October  2021, 15(5): 1099-1119. 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  October 2021 Early access  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 and Imaging, 2021, 15 (5) : 1099-1119. 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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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. 

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[23]

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

[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.

[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.

[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.

[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.

[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.

[29]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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. 

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[23]

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

[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.

[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.

[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.

[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.

[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.

[29]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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 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 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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