA. PascarellaF. PitolliE. Somersalo and B. Vantaggi, Brain activity mapping from meg data via a hierarchical bayesian algorithm with automatic depth weighting, Brain topography, 32 (2019), 363-393.  doi: 10.1007/s10548-018-0670-7.  Google Scholar

[7]

D. Calvetti, M. Pragliola, E. Somersalo and A. Strang, Sparse reconstructions from few noisy data: Analysis of hierarchical bayesian models with generalized gamma hyperpriors, Inverse Problems, 36 (2020), 025010, 29 pp. doi: 10.1088/1361-6420/ab4d92.  Google Scholar

[8]

D. Calvetti, E. Somersalo and A. Strang, Hierachical bayesian models and sparsity: l 2-magic, Inverse Problems, 35 (2019), 035003, 26 pp. doi: 10.1088/1361-6420/aaf5ab.  Google Scholar

[9]

F. CostaH. BatatiaT. OberlinC. D'Giano and J. Tourneret, Bayesian EEG source localization using a structured sparsity prior, NeuroImage, Elsevier, 144 (2017), 142-152.  doi: 10.1016/j.neuroimage.2016.08.064.  Google Scholar

[10]

P. Del MoralA. Doucet and A. Jasra, Sequential monte carlo samplers, Journal of the Royal Statistical Society: Series B (Statistical Methodology), 68 (2006), 411-436.  doi: 10.1111/j.1467-9868.2006.00553.x.  Google Scholar

[11]

R. Douc and O. Cappé, Comparison of resampling schemes for particle filtering, ISPA 2005. Proceedings of the 4th International Symposium on Image and Signal Processing and Analysis, 2005., IEEE, (2005), 64–69. doi: 10.1109/ISPA.2005.195385.  Google Scholar

[12]

M. M. DunlopM. A. IglesiasA and M. Stuart, Hierarchical bayesian level set inversion, Statistics and Computing, 27 (2017), 1555-1584.  doi: 10.1007/s11222-016-9704-8.  Google Scholar

[13]

S. N. Evans and Philip B. Stark, Inverse problems as statistics, Inverse problems, 18 (2002), R55–R97. doi: 10.1088/0266-5611/18/4/201.  Google Scholar

[14]

A. GanesanM. RigbyA. Zammit-MangionA. ManningR. PrinnP. FraserC. HarthK. KimP. Krummel and S. Li, Characterization of uncertainties in atmospheric trace gas inversions using hierarchical bayesian methods, Atmos. Chem. Phys, 14 (2014), 3855-3864.   Google Scholar

[15]

A. Gramfort, M. Kowalski and M. Hämäläinen, Mixed-norm estimates for the m/eeg inverse problem using accelerated gradient methods, Physics in Medicine and Biology, 57 (2012), 1937. doi: 10.1088/0031-9155/57/7/1937.  Google Scholar

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A. Gramfort, M. Luessi, E. Larson, D. A. Engemann, D. Strohmeier, C. Brodbeck, R. Goj, M. Jas, T. Brooks, L. Parkkonen and et al., MNE software for processing MEG and EEG data, Frontiers in Neuroscience, 7 (2013), 267. doi: 10.3389/fnins.2013.00267.  Google Scholar

[17]

A. GramfortM. LuessiE. LarsonD. A. EngemannD. StrohmeierC. BrodbeckL. Parkkonen and M. S. Hämäläinen, MNE software for processing MEG and EEG data, Neuroimage, 86 (2014), 446-460.  doi: 10.1016/j.neuroimage.2013.10.027.  Google Scholar

[18]

A. GramfortD. StrohmeierJ. HaueisenM. S. Hämäläinen and M. Kowalski, Time-frequency mixed-norm estimates: Sparse M/EEG imaging with non-stationary source activations, NeuroImage, 70 (2013), 410-422.  doi: 10.1016/j.neuroimage.2012.12.051.  Google Scholar

[19]

P. J. Green, Reversible jump markov chain monte carlo computation and Bayesian model determination, Biometrika, 82 (1995), 711-732.  doi: 10.1093/biomet/82.4.711.  Google Scholar

[20]

M. Hämäläinen, R. Hari, R. J. Ilmoniemi, J. Knuutila and O.i V. Lounasmaa, Magnetoencephalography – theory, instrumentation, and applications to noninvasive studies of the working human brain, Reviews of Modern Physics, 65 (1993), 413. Google Scholar

