Download - Signal decompositions using trans-dimensional Bayesian methods: Alireza Roodaki's Ph.D. defense
Relabeling and summarizing posterior distributionsProposed approach
ResultsConclusion
Signal decompositions usingtrans-dimensional Bayesian methods
Alireza Roodaki
Ph.D. Thesis Defense
Department of Signal Processing and Electronic Systems
2012, May 14th
Advisors: Julien Bect and Gilles Fleury
Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 1/ 52
Relabeling and summarizing posterior distributionsProposed approach
ResultsConclusion
Example 1: Detection and estimation of muons in theAuger project
Figure: A conceptual shower(http://auger.org).
Ultra high energy particlescoming from space(E ∼ 1019eV)
How and where?
What is their composition(Proton, Iron)?
Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 1/ 52
Relabeling and summarizing posterior distributionsProposed approach
ResultsConclusion
Example 1: Detection and estimation of muons in theAuger project
Figure: A conceptual shower(http://auger.org).
Ultra high energy particlescoming from space(E ∼ 1019eV)
How and where?
What is their composition(Proton, Iron)?
Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 1/ 52
Relabeling and summarizing posterior distributionsProposed approach
ResultsConclusion
Example 1: Detection and estimation of muons in theAuger project
Figure: A conceptual shower(http://auger.org).
Ultra high energy particlescoming from space(E ∼ 1019eV)
How and where?
What is their composition(Proton, Iron)?
Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 1/ 52
Relabeling and summarizing posterior distributionsProposed approach
ResultsConclusion
Example 1: Detection and estimation of muons in theAuger project (Contd.)
Figure: A conceptual shower anddetectors (water tanks)(http://auger.org).
muons are generatedwhen particles cross theatmosphere
the number k of muonsand their arrival times tµare indicators of the originand composition of particle
Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 2/ 52
Relabeling and summarizing posterior distributionsProposed approach
ResultsConclusion
Example 1: Detection and estimation of muons in theAuger project (Contd.)
Figure: A conceptual shower anddetectors (water tanks)(http://auger.org).
muons are generatedwhen particles cross theatmosphere
the number k of muonsand their arrival times tµare indicators of the originand composition of particle
Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 2/ 52
Relabeling and summarizing posterior distributionsProposed approach
ResultsConclusion
Example 1: Detection and estimation of muons in theAuger project (Contd.)
Figure: A conceptual shower anddetectors (water tanks)(http://auger.org).
muons are generatedwhen particles cross theatmosphere
the number k of muonsand their arrival times tµare indicators of the originand composition of particle
Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 2/ 52
Relabeling and summarizing posterior distributionsProposed approach
ResultsConclusion
Example 1: Detection and estimation of muons in theAuger project (Contd.)
Figure: Water tank detector.
Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 3/ 52
Relabeling and summarizing posterior distributionsProposed approach
ResultsConclusion
Example 1: Detection and estimation of muons in theAuger project (Contd.)
#P
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nsity
100 200 300 400 500 6000
0.5
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Figure: Observed signal (n)
Prof. Balázs Kégl from LAL, University of Paris 11.Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 4/ 52
Relabeling and summarizing posterior distributionsProposed approach
ResultsConclusion
Example 1: Detection and estimation of muons in theAuger project (Contd.)
#P
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t [ns]
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nsity
100 200 300 400 500 6000
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Figure: Observed signal (n)
Prof. Balázs Kégl from LAL, University of Paris 11.Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 4/ 52
Relabeling and summarizing posterior distributionsProposed approach
ResultsConclusion
Example 2: Spectral analysis
time
Pow
er
radial frequency0 0.5 1 1.5 2 2.5 3
0 10 20 30 40 50 60
0
50
100
150
−10
0
10
Figure: Observed signal (top) and itsperiodogram (bottom).
Applications
RADAR / SONAR
Array signal processing
Vibration analysis
. . .
Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 5/ 52
Relabeling and summarizing posterior distributionsProposed approach
ResultsConclusion
Example 2: Spectral analysis (Cont.)
Detection and estimation of sinusoids in white noisemodel the observed signal y by sinusoidal components
Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 6/ 52
Relabeling and summarizing posterior distributionsProposed approach
ResultsConclusion
Example 2: Spectral analysis (Cont.)
Detection and estimation of sinusoids in white noisemodel the observed signal y by sinusoidal components
observed signal
Mk : y [i] =
k∑
j=1
(
aj cos[ωj i] + bj sin[ωj i])
+ n[i].
Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 6/ 52
Relabeling and summarizing posterior distributionsProposed approach
ResultsConclusion
Example 2: Spectral analysis (Cont.)
Detection and estimation of sinusoids in white noisemodel the observed signal y by sinusoidal components
observed signal
Mk : y [i] =
k∑
j=1
(
aj cos[ωj i] + bj sin[ωj i])
+ n[i].
Joint model selection and parameter estimation problem
Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 6/ 52
Relabeling and summarizing posterior distributionsProposed approach
ResultsConclusion
Trans-dimensional problems
Def. The problems in which the number of things that wedon′t know is one of the things that we don′t know [Green,
2003.]
Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 7/ 52
Relabeling and summarizing posterior distributionsProposed approach
ResultsConclusion
Trans-dimensional problems
Def. The problems in which the number of things that wedon′t know is one of the things that we don′t know [Green,
2003.]
space X =⋃
k∈K{k} ×Θk with points x = (k ,θk )
➠ k ∈ K denotes number of components
➠ θk ∈ Θk is a vector of component-specific
parameters
Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 7/ 52
Relabeling and summarizing posterior distributionsProposed approach
ResultsConclusion
Trans-dimensional problems
Def. The problems in which the number of things that wedon′t know is one of the things that we don′t know [Green,
2003.]
space X =⋃
k∈K{k} ×Θk with points x = (k ,θk )
➠ k ∈ K denotes number of components
➠ θk ∈ Θk is a vector of component-specific
parameters
Applications:➠ Spectral Analysis (Array signal processing)
➠ (Gaussian) Mixture modeling & Clustering
➠ Object detection and recognition
Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 7/ 52
Relabeling and summarizing posterior distributionsProposed approach
ResultsConclusion
Bayesian inference
Likelihood
p(x | y) =p(y | x) p(x)
∫
Xp(y | x ′) p(x ′)dx ′
Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 8/ 52
Relabeling and summarizing posterior distributionsProposed approach
ResultsConclusion
Bayesian inference
Likelihood
p(x | y) =p(y | x) p(x)
∫
Xp(y | x ′) p(x ′)dx ′
Prior distribution
Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 8/ 52
Relabeling and summarizing posterior distributionsProposed approach
ResultsConclusion
Bayesian inference
Likelihood
p(x | y) =p(y | x) p(x)
∫
Xp(y | x ′) p(x ′)dx ′
Prior distributionPosterior distribution
Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 8/ 52
Relabeling and summarizing posterior distributionsProposed approach
ResultsConclusion
Bayesian inference
Likelihood
p(x | y) =p(y | x) p(x)
∫
Xp(y | x ′) p(x ′)dx ′
Prior distributionPosterior distribution
x = (k ,θk )
Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 8/ 52
Relabeling and summarizing posterior distributionsProposed approach
ResultsConclusion
Bayesian inference
Likelihood
p(x | y) =p(y | x) p(x)
∫
Xp(y | x ′) p(x ′)dx ′
Prior distributionPosterior distribution
x = (k ,θk )
➠ both detection and estimation problems
Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 8/ 52
Relabeling and summarizing posterior distributionsProposed approach
ResultsConclusion
Bayesian inference
Likelihood
p(x | y) =p(y | x) p(x)
∫
Xp(y | x ′) p(x ′)dx ′
Prior distributionPosterior distribution
x = (k ,θk )
➠ both detection and estimation problems
high-dimensional / intractable integrals
Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 8/ 52
Relabeling and summarizing posterior distributionsProposed approach
ResultsConclusion
Markov Chain Monte Carlo (MCMC) methods
generate samples from the posterior distribution of interest(target distribution), say, π.
construct a Markov chain (x (1), . . . , x (M)) that under someconditions converges to π.
Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 9/ 52
Relabeling and summarizing posterior distributionsProposed approach
ResultsConclusion
Markov Chain Monte Carlo (MCMC) methods
generate samples from the posterior distribution of interest(target distribution), say, π.
construct a Markov chain (x (1), . . . , x (M)) that under someconditions converges to π.
