Exact Hamiltonian Monte Carlo for Truncated Multivariate Gaussians

Pakman, Ari; Paninski, Liam

doi:10.1080/10618600.2013.788448

Cited by 155 publications

(181 citation statements)

References 49 publications

(64 reference statements)

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“…Hamiltonian dynamics are usually simulated by discretization methods such as Euler or leapfrog methods (as will be shown in Section 4.2). However, when U (q) is quadratic, it has been shown [14] that it is possible to simulate exactly from f (q, p). This method is summarized below.…”

Section: Hamiltonian Monte Carlo Methodsmentioning

confidence: 99%

“…In this section, we summarize the Exact Hamiltonian Monte Carlo (EHMC) method investigated in [14] to sample from (4)…”

Section: Exact Hamiltonian Monte Carlomentioning

confidence: 99%

“…However, new HMC methods have been recently proposed to handle constrained variables [8,Chap. 5] [12][13][14] which allow HMCs to sample according to TMGDs restricted to the standard simplex.…”

Section: Introductionmentioning

confidence: 99%

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Sampling from a multivariate Gaussian distribution truncated on a simplex: A review

Altmann

McLaughlin

Dobigeon

2014

2014 IEEE Workshop on Statistical Signal Processing (SSP)

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In many Bayesian models, the posterior distribution of interest is a multivariate Gaussian distribution restricted to a specific domain. In particular, when the unknown parameters to be estimated can be considered as proportions or probabilities, they must satisfy positivity and sum-to-one constraints. This paper reviews recent Monte Carlo methods for sampling from multivariate Gaussian distributions restricted to the standard simplex. First, a classical Gibbs sampler is presented. Then, two Hamiltonian Monte Carlo methods are described and analyzed. In a similar fashion to the Gibbs sampler, the first method has a acceptance rate equal to one whereas the second requires an accept/reject procedure. The performance of the three methods are compared through the use of a few examples.Index Terms-Markov Chain Monte Carlo methods, truncated multivariate Gaussian distributions, Constrained Hamiltonian Monte Carlo.

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Section: Hamiltonian Monte Carlo Methodsmentioning

confidence: 99%

“…In this section, we summarize the Exact Hamiltonian Monte Carlo (EHMC) method investigated in [14] to sample from (4)…”

Section: Exact Hamiltonian Monte Carlomentioning

confidence: 99%

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Sampling from a multivariate Gaussian distribution truncated on a simplex: A review

Altmann

McLaughlin

Dobigeon

2014

2014 IEEE Workshop on Statistical Signal Processing (SSP)

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“…2 is a multivariate Gaussian distribution restricted to the simplex S, which can be sampled efficiently using the method recently proposed in [20]. Noise variance σ 2 : Sampling σ 2 from its conditional distribution is not straightforward.…”

Section: Bayesian Inference Using a Metropolis-within-gibbs Samplermentioning

confidence: 99%

Residual component analysis of hyperspectral images for joint nonlinear unmixing and nonlinearity detection

Altmann

Dobigeon

McLaughlin

et al. 2014

2014 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)

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This paper presents a nonlinear mixing model for joint hyperspectral image unmixing and nonlinearity detection. The proposed model assumes that the pixel reflectances are linear mixtures of endmembers, corrupted by an additional nonlinear term and an additive Gaussian noise. A Markov random field is considered for nonlinearity detection based on the spatial structure of the nonlinear terms. The observed image is segmented into regions where nonlinear terms, if present, share similar statistical properties. A Bayesian algorithm is proposed to estimate the parameters involved in the model yielding a joint nonlinear unmixing and nonlinearity detection algorithm. Simulations conducted with synthetic and real data show the accuracy of the proposed unmixing and nonlinearity detection strategy for the analysis of hyperspectral images.

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“…In particular we use a Gibbs sampler that cyclically samples a, τ 2 and (W, s). For the latter, we use an exact Hamiltonian Monte Carlo sampler based on the method of (Pakman and Paninski, 2013). The sign constraint from Dale's law can be imposed by simply restricting (2.23) to be non-zero only for W i ≥ 0 or W i ≤ 0.…”

Section: A Fully Bayesian Approachmentioning

confidence: 99%

Fast state-space methods for inferring dendritic synaptic connectivity

et al. 2013

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We present fast methods for filtering voltage measurements and performing optimal inference of the location and strength of synaptic connections in large dendritic trees. Given noisy, subsampled voltage observations we develop fast l 1 -penalized regression methods for Kalman state-space models of the neuron voltage dynamics. The value of the l 1 -penalty parameter is chosen using crossvalidation or, for low signal-to-noise ratio, a Mallows' C p -like criterion. Using low-rank approximations, we reduce the inference runtime from cubic to linear in the number of dendritic compartments. We also present an alternative, fully Bayesian approach to the inference problem using a spike-andslab prior. We illustrate our results with simulations on toy and real neuronal geometries. We consider observation schemes that either scan the dendritic geometry uniformly or measure linear combinations of voltages across several locations with random coefficients. For the latter, we show how to choose the coefficients to offset the correlation between successive measurements imposed by the neuron dynamics. This results in a "compressed sensing" observation scheme, with an important reduction in the number of measurements required to infer the synaptic weights.

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Exact Hamiltonian Monte Carlo for Truncated Multivariate Gaussians

Cited by 155 publications

References 49 publications

Sampling from a multivariate Gaussian distribution truncated on a simplex: A review

Sampling from a multivariate Gaussian distribution truncated on a simplex: A review

Residual component analysis of hyperspectral images for joint nonlinear unmixing and nonlinearity detection

Fast state-space methods for inferring dendritic synaptic connectivity

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