[21]

W. K. Hastings, Monte Carlo sampling methods using Markov chains and their applications, Biometrika, 57 (1970), 97-109.  doi: 10.1093/biomet/57.1.97.  Google Scholar

[22]

Y. Hu, W. Yang and J. Liu, Coarse-to-fine hyper-prior modeling for learned image compression, Proc. AAAI Conf. Artif. Intell., (2020), 1–8. doi: 10.1609/aaai.v34i07.6736.  Google Scholar

[23]

J. Kaipio and E. Somersalo, Statistical and Computational Inverse Problems, Applied Mathematical Sciences, 160, Springer Science & Business Media, 2006.  Google Scholar

[24]

J. Kaipio and E. Somersalo, Statistical inverse problems: discretization, model reduction and inverse crimes, Journal of Computational and Applied Mathematics, 198 (2007), 493-504.  doi: 10.1016/j.cam.2005.09.027.  Google Scholar

[25]

J. KaiserW. LutzenbergerH. PreisslH. Ackermann and N. Birbaumer, Right-hemisphere dominance for the processing of sound-source lateralization, Journal of Neuroscience, 20 (2000), 6631-6639.  doi: 10.1523/JNEUROSCI.20-17-06631.2000.  Google Scholar

[26]

S KnakeE. HalgrenH. ShiraishiK. HaraH. HamerP. GrantV. CarrD. FoxeS. Camposano and E. Busa, The value of multichannel meg and eeg in the presurgical evaluation of 70 epilepsy patients, Epilepsy Research, 69 (2006), 80-86.  doi: 10.1016/j.eplepsyres.2006.01.001.  Google Scholar

[27]

G. LuriaD. DuranE. VisaniS. SommarivaF. RotondiD. R. SebastianoF. PanzicaM. Piana and A. Sorrentino, Bayesian multi-dipole modelling in the frequency domain, Journal of Neuroscience Methods, 312 (2019), 27-36.  doi: 10.1016/j.jneumeth.2018.11.007.  Google Scholar

[28]

J. C MosherR. M Leahy and P. S Lewis, Eeg and meg: Forward solutions for inverse methods, IEEE Transactions on Biomedical Engineering, 46 (1999), 245-259.  doi: 10.1109/10.748978.  Google Scholar

[29]

E. Niedermeyer and F. Lopes da Silva, Electroencephalography: Basic Principles, Clinical Applications, and Related Fields, Lippincott Williams & Wilkins, 2005. Google Scholar

[30]

A. NummenmaaT. AuranenM. S. HämäläinenI. P. JääskeläinenJ. LampinenM. Sams and A. Vehtari, Hierarchical Bayesian estimates of distributed MEG sources: Theoretical aspects and comparison of variational and MCMC methods, NeuroImage, 35 (2007), 669-685.  doi: 10.1016/j.neuroimage.2006.05.001.  Google Scholar

[31]

S. Pursiainen, A. Sorrentino, C. Campi and M. Piana, Forward simulation and inverse dipole localization with the lowest order Raviart-Thomas elements for electroencephalography, Inverse Problems, 27 (2011), 045003, 17 pp. doi: 10.1088/0266-5611/27/4/045003.  Google Scholar

[32]

V. Rimpiläinen, A. Koulouri, F. Lucka, J. P. Kaipio and C. H. Wolters, Bayesian modelling of skull conductivity uncertainties in eeg source imaging, EMBEC & NBC 2017, Springer, (2017), 892–895. Google Scholar

[33]

J. Sarvas, Basic mathematical and electromagnetic concepts of the biomagnetic inverse problem, Phys. Med. Biol., 32 (1987), 11-22.  doi: 10.1088/0031-9155/32/1/004.  Google Scholar

[34]

D. SchuhmacherB.-T. Vo and B.-N. Vo, A consistent metric for performance evaluation of multi-object filters, IEEE Transactions on Signal Processing, 56 (2008), 3447-3457.  doi: 10.1109/TSP.2008.920469.  Google Scholar

[35]

F. SciacchitanoS. Lugaro and A. Sorrentino, Sparse bayesian imaging of solar flares, SIAM Journal on Imaging Sciences, 12 (2019), 319-343.  doi: 10.1137/18M1204103.  Google Scholar

[36]