Famous algorithms:➠ Metropolis-Hastings (MH) sampler [Metropolis, et al.
1953, Hastings, 1970.]
➠ Gibbs sampler [Geman and Geman, 1984.]
➠ RJ-MCMC sampler [Green, 1995.]
[Robert and Casella, 2004.]
Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 9/ 52
Relabeling and summarizing posterior distributionsProposed approach
ResultsConclusion
Markov Chain Monte Carlo (MCMC) methods
generate samples from the posterior distribution of interest(target distribution), say, π.
construct a Markov chain (x (1), . . . , x (M)) that under someconditions converges to π.
Famous algorithms:➠ Metropolis-Hastings (MH) sampler [Metropolis, et al.
1953, Hastings, 1970.]
➠ Gibbs sampler [Geman and Geman, 1984.]
➠ RJ-MCMC sampler [Green, 1995.]
[Robert and Casella, 2004.]
Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 9/ 52
Relabeling and summarizing posterior distributionsProposed approach
ResultsConclusion
Example 2: spectral analysis (Cont.)
RJ-MCMC sampler ⇒ variable dimensional samples
ωk
k
Iteration number160 170 180 190 200
2
34
0.4
0.6
0.8
Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 10/ 52
Outline
1 Relabeling and summarizing posterior distributionsLabel-switching issueVariable-dimensional summarization
Outline
1 Relabeling and summarizing posterior distributionsLabel-switching issueVariable-dimensional summarization
2 Proposed approachAn original variable-dimensional parametric modelEstimating the model parameters (SEM-type algorithms)Robustifying strategies
Outline
1 Relabeling and summarizing posterior distributionsLabel-switching issueVariable-dimensional summarization
2 Proposed approachAn original variable-dimensional parametric modelEstimating the model parameters (SEM-type algorithms)Robustifying strategies
3 ResultsDetection and estimation of sinusoids in white noiseDetection and estimation of muons in the Auger project
Outline
1 Relabeling and summarizing posterior distributionsLabel-switching issueVariable-dimensional summarization
2 Proposed approachAn original variable-dimensional parametric modelEstimating the model parameters (SEM-type algorithms)Robustifying strategies
3 ResultsDetection and estimation of sinusoids in white noiseDetection and estimation of muons in the Auger project
4 Conclusion
Outline
1 Relabeling and summarizing posterior distributionsLabel-switching issueVariable-dimensional summarization
2 Proposed approachAn original variable-dimensional parametric modelEstimating the model parameters (SEM-type algorithms)Robustifying strategies
3 ResultsDetection and estimation of sinusoids in white noiseDetection and estimation of muons in the Auger project
4 Conclusion
Relabeling and summarizing posterior distributionsProposed approach
ResultsConclusion
Label-switching issueVariable-dimensional summarization
summarizing posterior distributions
Posterior = all the information
Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 12/ 52
Relabeling and summarizing posterior distributionsProposed approach
ResultsConclusion
Label-switching issueVariable-dimensional summarization
summarizing posterior distributions
Posterior = all the information➠ It is a complex mathematical object (not easy to
manipulate)
Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 12/ 52
Relabeling and summarizing posterior distributionsProposed approach
ResultsConclusion
Label-switching issueVariable-dimensional summarization
summarizing posterior distributions
Posterior = all the information➠ It is a complex mathematical object (not easy to
manipulate)
(RJ)-MCMC sampler
Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 12/ 52
Relabeling and summarizing posterior distributionsProposed approach
ResultsConclusion
Label-switching issueVariable-dimensional summarization
summarizing posterior distributions
Posterior = all the information➠ It is a complex mathematical object (not easy to
manipulate)
(RJ)-MCMC sampler➠ What to do with the generated samples?
Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 12/ 52
Relabeling and summarizing posterior distributionsProposed approach
ResultsConclusion
Label-switching issueVariable-dimensional summarization
summarizing posterior distributions
Posterior = all the information➠ It is a complex mathematical object (not easy to
manipulate)
(RJ)-MCMC sampler➠ What to do with the generated samples?
Summarizationhuman readable summaries
Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 12/ 52
Relabeling and summarizing posterior distributionsProposed approach
ResultsConclusion
Label-switching issueVariable-dimensional summarization
summarizing posterior distributions
Posterior = all the information➠ It is a complex mathematical object (not easy to
manipulate)
(RJ)-MCMC sampler➠ What to do with the generated samples?
Summarizationhuman readable summaries
interpretable figures (e.g., histograms)
Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 12/ 52
Relabeling and summarizing posterior distributionsProposed approach
ResultsConclusion
Label-switching issueVariable-dimensional summarization
summarizing posterior distributions
Posterior = all the information➠ It is a complex mathematical object (not easy to
manipulate)
(RJ)-MCMC sampler➠ What to do with the generated samples?
Summarizationhuman readable summaries
interpretable figures (e.g., histograms)
statistical measures (e.g., mean and variance)
Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 12/ 52
Relabeling and summarizing posterior distributionsProposed approach
ResultsConclusion
Label-switching issueVariable-dimensional summarization
Fixed-dimensional problems
uni-modal uni-variate case
Samples
p
0 2 40
0.2
0.4
0.6
report location (mean and median) and dispersion(variance and confidence intervals) parameters
µ σ2
2.0 0.25
Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 13/ 52
Relabeling and summarizing posterior distributionsProposed approach
ResultsConclusion
Label-switching issueVariable-dimensional summarization
Label-switching issue
Additive mixture: lack of identifiability
the likelihood is invariant under relabeling of components
Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 14/ 52
Relabeling and summarizing posterior distributionsProposed approach
ResultsConclusion
Label-switching issueVariable-dimensional summarization
Label-switching issue
Additive mixture: lack of identifiability
the likelihood is invariant under relabeling of components
the posterior distribution is invariant under permutation ofcomponents
Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 14/ 52
Relabeling and summarizing posterior distributionsProposed approach
ResultsConclusion
Label-switching issueVariable-dimensional summarization
Label-switching issue
Additive mixture: lack of identifiability
the likelihood is invariant under relabeling of components
the posterior distribution is invariant under permutation ofcomponents
Comp. #1
Comp. #2
ω
Comp. #3
0.5 0.75 10
100
100
10Marginal posteriors ofcomponent-specificparameters are identical!
Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 14/ 52
Relabeling and summarizing posterior distributionsProposed approach
ResultsConclusion
Label-switching issueVariable-dimensional summarization
Label-switching issue
Additive mixture: lack of identifiability
the likelihood is invariant under relabeling of components
the posterior distribution is invariant under permutation ofcomponents
Comp. #1
Comp. #2
ω
Comp. #3
0.5 0.75 10
100
100
10Marginal posteriors ofcomponent-specificparameters are identical!How to summarize theposterior information?
Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 14/ 52
Relabeling and summarizing posterior distributionsProposed approach
ResultsConclusion
Label-switching issueVariable-dimensional summarization
strategies to deal with label-switching
imposing artificial “identifiability constraints”Exp: sorting the components [Richardson and Green, 1997.]
Comp. #1
Comp. #2
Comp. #3
0.5 0.75 10
10
0
10
010
Figure: components are sorted based on ω.
Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 15/ 52
Relabeling and summarizing posterior distributionsProposed approach
ResultsConclusion
Label-switching issueVariable-dimensional summarization
strategies to deal with label-switching
imposing artificial “identifiability constraints”Exp: sorting the components [Richardson and Green, 1997.]
Comp. #1
Comp. #2
Comp. #3
0.5 0.75 10
10
0
10
010
Figure: components are sorted based on ω.
relabeling algorithms [Celeux, et al. 1998, Stephens, 2000, Jasra,
et al, 2005, Sperrin et al, 2010, Yao, 2011.].
Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 15/ 52
Relabeling and summarizing posterior distributionsProposed approach
ResultsConclusion
Label-switching issueVariable-dimensional summarization
Example 2: spectral analysis (Cont.)Variable-dimensional posterior distribution
k
p(k |y) ω0.5 0.75 10 0.3 0.6
2
3
4
Figure: Posteriors of k and sorted radial frequencies, ωk , given k .Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 16/ 52
Relabeling and summarizing posterior distributionsProposed approach
ResultsConclusion
Label-switching issueVariable-dimensional summarization
Classical Bayesian approaches
Bayesian Model Selection (BMS)
One model is selected (estimated) by looking at the MAP,i.e. k = argmax p(k |y).