S. Sommariva and A. Sorrentino, Sequential monte carlo samplers for semi-linear inverse problems and application to magnetoencephalography, Inverse Problems, 30 (2014), 114020, 23 pp. doi: 10.1088/0266-5611/30/11/114020.  Google Scholar

[37]

A. Sorrentino, G. Luria and R. Aramini, Bayesian multi-dipole modelling of a single topography in meg by adaptive sequential monte carlo samplers, Inverse Problems, 30 (2014), 045010, 22 pp. doi: 10.1088/0266-5611/30/4/045010.  Google Scholar

show all references

References:
[1]

Z. A. Acar and S. Makeig, Neuroelectromagnetic forward head modeling toolbox, Journal of Neuroscience Methods, 190 (2010), 258-270.  doi: 10.1016/j.jneumeth.2010.04.031.  Google Scholar

[2]

C. AguerrebereA. AlmansaJ. DelonY. Gousseau and P. Musé, A bayesian hyperprior approach for joint image denoising and interpolation, with an application to hdr imaging, IEEE Transactions on Computational Imaging, 3 (2017), 633-646.  doi: 10.1109/TCI.2017.2704439.  Google Scholar

[3]

A. F. Ansari and H. Soh, Hyperprior induced unsupervised disentanglement of latent representations, Proceedings of the AAAI Conference on Artificial Intelligence, 33 (2019), 3175-3182.  doi: 10.1609/aaai.v33i01.33013175.  Google Scholar

[4]

J. Ballé, D. Minnen, S. Singh, S. J. Hwang and N. Johnston, Variational image compression with a scale hyperprior, preprint, (2018), arXiv: 1802.01436. Google Scholar

[5]

D. CalvettiH. HakulaS. Pursiainen and E. Somersalo, Conditionally gaussian hypermodels for cerebral source localization, SIAM Journal on Imaging Sciences, 2 (2009), 879-909.  doi: 10.1137/080723995.  Google Scholar

[6]

D. CalvettiA. PascarellaF. PitolliE. Somersalo and B. Vantaggi, Brain activity mapping from meg data via a hierarchical bayesian algorithm with automatic depth weighting, Brain topography, 32 (2019), 363-393.  doi: 10.1007/s10548-018-0670-7.  Google Scholar

[7]

D. Calvetti, M. Pragliola, E. Somersalo and A. Strang, Sparse reconstructions from few noisy data: Analysis of hierarchical bayesian models with generalized gamma hyperpriors, Inverse Problems, 36 (2020), 025010, 29 pp. doi: 10.1088/1361-6420/ab4d92.  Google Scholar

[8]

D. Calvetti, E. Somersalo and A. Strang, Hierachical bayesian models and sparsity: l 2-magic, Inverse Problems, 35 (2019), 035003, 26 pp. doi: 10.1088/1361-6420/aaf5ab.  Google Scholar

[9]

F. CostaH. BatatiaT. OberlinC. D'Giano and J. Tourneret, Bayesian EEG source localization using a structured sparsity prior, NeuroImage, Elsevier, 144 (2017), 142-152.  doi: 10.1016/j.neuroimage.2016.08.064.  Google Scholar

[10]

P. Del MoralA. Doucet and A. Jasra, Sequential monte carlo samplers, Journal of the Royal Statistical Society: Series B (Statistical Methodology), 68 (2006), 411-436.  doi: 10.1111/j.1467-9868.2006.00553.x.  Google Scholar

[11]

R. Douc and O. Cappé, Comparison of resampling schemes for particle filtering, ISPA 2005. Proceedings of the 4th International Symposium on Image and Signal Processing and Analysis, 2005., IEEE, (2005), 64–69. doi: 10.1109/ISPA.2005.195385.  Google Scholar

[12]

M. M. DunlopM. A. IglesiasA and M. Stuart, Hierarchical bayesian level set inversion, Statistics and Computing, 27 (2017), 1555-1584.  doi: 10.1007/s11222-016-9704-8.  Google Scholar

[13]

S. N. Evans and Philip B. Stark, Inverse problems as statistics, Inverse problems, 18 (2002), R55–R97. doi: 10.1088/0266-5611/18/4/201.  Google Scholar

[14]

A. GanesanM. RigbyA. Zammit-MangionA. ManningR. PrinnP. FraserC. HarthK. KimP. Krummel and S. Li, Characterization of uncertainties in atmospheric trace gas inversions using hierarchical bayesian methods, Atmos. Chem. Phys, 14 (2014), 3855-3864.   Google Scholar