Component-specific parameters are summarized givenk = k .
Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 17/ 52
Relabeling and summarizing posterior distributionsProposed approach
ResultsConclusion
Label-switching issueVariable-dimensional summarization
Bayesian Model Selection (BMS)
k
p(k |y) ω
0.5 0.75 10 0.3 0.6
3
2
3
4
Figure: The model with k = 2 is selected.
Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 18/ 52
Relabeling and summarizing posterior distributionsProposed approach
ResultsConclusion
Label-switching issueVariable-dimensional summarization
Classical Bayesian approaches
Bayesian Model Selection (BMS)
One model is selected (estimated) by looking at the MAP,i.e. k = argmax p(k |y).
Component-specific parameters are summarized givenk = k .
Bayesian Model Averaging (BMA)
Use the information from all possible models:p(∆|y) =
∑
k p(∆|k , y)p(k |y)
However, ∆ cannot be ωk as its size changes from modelto model.
Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 19/ 52
Relabeling and summarizing posterior distributionsProposed approach
ResultsConclusion
Label-switching issueVariable-dimensional summarization
Bayesian Model Averaging (BMA)
Binned data representation (∆ = N(Bj)):
E(N(Bj) | y) =
kmax∑
k=1
E(N(Bj) | k , y) · p(k | y)
wherej = 1, . . . ,Nbinand E(N(Bj)) is theexpected number ofcomponents in binBj .
expe
cted
nbr
com
p
ω0 0.5 1 1.5 2 2.5 3
0
.25
.5
.75
1
Figure: Expected number of componentsusing BMA.
Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 20/ 52
Relabeling and summarizing posterior distributionsProposed approach
ResultsConclusion
Label-switching issueVariable-dimensional summarization
Are BMS and BMA approaches satisfactory?
Bayesian Model Selection (BMS)
➠ selects a model ⇒ component-specific parameters
➠ losing information from the discarded models
➠ ignoring the uncertainties about the presence ofcomponents.
Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 21/ 52
Relabeling and summarizing posterior distributionsProposed approach
ResultsConclusion
Label-switching issueVariable-dimensional summarization
Are BMS and BMA approaches satisfactory? (Cont.)
Bayesian Model Averaging (BMA)
➠ appropriate for signal reconstruction and prediction
➠ does not provide information about component-specificparameters
Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 22/ 52
Relabeling and summarizing posterior distributionsProposed approach
ResultsConclusion
Label-switching issueVariable-dimensional summarization
A novel approach is needed!
A novel approach is needed!
Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 23/ 52
Relabeling and summarizing posterior distributionsProposed approach
ResultsConclusion
Label-switching issueVariable-dimensional summarization
A novel approach is needed!
A novel approach is needed!
Properties of an “ideal” approach
information from all (plausible) models➠ interpretable summaries for component-specific
parameters
uncertainties about the presence of components
Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 23/ 52
Outline
1 Relabeling and summarizing posterior distributionsLabel-switching issueVariable-dimensional summarization
2 Proposed approachAn original variable-dimensional parametric modelEstimating the model parameters (SEM-type algorithms)Robustifying strategies
3 ResultsDetection and estimation of sinusoids in white noiseDetection and estimation of muons in the Auger project
4 Conclusion
Relabeling and summarizing posterior distributionsProposed approach
ResultsConclusion
An original variable-dimensional parametric modelEstimating the model parameters (SEM-type algorithms)Robustifying strategies
Big picture: relabeling and summarizing posteriordistributions
True posterior f = p(· | y)
0 1 2 30
1
2
Approximate posterior qη
0 1 2 30
1
2
Parametric family {qη, η ∈ N}
Measure of “distance”
[Stephens, 2000.]
Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 24/ 52
Relabeling and summarizing posterior distributionsProposed approach
ResultsConclusion
An original variable-dimensional parametric modelEstimating the model parameters (SEM-type algorithms)Robustifying strategies
Big picture: relabeling and summarizing posteriordistributions
True posterior f = p(· | y)
0 1 2 30
1
2
Samples
0 1 2 30
1
2
Approximate posterior qη
0 1 2 30
1
2
Parametric family {qη, η ∈ N}
Measure of “distance”
Samples
[Stephens, 2000.]
Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 24/ 52
Relabeling and summarizing posterior distributionsProposed approach
ResultsConclusion
An original variable-dimensional parametric modelEstimating the model parameters (SEM-type algorithms)Robustifying strategies
Fixed-dimensional problems
uni-modal uni-variate case
Samples
p
0 2 40
0.2
0.4
0.6
report location (mean and median) and dispersion(variance and confidence intervals) parameters
µ σ2
2.0 0.25
Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 25/ 52
Relabeling and summarizing posterior distributionsProposed approach
ResultsConclusion
An original variable-dimensional parametric modelEstimating the model parameters (SEM-type algorithms)Robustifying strategies
variable-dimensional approximate posterior
An original (variable-dimensional) parametric model qη
Four main requirements:1 Must be defined on the same space X =
⋃
k∈K{k} ×Θk
Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 26/ 52
Relabeling and summarizing posterior distributionsProposed approach
ResultsConclusion
An original variable-dimensional parametric modelEstimating the model parameters (SEM-type algorithms)Robustifying strategies
variable-dimensional approximate posterior
An original (variable-dimensional) parametric model qη
Four main requirements:1 Must be defined on the same space X =
⋃
k∈K{k} ×Θk
2 Must be permutation invariance
Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 26/ 52
Relabeling and summarizing posterior distributionsProposed approach
ResultsConclusion
An original variable-dimensional parametric modelEstimating the model parameters (SEM-type algorithms)Robustifying strategies
variable-dimensional approximate posterior
An original (variable-dimensional) parametric model qη
Four main requirements:1 Must be defined on the same space X =
⋃
k∈K{k} ×Θk
2 Must be permutation invariance3 Must be “simple” (small number of parameters)
Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 26/ 52
Relabeling and summarizing posterior distributionsProposed approach
ResultsConclusion
An original variable-dimensional parametric modelEstimating the model parameters (SEM-type algorithms)Robustifying strategies
variable-dimensional approximate posterior
An original (variable-dimensional) parametric model qη
Four main requirements:1 Must be defined on the same space X =
⋃
k∈K{k} ×Θk
2 Must be permutation invariance3 Must be “simple” (small number of parameters)4 Must be able to capture the main features of the posterior
distributions typically met in practice.
Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 26/ 52
Relabeling and summarizing posterior distributionsProposed approach
ResultsConclusion
An original variable-dimensional parametric modelEstimating the model parameters (SEM-type algorithms)Robustifying strategies
A generative model point of view
1 2
· · ·
L
x = (k ,θk ) ∈ X =⋃
k{k} ×Θk
Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 27/ 52
Relabeling and summarizing posterior distributionsProposed approach
ResultsConclusion
An original variable-dimensional parametric modelEstimating the model parameters (SEM-type algorithms)Robustifying strategies
A generative model point of view
u11 u2
2
· · ·
uLL
x = (k ,θk ) ∈ X =⋃
k{k} ×Θk
ul ∈ Θ
Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 27/ 52
Relabeling and summarizing posterior distributionsProposed approach
ResultsConclusion
An original variable-dimensional parametric modelEstimating the model parameters (SEM-type algorithms)Robustifying strategies
A generative model point of view
ξ1 u11
ξ2 u22
· · ·
ξL uLL
x = (k ,θk ) ∈ X =⋃
k{k} ×Θk
ul ∈ Θ
ξl ∈ {0, 1}
Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 27/ 52
Relabeling and summarizing posterior distributionsProposed approach
ResultsConclusion
An original variable-dimensional parametric modelEstimating the model parameters (SEM-type algorithms)Robustifying strategies
A generative model point of view
ξ1 u11
ξ2 u22
· · ·
ξL uLL
{u l | ξl = 1}
x = (k ,θk ) ∈ X =⋃
k{k} ×Θk
ul ∈ Θ
ξl ∈ {0, 1}
Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 27/ 52
Relabeling and summarizing posterior distributionsProposed approach
ResultsConclusion
An original variable-dimensional parametric modelEstimating the model parameters (SEM-type algorithms)Robustifying strategies
A generative model point of view
ξ1 u11
ξ2 u22
· · ·
ξL uLL
{u l | ξl = 1}
random arrangement
x = (k ,θk ) ∈ X =⋃
k{k} ×Θk
ul ∈ Θ
ξl ∈ {0, 1}
Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 27/ 52
Relabeling and summarizing posterior distributionsProposed approach
ResultsConclusion
An original variable-dimensional parametric modelEstimating the model parameters (SEM-type algorithms)Robustifying strategies
A generative model point of view
ξ1 u1
π1
1ξ2 u2
π2
2
· · ·
ξL uL
πL
L
{u l | ξl = 1}
random arrangement
x = (k ,θk ) ∈ X =⋃
k{k} ×Θk
ul ∈ Θ
ξl ∈ {0, 1}
ξl ∼ B(πl)
Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 27/ 52
Relabeling and summarizing posterior distributionsProposed approach
ResultsConclusion
An original variable-dimensional parametric modelEstimating the model parameters (SEM-type algorithms)Robustifying strategies
A generative model point of view
ξ1 u1
π1µ1 Σ1
1ξ2 u2
π2µ2 Σ2
2
· · ·
ξL uL
πLµL ΣL
L
{u l | ξl = 1}
random arrangement
x = (k ,θk ) ∈ X =⋃
k{k} ×Θk
ul ∈ Θ
ξl ∈ {0, 1}
ξl ∼ B(πl)
ul ∼ N (µl ,Σl)
Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 27/ 52
Relabeling and summarizing posterior distributionsProposed approach
ResultsConclusion
An original variable-dimensional parametric modelEstimating the model parameters (SEM-type algorithms)Robustifying strategies
A generative model point of view
ξ1 u1
π1µ1 Σ1
1ξ2 u2
π2µ2 Σ2
2
· · ·
ξL uL
πLµL ΣL
L
{u l | ξl = 1}
random arrangement
x = (k ,θk ) ∈ X =⋃
k{k} ×Θk
ul ∈ Θ
ξl ∈ {0, 1}
ξl ∼ B(πl)
ul ∼ N (µl ,Σl)
ηl = {πl ,µl ,Σl}
Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 27/ 52
Relabeling and summarizing posterior distributionsProposed approach
ResultsConclusion
An original variable-dimensional parametric modelEstimating the model parameters (SEM-type algorithms)Robustifying strategies
A generative model point of view
1 for l = 1, . . . , Lgenerate binary number ξl ∼ B (πl) end
Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 28/ 52
Relabeling and summarizing posterior distributionsProposed approach
ResultsConclusion
An original variable-dimensional parametric modelEstimating the model parameters (SEM-type algorithms)Robustifying strategies
A generative model point of view
1 for l = 1, . . . , Lgenerate binary number ξl ∼ B (πl) end
2 set k =∑L
l=1 ξl ;3 for each l such that ξl = 1
generate random sample ul ∼ N (µl ,Σl) end
4 Random arrangement ⇒ θk
Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 28/ 52
Relabeling and summarizing posterior distributionsProposed approach
ResultsConclusion
An original variable-dimensional parametric modelEstimating the model parameters (SEM-type algorithms)Robustifying strategies
A generative model point of view
1 for l = 1, . . . , Lgenerate binary number ξl ∼ B (πl) end
2 set k =∑L
l=1 ξl ;3 for each l such that ξl = 1
generate random sample ul ∼ N (µl ,Σl) end
4 Random arrangement ⇒ θk
Example:
L = 3π = (0.4, 0.9, 0.7)µ = (0.2, 0.5, 1)s2 = (0.05, 0.02, 0.1)
0.4 0.8 1.2
6
12
18
Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 28/ 52
Relabeling and summarizing posterior distributionsProposed approach
ResultsConclusion
An original variable-dimensional parametric modelEstimating the model parameters (SEM-type algorithms)Robustifying strategies
A generative model point of view
1 for l = 1, . . . , Lgenerate binary number ξl ∼ B (πl) end
2 set k =∑L
l=1 ξl ;3 for each l such that ξl = 1
generate random sample ul ∼ N (µl ,Σl) end
4 Random arrangement ⇒ θk
Example:
L = 3π = (0.4, 0.9, 0.7)µ = (0.2, 0.5, 1)s2 = (0.05, 0.02, 0.1)
ξ = (0, 1, 1) ⇒ k = 2 &θk = (0.52, 1.05)
0.4 0.8 1.2
6
12
18
Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 28/ 52
Relabeling and summarizing posterior distributionsProposed approach
ResultsConclusion
An original variable-dimensional parametric modelEstimating the model parameters (SEM-type algorithms)Robustifying strategies
A generative model point of view
1 for l = 1, . . . , Lgenerate binary number ξl ∼ B (πl) end
2 set k =∑L
l=1 ξl ;3 for each l such that ξl = 1
generate random sample ul ∼ N (µl ,Σl) end
4 Random arrangement ⇒ θk
Example:
L = 3π = (0.4, 0.9, 0.7)µ = (0.2, 0.5, 1)s2 = (0.05, 0.02, 0.1)
ξ = (0, 1, 0) ⇒ k = 1 &θk = 0.49
0.4 0.8 1.2
6
12
18
Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 28/ 52
Relabeling and summarizing posterior distributionsProposed approach
ResultsConclusion
An original variable-dimensional parametric modelEstimating the model parameters (SEM-type algorithms)Robustifying strategies
A generative model point of view
1 for l = 1, . . . , Lgenerate binary number ξl ∼ B (πl) end
2 set k =∑L
l=1 ξl ;3 for each l such that ξl = 1
generate random sample ul ∼ N (µl ,Σl) end
4 Random arrangement ⇒ θk
Example:
L = 3π = (0.4, 0.9, 0.7)µ = (0.2, 0.5, 1)s2 = (0.05, 0.02, 0.1)
ξ = (1, 0, 1) ⇒ k = 2 &θk = (0.27, 1.03)
0.4 0.8 1.2
6
12
18
Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 28/ 52
Relabeling and summarizing posterior distributionsProposed approach
ResultsConclusion
An original variable-dimensional parametric modelEstimating the model parameters (SEM-type algorithms)Robustifying strategies
A generative model point of view
1 for l = 1, . . . , Lgenerate binary number ξl ∼ B (πl) end
2 set k =∑L
l=1 ξl ;3 for each l such that ξl = 1
generate random sample ul ∼ N (µl ,Σl) end
4 Random arrangement ⇒ θk
Example:
L = 3π = (0.4, 0.9, 0.7)µ = (0.2, 0.5, 1)s2 = (0.05, 0.02, 0.1)
ξ = (0, 0, 1) ⇒ k = 1 &θk = 1.05
0.4 0.8 1.2
6
12
18
Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 28/ 52
Relabeling and summarizing posterior distributionsProposed approach
ResultsConclusion
An original variable-dimensional parametric modelEstimating the model parameters (SEM-type algorithms)Robustifying strategies
A generative model point of view
1 for l = 1, . . . , Lgenerate binary number ξl ∼ B (πl) end
2 set k =∑L
l=1 ξl ;3 for each l such that ξl = 1
generate random sample ul ∼ N (µl ,Σl) end
4 Random arrangement ⇒ θk
Example:
L = 3π = (0.4, 0.9, 0.7)µ = (0.2, 0.5, 1)s2 = (0.05, 0.02, 0.1)
ξ = (1, 1, 1) ⇒ k = 3 &θk = (0.27, 0.53, 1.15)
0.4 0.8 1.2
6
12
18
Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 28/ 52
Relabeling and summarizing posterior distributionsProposed approach
ResultsConclusion
An original variable-dimensional parametric modelEstimating the model parameters (SEM-type algorithms)Robustifying strategies
A generative model point of view
1 for l = 1, . . . , Lgenerate binary number ξl ∼ B (πl) end
2 set k =∑L
l=1 ξl ;3 for each l such that ξl = 1
generate random sample ul ∼ N (µl ,Σl) end
4 Random arrangement ⇒ θk
Example:
L = 3π = (0.4, 0.9, 0.7)µ = (0.2, 0.5, 1)s2 = (0.05, 0.02, 0.1)
ξ = (0, 0, 0) ⇒ k = 0 &θk = ()
0.4 0.8 1.2
6
12
18
Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 28/ 52
Relabeling and summarizing posterior distributionsProposed approach
ResultsConclusion
An original variable-dimensional parametric modelEstimating the model parameters (SEM-type algorithms)Robustifying strategies
Big picture: relabeling and summarizing posteriordistributions
True posterior f = p(· | y)
0 1 2 30
1
2
Samples
0 1 2 30
1
2
Approximate posterior qη
0 1 2 30
1
2
Parametric family {qη, η ∈ N}
Measure of “distance”
Samples
[Stephens, 2000.]
Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 29/ 52
Relabeling and summarizing posterior distributionsProposed approach
ResultsConclusion
An original variable-dimensional parametric modelEstimating the model parameters (SEM-type algorithms)Robustifying strategies
fitting the parametric model qη to the posterior f
minimizing the Kullback-Leibler divergence
J (η) , DKL (f (x)‖qη(x)) =
∫
f (x) logf (x)
qη(x)dx
Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 30/ 52
Relabeling and summarizing posterior distributionsProposed approach
ResultsConclusion
An original variable-dimensional parametric modelEstimating the model parameters (SEM-type algorithms)Robustifying strategies
fitting the parametric model qη to the posterior f
minimizing the Kullback-Leibler divergence
J (η) , DKL (f (x)‖qη(x)) =
∫
f (x) logf (x)
qη(x)dx
A key point: samples x (i), i = 1, 2, · · · ,M, are generatedfrom f , so
J (η) ≃ −1M
M∑
i=1
log(
qη(x (i)))
+ Const.
Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 30/ 52
Relabeling and summarizing posterior distributionsProposed approach
ResultsConclusion
An original variable-dimensional parametric modelEstimating the model parameters (SEM-type algorithms)Robustifying strategies
fitting the parametric model qη to the posterior f
minimizing the Kullback-Leibler divergence
J (η) , DKL (f (x)‖qη(x)) =
∫
f (x) logf (x)
qη(x)dx
A key point: samples x (i), i = 1, 2, · · · ,M, are generatedfrom f , so
J (η) ≃ −1M
M∑
i=1
log(
qη(x (i)))
+ Const.
Objective: η = argmaxη∑M
i=1 log(
qη(x (i)))
.
Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 30/ 52
Relabeling and summarizing posterior distributionsProposed approach
ResultsConclusion
An original variable-dimensional parametric modelEstimating the model parameters (SEM-type algorithms)Robustifying strategies
Expectation Maximization (EM)
latent (hidden) variable:➠ Binary indicator vector ξ
➠ Random permutation
Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 31/ 52
Relabeling and summarizing posterior distributionsProposed approach
ResultsConclusion
An original variable-dimensional parametric modelEstimating the model parameters (SEM-type algorithms)Robustifying strategies
Expectation Maximization (EM)
latent (hidden) variable:➠ Binary indicator vector ξ
➠ Random permutation
➠ Define z = (z1, . . . , zk ) as an allocation vector for x
➠ zj = l ⇒ x j comes from component l
Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 31/ 52
Relabeling and summarizing posterior distributionsProposed approach
ResultsConclusion
An original variable-dimensional parametric modelEstimating the model parameters (SEM-type algorithms)Robustifying strategies
Expectation Maximization (EM)
latent (hidden) variable:➠ Binary indicator vector ξ
➠ Random permutation
➠ Define z = (z1, . . . , zk ) as an allocation vector for x
➠ zj = l ⇒ x j comes from component l
idea: use EM to maximize the likelihood.
Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 31/ 52
Relabeling and summarizing posterior distributionsProposed approach
ResultsConclusion
An original variable-dimensional parametric modelEstimating the model parameters (SEM-type algorithms)Robustifying strategies
Expectation Maximization (EM)
latent (hidden) variable:➠ Binary indicator vector ξ
➠ Random permutation
➠ Define z = (z1, . . . , zk ) as an allocation vector for x
➠ zj = l ⇒ x j comes from component l
idea: use EM to maximize the likelihood.
The E-step is computationally expensive!
For example, assuming L = 15 and k (i) = 10, then, itcontains 1.1 × 1010 terms.
Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 31/ 52
Relabeling and summarizing posterior distributionsProposed approach
ResultsConclusion
An original variable-dimensional parametric modelEstimating the model parameters (SEM-type algorithms)Robustifying strategies
Stochastic EM (SEM)
at iteration (r + 1)
Stochastic (S)-step: for i = 1, . . . ,Mgenerate z(i) from p( · | x (i), η(r)) end
M-step: η(r+1) = argmaxη∑M
i=1 log(p(x (i), z(i) |η))
[Broniatowski, et al. 1983. Celeux and Diebolt, 1986. Celeux and Diebolt,
1993.]
Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 32/ 52
Relabeling and summarizing posterior distributionsProposed approach
ResultsConclusion
An original variable-dimensional parametric modelEstimating the model parameters (SEM-type algorithms)Robustifying strategies
Stochastic EM (SEM)
at iteration (r + 1)
Stochastic (S)-step: for i = 1, . . . ,Mgenerate z(i) from p( · | x (i), η(r)) end
M-step: η(r+1) = argmaxη∑M
i=1 log(p(x (i), z(i) |η))
[Broniatowski, et al. 1983. Celeux and Diebolt, 1986. Celeux and Diebolt,
1993.]
Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 32/ 52
Relabeling and summarizing posterior distributionsProposed approach
ResultsConclusion
An original variable-dimensional parametric modelEstimating the model parameters (SEM-type algorithms)Robustifying strategies
Stochastic EM (SEM)
at iteration (r + 1)
Stochastic (S)-step: for i = 1, . . . ,Mgenerate z(i) from p( · | x (i), η(r)) end
M-step: η(r+1) = argmaxη∑M
i=1 log(p(x (i), z(i) |η))
[Broniatowski, et al. 1983. Celeux and Diebolt, 1986. Celeux and Diebolt,
1993.]
Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 32/ 52
Relabeling and summarizing posterior distributionsProposed approach
ResultsConclusion
An original variable-dimensional parametric modelEstimating the model parameters (SEM-type algorithms)Robustifying strategies
Stochastic EM (SEM)
at iteration (r + 1)
Stochastic (S)-step: for i = 1, . . . ,Mgenerate z(i) from p( · | x (i), η(r)) end
M-step: η(r+1) = argmaxη∑M
i=1 log(p(x (i), z(i) |η))
[Broniatowski, et al. 1983. Celeux and Diebolt, 1986. Celeux and Diebolt,
1993.]
S-step
To draw z(i) ∼ p( · | x (i), η(r)) we developed an I-MH sampler.
Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 32/ 52
Relabeling and summarizing posterior distributionsProposed approach
ResultsConclusion
An original variable-dimensional parametric modelEstimating the model parameters (SEM-type algorithms)Robustifying strategies
Example 2: spectral analysis (Cont.)Variable-dimensional posterior distribution
k
p(k |y) ω0.5 0.75 10 0.3 0.6
2
3
4
Figure: Posteriors of k and sorted radial frequencies, ωk , given k .Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 33/ 52
Relabeling and summarizing posterior distributionsProposed approach
ResultsConclusion
An original variable-dimensional parametric modelEstimating the model parameters (SEM-type algorithms)Robustifying strategies
Robustifying solutions
Add a Poisson point process component
To capture the “outliers”
λ is the mean parameter
points are uniformly distributed on Θ
Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 34/ 52
Relabeling and summarizing posterior distributionsProposed approach
ResultsConclusion
An original variable-dimensional parametric modelEstimating the model parameters (SEM-type algorithms)Robustifying strategies
Robustifying solutions
Add a Poisson point process component
To capture the “outliers”
λ is the mean parameter
points are uniformly distributed on Θ
Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 34/ 52
Relabeling and summarizing posterior distributionsProposed approach
ResultsConclusion
An original variable-dimensional parametric modelEstimating the model parameters (SEM-type algorithms)Robustifying strategies
Robustifying solutions
Add a Poisson point process component
To capture the “outliers”
λ is the mean parameter
points are uniformly distributed on Θ
Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 34/ 52
Relabeling and summarizing posterior distributionsProposed approach
ResultsConclusion
An original variable-dimensional parametric modelEstimating the model parameters (SEM-type algorithms)Robustifying strategies
Robustifying solutions
Add a Poisson point process component
To capture the “outliers”
λ is the mean parameter
points are uniformly distributed on Θ
Other possibilities
Robust estimates in the M-step.➠ Median instead of mean
➠ interquartile range instead of variance
Using another divergence measure.
Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 34/ 52
Outline
1 Relabeling and summarizing posterior distributionsLabel-switching issueVariable-dimensional summarization
2 Proposed approachAn original variable-dimensional parametric modelEstimating the model parameters (SEM-type algorithms)Robustifying strategies
3 ResultsDetection and estimation of sinusoids in white noiseDetection and estimation of muons in the Auger project
4 Conclusion
Relabeling and summarizing posterior distributionsProposed approach
ResultsConclusion
Detection and estimation of sinusoids in white noiseDetection and estimation of muons in the Auger project
Example 2: Spectral analysis
time
Pow
er
radial frequency0 0.5 1 1.5 2 2.5 3
0 10 20 30 40 50 60
0
50
100
150
−10
0
10
Figure: Observed signal (top) and its periodogram (bottom).
Mk : y [i] =k
∑
j=1
(
aj cos[ωj i] + bj sin[ωj i])
+ n[i].
[Andrieu and Doucet, 1999.]Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 35/ 52
Relabeling and summarizing posterior distributionsProposed approach
ResultsConclusion
Detection and estimation of sinusoids in white noiseDetection and estimation of muons in the Auger project
Example 2: spectral analysis (Cont.)Variable-dimensional posterior distribution
k
p(k |y) ω0.5 0.75 10 0.3 0.6
2
3
4
Figure: Posteriors of k and sorted radial frequencies, ωk , given k .Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 36/ 52
Relabeling and summarizing posterior distributionsProposed approach
ResultsConclusion
Detection and estimation of sinusoids in white noiseDetection and estimation of muons in the Auger project
S-step: randomized allocation procedure
0.72 0.62 0.73 0.83 0.67 0.64 0.64 0.74 0.72
z1 z2 z1 z2 z3 z4 z1 z2 z3
M-step
ω
norm
.de
nsity
0.5 0.75 10
0.5
1
λ = 0.1 Iter = 0
Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 37/ 52
Relabeling and summarizing posterior distributionsProposed approach
ResultsConclusion
Detection and estimation of sinusoids in white noiseDetection and estimation of muons in the Auger project
S-step: randomized allocation procedure
0.72 0.62 0.73 0.83 0.67 0.64 0.64 0.74 0.72
3 1 4 2 3 1 1 4 2
M-step
ω
norm
.de
nsity
0.5 0.75 10
0.5
1
λ = 0.1 Iter = 0
Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 37/ 52
Relabeling and summarizing posterior distributionsProposed approach
ResultsConclusion
Detection and estimation of sinusoids in white noiseDetection and estimation of muons in the Auger project
S-step: randomized allocation procedure
0.72 0.62 0.73 0.83 0.67 0.64 0.64 0.74 0.72
3 1 4 2 3 1 1 4 2
M-step
ω
norm
.de
nsity
0.5 0.75 10
0.5
1
λ = 0.167 Iter = 1
Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 37/ 52
Relabeling and summarizing posterior distributionsProposed approach
ResultsConclusion
Detection and estimation of sinusoids in white noiseDetection and estimation of muons in the Auger project
S-step: randomized allocation procedure
0.72 0.62 0.73 0.83 0.67 0.64 0.64 0.74 0.72
3 1 4 2 3 1 1 3 2
M-step
ω
norm
.de
nsity
0.5 0.75 10
0.5
1
λ = 0.167 Iter = 1
Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 37/ 52
Relabeling and summarizing posterior distributionsProposed approach
ResultsConclusion
Detection and estimation of sinusoids in white noiseDetection and estimation of muons in the Auger project
S-step: randomized allocation procedure
0.72 0.62 0.73 0.83 0.67 0.64 0.64 0.74 0.72
3 1 4 2 3 1 1 3 2
M-step
ω
norm
.de
nsity
0.5 0.75 10
0.5
1
λ = 0.226 Iter = 2
Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 37/ 52
Relabeling and summarizing posterior distributionsProposed approach
ResultsConclusion
Detection and estimation of sinusoids in white noiseDetection and estimation of muons in the Auger project
S-step: randomized allocation procedure
0.72 0.62 0.73 0.83 0.67 0.64 0.64 0.74 0.72
3 1 2 4 3 1 1 2 3
M-step
ω
norm
.de
nsity
0.5 0.75 10
0.5
1
λ = 0.226 Iter = 3
Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 37/ 52
Relabeling and summarizing posterior distributionsProposed approach
ResultsConclusion
Detection and estimation of sinusoids in white noiseDetection and estimation of muons in the Auger project
S-step: randomized allocation procedure
0.72 0.62 0.73 0.83 0.67 0.64 0.64 0.74 0.72
3 1 2 4 3 1 1 2 3
M-step
ω
norm
.de
nsity
0.5 0.75 10
0.5
1
λ = 0.236 Iter = 4
Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 37/ 52
Relabeling and summarizing posterior distributionsProposed approach
ResultsConclusion
Detection and estimation of sinusoids in white noiseDetection and estimation of muons in the Auger project
S-step: randomized allocation procedure
0.72 0.62 0.73 0.83 0.67 0.64 0.64 0.74 0.72
3 1 3 4 2 1 1 3 2
M-step
ω
norm
.de
nsity
0.5 0.75 10
0.5
1
λ = 0.236 Iter = 4
Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 37/ 52
Relabeling and summarizing posterior distributionsProposed approach
ResultsConclusion
Detection and estimation of sinusoids in white noiseDetection and estimation of muons in the Auger project
S-step: randomized allocation procedure
0.72 0.62 0.73 0.83 0.67 0.64 0.64 0.74 0.72
3 1 3 4 2 1 1 3 2
M-step
ω
norm
.de
nsity
0.5 0.75 10
0.5
1
λ = 0.246 Iter = 5
Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 37/ 52
Relabeling and summarizing posterior distributionsProposed approach
ResultsConclusion
Detection and estimation of sinusoids in white noiseDetection and estimation of muons in the Auger project
S-step: randomized allocation procedure
0.72 0.62 0.73 0.83 0.67 0.64 0.64 0.74 0.72
3 1 3 4 2 1 1 3 2
M-step
ω
norm
.de
nsity
0.5 0.75 10
0.5
1
λ = 0.246 Iter = 5
Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 37/ 52
Relabeling and summarizing posterior distributionsProposed approach
ResultsConclusion
Detection and estimation of sinusoids in white noiseDetection and estimation of muons in the Auger project
S-step: randomized allocation procedure
0.72 0.62 0.73 0.83 0.67 0.64 0.64 0.74 0.72
3 1 3 4 2 1 1 3 2
M-step
ω
norm
.de
nsity
0.5 0.75 10
0.5
1
λ = 0.248 Iter = 6
Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 37/ 52
Relabeling and summarizing posterior distributionsProposed approach
ResultsConclusion
Detection and estimation of sinusoids in white noiseDetection and estimation of muons in the Auger project
S-step: randomized allocation procedure
0.72 0.62 0.73 0.83 0.67 0.64 0.64 0.74 0.72
3 1 3 4 2 1 1 3 2
M-step
ω
norm
.de
nsity
0.5 0.75 10
0.5
1
λ = 0.248 Iter = 6
Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 37/ 52
Relabeling and summarizing posterior distributionsProposed approach
ResultsConclusion
Detection and estimation of sinusoids in white noiseDetection and estimation of muons in the Auger project
S-step: randomized allocation procedure
0.72 0.62 0.73 0.83 0.67 0.64 0.64 0.74 0.72
3 1 3 4 2 1 1 3 2
M-step
ω
norm
.de
nsity
0.5 0.75 10
0.5
1
λ = 0.251 Iter = 7
Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 37/ 52
Relabeling and summarizing posterior distributionsProposed approach
ResultsConclusion
Detection and estimation of sinusoids in white noiseDetection and estimation of muons in the Auger project
S-step: randomized allocation procedure
0.72 0.62 0.73 0.83 0.67 0.64 0.64 0.74 0.72
3 1 3 4 2 1 1 3 2
M-step
ω
norm
.de
nsity
0.5 0.75 10
0.5
1
λ = 0.251 Iter = 7
Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 37/ 52