[15]

A. Gramfort, M. Kowalski and M. Hämäläinen, Mixed-norm estimates for the m/eeg inverse problem using accelerated gradient methods, Physics in Medicine and Biology, 57 (2012), 1937. doi: 10.1088/0031-9155/57/7/1937.  Google Scholar

[16]

A. Gramfort, M. Luessi, E. Larson, D. A. Engemann, D. Strohmeier, C. Brodbeck, R. Goj, M. Jas, T. Brooks, L. Parkkonen and et al., MNE software for processing MEG and EEG data, Frontiers in Neuroscience, 7 (2013), 267. doi: 10.3389/fnins.2013.00267.  Google Scholar

[17]

A. GramfortM. LuessiE. LarsonD. A. EngemannD. StrohmeierC. BrodbeckL. Parkkonen and M. S. Hämäläinen, MNE software for processing MEG and EEG data, Neuroimage, 86 (2014), 446-460.  doi: 10.1016/j.neuroimage.2013.10.027.  Google Scholar

[18]

A. GramfortD. StrohmeierJ. HaueisenM. S. Hämäläinen and M. Kowalski, Time-frequency mixed-norm estimates: Sparse M/EEG imaging with non-stationary source activations, NeuroImage, 70 (2013), 410-422.  doi: 10.1016/j.neuroimage.2012.12.051.  Google Scholar

[19]

P. J. Green, Reversible jump markov chain monte carlo computation and Bayesian model determination, Biometrika, 82 (1995), 711-732.  doi: 10.1093/biomet/82.4.711.  Google Scholar

[20]

M. Hämäläinen, R. Hari, R. J. Ilmoniemi, J. Knuutila and O.i V. Lounasmaa, Magnetoencephalography – theory, instrumentation, and applications to noninvasive studies of the working human brain, Reviews of Modern Physics, 65 (1993), 413. Google Scholar

[21]

W. K. Hastings, Monte Carlo sampling methods using Markov chains and their applications, Biometrika, 57 (1970), 97-109.  doi: 10.1093/biomet/57.1.97.  Google Scholar

[22]

Y. Hu, W. Yang and J. Liu, Coarse-to-fine hyper-prior modeling for learned image compression, Proc. AAAI Conf. Artif. Intell., (2020), 1–8. doi: 10.1609/aaai.v34i07.6736.  Google Scholar

[23]

J. Kaipio and E. Somersalo, Statistical and Computational Inverse Problems, Applied Mathematical Sciences, 160, Springer Science & Business Media, 2006.  Google Scholar

[24]

J. Kaipio and E. Somersalo, Statistical inverse problems: discretization, model reduction and inverse crimes, Journal of Computational and Applied Mathematics, 198 (2007), 493-504.  doi: 10.1016/j.cam.2005.09.027.  Google Scholar

[25]

J. KaiserW. LutzenbergerH. PreisslH. Ackermann and N. Birbaumer, Right-hemisphere dominance for the processing of sound-source lateralization, Journal of Neuroscience, 20 (2000), 6631-6639.  doi: 10.1523/JNEUROSCI.20-17-06631.2000.  Google Scholar

[26]

S KnakeE. HalgrenH. ShiraishiK. HaraH. HamerP. GrantV. CarrD. FoxeS. Camposano and E. Busa, The value of multichannel meg and eeg in the presurgical evaluation of 70 epilepsy patients, Epilepsy Research, 69 (2006), 80-86.  doi: 10.1016/j.eplepsyres.2006.01.001.  Google Scholar

[27]

G. LuriaD. DuranE. VisaniS. SommarivaF. RotondiD. R. SebastianoF. PanzicaM. Piana and A. Sorrentino, Bayesian multi-dipole modelling in the frequency domain, Journal of Neuroscience Methods, 312 (2019), 27-36.  doi: 10.1016/j.jneumeth.2018.11.007.  Google Scholar

[28]

J. C MosherR. M Leahy and P. S Lewis, Eeg and meg: Forward solutions for inverse methods, IEEE Transactions on Biomedical Engineering, 46 (1999), 245-259.  doi: 10.1109/10.748978.  Google Scholar

[29]

E. Niedermeyer and F. Lopes da Silva, Electroencephalography: Basic Principles, Clinical Applications, and Related Fields, Lippincott Williams & Wilkins, 2005. Google Scholar

[30]

A. NummenmaaT. AuranenM. S. HämäläinenI. P. JääskeläinenJ. LampinenM. Sams and A. Vehtari, Hierarchical Bayesian estimates of distributed MEG sources: Theoretical aspects and comparison of variational and MCMC methods, NeuroImage, 35 (2007), 669-685.  doi: 10.1016/j.neuroimage.2006.05.001.  Google Scholar

[31]

S. Pursiainen, A. Sorrentino, C. Campi and M. Piana, Forward simulation and inverse dipole localization with the lowest order Raviart-Thomas elements for electroencephalography, Inverse Problems, 27 (2011), 045003, 17 pp. doi: 10.1088/0266-5611/27/4/045003.  Google Scholar

[32]

V. Rimpiläinen, A. Koulouri, F. Lucka, J. P. Kaipio and C. H. Wolters, Bayesian modelling of skull conductivity uncertainties in eeg source imaging, EMBEC & NBC 2017, Springer, (2017), 892–895. Google Scholar

[33]

J. Sarvas, Basic mathematical and electromagnetic concepts of the biomagnetic inverse problem, Phys. Med. Biol., 32 (1987), 11-22.  doi: 10.1088/0031-9155/32/1/004.  Google Scholar

[34]

D. SchuhmacherB.-T. Vo and B.-N. Vo, A consistent metric for performance evaluation of multi-object filters, IEEE Transactions on Signal Processing, 56 (2008), 3447-3457.  doi: 10.1109/TSP.2008.920469.  Google Scholar

[35]

F. SciacchitanoS. Lugaro and A. Sorrentino, Sparse bayesian imaging of solar flares, SIAM Journal on Imaging Sciences, 12 (2019), 319-343.  doi: 10.1137/18M1204103.  Google Scholar

[36]

S. Sommariva and A. Sorrentino, Sequential monte carlo samplers for semi-linear inverse problems and application to magnetoencephalography, Inverse Problems, 30 (2014), 114020, 23 pp. doi: 10.1088/0266-5611/30/11/114020.  Google Scholar

[37]

A. Sorrentino, G. Luria and R. Aramini, Bayesian multi-dipole modelling of a single topography in meg by adaptive sequential monte carlo samplers, Inverse Problems, 30 (2014), 045010, 22 pp. doi: 10.1088/0266-5611/30/4/045010.  Google Scholar

Figure 1.  An example of simulated EEG (top) and MEG (bottom) recordings
Figure 2.  Confusion matrices for the estimated number of dipoles, for three different values of the prior scale factor $ k $: in the top panel results obtained with simulated EEG data, in the bottom panel results obtained with simulated MEG data
Figure 3.  Boxplots of the OSPA metric, quantifying the distance between the true and estimated dipole configurations, for three different values of the prior scale factor $ k $: in red the SESAME results, in blue the h-SESAME results; in the top panel results obtained with simulated EEG data, in the bottom panel results obtained with simulated MEG data
Figure 4.  Variance of the posterior probability map with respect to different values of the prior scale factor $ k $, as defined in eq. (16): in red the SESAME results, in blue the h-SESAME results
Figure 5.  Estimated value of the prior width $ \sigma_q $, for different values of the prior scale factor $ k $
Figure 6.  Experimental MEG data: averaged response to auditory stimuli; data taken from the sample open dataset within the MNE–Python package
Figure 6 with different values of the prior scale factor: $ k = 0.1 $ (top row), $ k = 1 $ (middle row) and $ k = 10 $ (bottom row)">Figure 7.  Posterior probability maps (left) and estimated source time courses (right) obtained by SESAME when applied to the experimental data shown in Figure 6 with different values of the prior scale factor: $ k = 0.1 $ (top row), $ k = 1 $ (middle row) and $ k = 10 $ (bottom row)
Figure 6 with different values of the prior scale factor: $ k = 0.1 $ (top row), $ k = 1 $ (middle row) and $ k = 10 $ (bottom row)">Figure 8.  Posterior probability maps (left) and estimated source time courses (right) obtained by h-SESAME when applied to the experimental data shown in Figure 6 with different values of the prior scale factor: $ k = 0.1 $ (top row), $ k = 1 $ (middle row) and $ k = 10 $ (bottom row)
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