Relabeling and summarizing posterior distributionsProposed approach
ResultsConclusion
Detection and estimation of sinusoids in white noiseDetection and estimation of muons in the Auger project
S-step: randomized allocation procedure
0.72 0.62 0.73 0.83 0.67 0.64 0.64 0.74 0.72
3 1 3 4 2 1 1 3 2
M-step
ω
norm
.de
nsity
0.5 0.75 10
0.5
1
λ = 0.248 Iter = 8
Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 37/ 52
Relabeling and summarizing posterior distributionsProposed approach
ResultsConclusion
Detection and estimation of sinusoids in white noiseDetection and estimation of muons in the Auger project
S-step: randomized allocation procedure
0.72 0.62 0.73 0.83 0.67 0.64 0.64 0.74 0.72
3 1 3 4 2 1 1 3 2
M-step
ω
norm
.de
nsity
0.5 0.75 10
0.5
1
λ = 0.248 Iter = 8
Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 37/ 52
Relabeling and summarizing posterior distributionsProposed approach
ResultsConclusion
Detection and estimation of sinusoids in white noiseDetection and estimation of muons in the Auger project
S-step: randomized allocation procedure
0.72 0.62 0.73 0.83 0.67 0.64 0.64 0.74 0.72
3 1 3 4 2 1 1 3 2
M-step
ω
norm
.de
nsity
0.5 0.75 10
0.5
1
λ = 0.256 Iter = 9
Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 37/ 52
Relabeling and summarizing posterior distributionsProposed approach
ResultsConclusion
Detection and estimation of sinusoids in white noiseDetection and estimation of muons in the Auger project
S-step: randomized allocation procedure
0.72 0.62 0.73 0.83 0.67 0.64 0.64 0.74 0.72
3 1 3 4 2 1 1 3 2
M-step
ω
norm
.de
nsity
0.5 0.75 10
0.5
1
λ = 0.256 Iter = 9
Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 37/ 52
Relabeling and summarizing posterior distributionsProposed approach
ResultsConclusion
Detection and estimation of sinusoids in white noiseDetection and estimation of muons in the Auger project
S-step: randomized allocation procedure
0.72 0.62 0.73 0.83 0.67 0.64 0.64 0.74 0.72
3 1 3 4 2 1 1 3 2
M-step
ω
norm
.de
nsity
0.5 0.75 10
0.5
1
λ = 0.255 Iter = 10
Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 37/ 52
Relabeling and summarizing posterior distributionsProposed approach
ResultsConclusion
Detection and estimation of sinusoids in white noiseDetection and estimation of muons in the Auger project
Results: sinusoid detection (convergence)µ s2
π λ
J
SEM iteration0 20 40 60 80 100
−3−2−1
01
0
0.25
0.5
0.250.5
0.751
10−4
10−3
10−2
0.60.650.7
0.75
Figure: parameter evolutions vs SEM iterations
Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 38/ 52
Relabeling and summarizing posterior distributionsProposed approach
ResultsConclusion
Detection and estimation of sinusoids in white noiseDetection and estimation of muons in the Auger project
Results: sinusoid detection (validation)k
0.5 0.7 0.90
0.51
2
3
4
Figure: Marginal posteriors of sorted radial frequencies (top) vs.normalized densities of the fitted Gaussian components (bottom).
Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 39/ 52
Relabeling and summarizing posterior distributionsProposed approach
ResultsConclusion
Detection and estimation of sinusoids in white noiseDetection and estimation of muons in the Auger project
Results: sinusoid detection (relabeling properties)
µ1 = 0.62, s1 = 0.019, π1 = 1.00In
tens
ityµ2 = 0.68, s2 = 0.056, π2 = 0.29
µ3 = 0.73, s3 = 0.011, π3 = 0.98
ω
Inte
nsity
λ = 0.24
ω0 1 2 30.5 0.75 1
0.5 0.75 10.5 0.75 1
0.5
1
10
20
30
40
5
10
20
30
Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 40/ 52
Relabeling and summarizing posterior distributionsProposed approach
ResultsConclusion
Detection and estimation of sinusoids in white noiseDetection and estimation of muons in the Auger project
Results: sinusoid detection (goodness-of-fit)
inte
nsity
ω
0.4 0.6 0.8 10
10
20
30
40
Figure: Estimated intensity of radial frequencies (Histogram : BMA,Curve : parametric model).
Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 41/ 52
Relabeling and summarizing posterior distributionsProposed approach
ResultsConclusion
Detection and estimation of sinusoids in white noiseDetection and estimation of muons in the Auger project
Results: sinusoid detection (goodness-of-fit)
k0 2 4 6 8 10
0
0
0.2
0.4
0.6
Figure: Posterior distribution of k (black) vs. its approximate version(gray).
Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 42/ 52
Relabeling and summarizing posterior distributionsProposed approach
ResultsConclusion
Detection and estimation of sinusoids in white noiseDetection and estimation of muons in the Auger project
Results: sinusoid detection (Comparison with BMS)
Comp. µ s π µBMS sBMS ωtruek
1 0.620 0.019 1 0.617 0.016 0.6282 0.686 0.056 0.29 — — 0.6773 0.727 0.011 0.98 0.727 0.012 0.726
Table: summaries of the variable-dimensional posterior distribution;the proposed method vs. BMS.
Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 43/ 52
Relabeling and summarizing posterior distributionsProposed approach
ResultsConclusion
Detection and estimation of sinusoids in white noiseDetection and estimation of muons in the Auger project
Results: sinusoid detection (Comparison with BMS)
Comp. µ s π µBMS sBMS ωtruek
1 0.620 0.019 1 0.617 0.016 0.6282 0.686 0.056 0.29 — — 0.6773 0.727 0.011 0.98 0.727 0.012 0.726
Table: summaries of the variable-dimensional posterior distribution;the proposed method vs. BMS.
Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 43/ 52
Relabeling and summarizing posterior distributionsProposed approach
ResultsConclusion
Detection and estimation of sinusoids in white noiseDetection and estimation of muons in the Auger project
Results: sinusoid detection (Comparison with BMS)
Comp. µ s π µBMS sBMS ωtruek
1 0.620 0.019 1 0.617 0.016 0.6282 0.686 0.056 0.29 — — 0.6773 0.727 0.011 0.98 0.727 0.012 0.726
Table: summaries of the variable-dimensional posterior distribution;the proposed method vs. BMS.
Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 43/ 52
Relabeling and summarizing posterior distributionsProposed approach
ResultsConclusion
Detection and estimation of sinusoids in white noiseDetection and estimation of muons in the Auger project
Results: sinusoid detection (Comparison with BMS)
Comp. µ s π µBMS sBMS ωtruek
1 0.620 0.019 1 0.617 0.016 0.6282 0.686 0.056 0.29 — — 0.6773 0.727 0.011 0.98 0.727 0.012 0.726
Table: summaries of the variable-dimensional posterior distribution;the proposed method vs. BMS.
The summary obtained by the proposed approach is richer thanthe one of the BMS approach.
Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 43/ 52
Relabeling and summarizing posterior distributionsProposed approach
ResultsConclusion
Detection and estimation of sinusoids in white noiseDetection and estimation of muons in the Auger project
Results: Auger (observatory)
Figure: A conceptual shower (http://auger.org).
Prof. Balázs Kégl from LAL, University of Paris 11.
Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 44/ 52
Relabeling and summarizing posterior distributionsProposed approach
ResultsConclusion
Detection and estimation of sinusoids in white noiseDetection and estimation of muons in the Auger project
Results: Auger (Observed signal)
#P
E
t [ns]
inte
nsity
100 200 300 400 500 600
0
0
0.5
1
1.5
0
10
20
Figure: Observed signal (n) made up of five muons.
Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 45/ 52
Relabeling and summarizing posterior distributionsProposed approach
ResultsConclusion
Detection and estimation of sinusoids in white noiseDetection and estimation of muons in the Auger project
Results: Auger (RJ-MCMC samples)
k
p(k |n) t [ns]100 200 300 400 5000 0.25 0.5
0.3
4
5
6
7
Figure: Posteriors of k and sorted arrival times, tµ, given k .
Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 46/ 52
Relabeling and summarizing posterior distributionsProposed approach
ResultsConclusion
Detection and estimation of sinusoids in white noiseDetection and estimation of muons in the Auger project
Results: Auger (choice of L)
L = 6
norm
aliz
edde
nsity
L = 7
L = 8
100 200 300 400 5000
0.5
1
0
0.5
1
0
0.5
1
Figure: Normalized densities of the fitted Gaussian components.
Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 47/ 52
Relabeling and summarizing posterior distributionsProposed approach
ResultsConclusion
Detection and estimation of sinusoids in white noiseDetection and estimation of muons in the Auger project
Results: Auger (relabeling properties)µ1 = 99.55, s1 = 3.882, π1 = 1.00
pdfµ2 = 173.28, s2 = 7.715, π2 = 0.18
µ3 = 173.41, s3 = 5.332, π3 = 1.00
µ4 = 237.78, s4 = 8.531, π4 = 0.39
µ5 = 261.18, s5 = 6.992, π5 = 0.93
µ6 = 504.32, s6 = 5.499, π6 = 1.00
t [ns]
inte
nsity
λ = 0.42
100 200 300 400 5000
0.01
0
0.1
0
0.1
0
0.1
0
0.1
0
0.1
0
0.1
Figure:Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 48/ 52
Relabeling and summarizing posterior distributionsProposed approach
ResultsConclusion
Detection and estimation of sinusoids in white noiseDetection and estimation of muons in the Auger project
Results: Auger (choice of L)
λ = 1.91
inte
nsity
L = 3
λ = 1.01
inte
nsity
L = 4
λ = 0.43
inte
nsity
t [ns]
L = 6
λ = 0.27
inte
nsity
t [ns]
L = 8
100 200 300 400 500100 200 300 400 5000
0.02
0.04
0
0.02
0.04
0
0.02
0.04
0
0.02
0.04
Figure: Residuals of the fitted model for different values of L.
Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 49/ 52
Outline
1 Relabeling and summarizing posterior distributionsLabel-switching issueVariable-dimensional summarization
2 Proposed approachAn original variable-dimensional parametric modelEstimating the model parameters (SEM-type algorithms)Robustifying strategies
3 ResultsDetection and estimation of sinusoids in white noiseDetection and estimation of muons in the Auger project
4 Conclusion
Relabeling and summarizing posterior distributionsProposed approach
ResultsConclusion
Conclusion & Future work
Problem?relabeling and summarizing variable-dimensionalposteriors
Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 50/ 52
Relabeling and summarizing posterior distributionsProposed approach
ResultsConclusion
Conclusion & Future work
Problem?relabeling and summarizing variable-dimensionalposteriors
BMS and BMA have limitations
Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 50/ 52
Relabeling and summarizing posterior distributionsProposed approach
ResultsConclusion
Conclusion & Future work
Problem?relabeling and summarizing variable-dimensionalposteriors
BMS and BMA have limitations
Label-switching ⇒ marginal posteriors are identical
Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 50/ 52
Relabeling and summarizing posterior distributionsProposed approach
ResultsConclusion
Conclusion & Future work
Problem?relabeling and summarizing variable-dimensionalposteriors
BMS and BMA have limitations
Label-switching ⇒ marginal posteriors are identical
ContributionWe proposed to approximate the variable-dimensionalposterior by an original parametric model
Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 50/ 52
Relabeling and summarizing posterior distributionsProposed approach
ResultsConclusion
Conclusion & Future work
Problem?relabeling and summarizing variable-dimensionalposteriors
BMS and BMA have limitations
Label-switching ⇒ marginal posteriors are identical
ContributionWe proposed to approximate the variable-dimensionalposterior by an original parametric model
Developed SEM-type algorithms to estimate theparameters
Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 50/ 52
Relabeling and summarizing posterior distributionsProposed approach
ResultsConclusion
Conclusion & Future work
Problem?relabeling and summarizing variable-dimensionalposteriors
BMS and BMA have limitations
Label-switching ⇒ marginal posteriors are identical
ContributionWe proposed to approximate the variable-dimensionalposterior by an original parametric model
Developed SEM-type algorithms to estimate theparameters
Designed an I-MH sampler to generate allocation vectors
Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 50/ 52
Relabeling and summarizing posterior distributionsProposed approach
ResultsConclusion
Conclusion & Future work
Problem?relabeling and summarizing variable-dimensionalposteriors
BMS and BMA have limitations
Label-switching ⇒ marginal posteriors are identical
ContributionWe proposed to approximate the variable-dimensionalposterior by an original parametric model
Developed SEM-type algorithms to estimate theparameters
Designed an I-MH sampler to generate allocation vectors
Robustness issue: Poisson point process
Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 50/ 52
Relabeling and summarizing posterior distributionsProposed approach
ResultsConclusion
Conclusion & Future work (Cont.)
The proposed approach
the label-switching issue
summaries for component-specific parameters
meaningful probabilities of presence
Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 51/ 52
Relabeling and summarizing posterior distributionsProposed approach
ResultsConclusion
Conclusion & Future work (Cont.)
The proposed approach
the label-switching issue
summaries for component-specific parameters
meaningful probabilities of presence
Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 51/ 52
Relabeling and summarizing posterior distributionsProposed approach
ResultsConclusion
Conclusion & Future work (Cont.)
The proposed approach
the label-switching issue
summaries for component-specific parameters
meaningful probabilities of presence
Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 51/ 52
Relabeling and summarizing posterior distributionsProposed approach
ResultsConclusion
Conclusion & Future work (Cont.)
The proposed approach
the label-switching issue
summaries for component-specific parameters
meaningful probabilities of presence
Perspectives
Choice of L
Theoretical properties of the SEM-type algorithm
Adaptive RJ-MCMC samplers
Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 51/ 52
Relabeling and summarizing posterior distributionsProposed approach
ResultsConclusion
Conclusion & Future work (Cont.)
The proposed approach
the label-switching issue
summaries for component-specific parameters
meaningful probabilities of presence
Perspectives
Choice of L
Theoretical properties of the SEM-type algorithm
Adaptive RJ-MCMC samplers
Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 51/ 52
Relabeling and summarizing posterior distributionsProposed approach
ResultsConclusion
Conclusion & Future work (Cont.)
The proposed approach
the label-switching issue
summaries for component-specific parameters
meaningful probabilities of presence
Perspectives
Choice of L
Theoretical properties of the SEM-type algorithm
Adaptive RJ-MCMC samplers
Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 51/ 52
Relabeling and summarizing posterior distributionsProposed approach
ResultsConclusion
List of publications
i) Alireza Roodaki, Julien Bect, and Gilles Fleury. Summarizing posteriordistributions in signal decomposition problems when the number of componentsis unknown. In ICASSP’12, Kyoto, Japan, 2012.
ii) Alireza Roodaki, Julien Bect, and Gilles Fleury. Note on the computation of theMetropolis-Hastings ratio for Birth-or-Death moves in trans-dimensional MCMCalgorithms for signal decomposition problems. submitted to IEEE Transaction onsignal processing.
iii) Alireza Roodaki, Julien Bect, and Gilles Fleury. Comparison of fully Bayesian andempirical Bayes approaches for joint Bayesian model selection and estimation ofsinusoids via reversible jump MCMC. ISBA’10, Benidorm, Spain, 2010.
iv) Alireza Roodaki, Julien Bect, and Gilles Fleury. An Empirical Bayes Approach forJoint Bayesian Model Selection and Estimation of Sinusoids via Reversible JumpMCMC. In: EUSIPCO’10, Aalborg , Denmark, 2010.
v) Alireza Roodaki, Julien Bect, and Gilles Fleury. On the joint Bayesian modelselection and estimation of sinusoids via reversible jump MCMC in low SNRsituations. In: ISSPA’10, Kuala Lumpur, Malaysia, 2010.
Thank you for your attention !
Alireza Roodaki (SUPELEC) Signal decompositions using trans-dimensional . . . 52/ 52