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If you have previously obtained access with your personal account, Please log in. If you previously purchased this article, Log in to Readcube. Log out of Readcube. Click on an option below to access. Log out of ReadCube. We propose a simple, fully Bayesian approach for multivariate receptor modeling that allows for flexible and consistent incorporation of a priori information. The model uses a generalization of the Dirichlet distribution as the prior distribution on source profiles that allows great flexibility in the specification of prior information.

A simulation study based on the Washington, DC airshed shows that the model compares favorably to Positive Matrix Factorization, a standard analysis approach used for pollution source apportionment. A significant advantage of the proposed approach compared to most popularly used methods is that the Bayesian framework yields complete distributional results for each parameter of interest including distributions for each element of the source profile and source contribution matrices. These distributions offer a great deal of power and versatility when addressing complex questions of interest to the researcher.

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Jeff W. Search for more papers by this author. William F. Christensen Corresponding Author E-mail address: william stat. Tools Request permission Export citation Add to favorites Track citation. Share Give access Share full text access. Share full text access.

John Salvatier: Bayesian inference with PyMC 3

Please review our Terms and Conditions of Use and check box below to share full-text version of article. Get access to the full version of this article. View access options below. You previously purchased this article through ReadCube. The mean and standard deviation were calculated for each loss function, based on 50 different data replicates to compare these frequentist risk measures. See [ 24 ] for loss measures in MCMC sampling. The risk measures are illustrated in Fig 5. We are surprised by the moderate performance of the graphical lasso in this simulation setting.

Even when the sample size increases, the risk measures do not diminish, and that is quite unexpected. Still this is out of tune with our method in which performance is more consistent with the sample size.

Both the MAP-estimate with diminished condition number combined with our decision-rule and Glasso stumble with AR 1 and dense precision matrices. Furthermore, the true matrices of these models have the highest condition numbers, at The dark horse in this simulation study is the Ledoit and Wolf estimator which gives the smallest frequentist risk with all measures in almost all cases.

It is also clearly the fastest method compared to the others. The only problem with this approach is the time which is used in the Gibbs sampling even with our moderate MCMC sample size. Our method takes mere seconds to perform in the same dimension. In Fig 6 , weakly-informative prior is compared to uninformative prior without decision-rule.

This is in line with the suggestions of [ 17 ]. Of particular note is the fact that the performance measures under quadratic loss decrease in every case. This included tens of thousands of sparse MAP-estimates.


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There may be a special case when there is a reason to assume that just a few off-diagonal elements of the precision matrix are non-zero, similar to the matrix d described at the beginning of the section 3. Our goal is to improve the estimator of the precision matrix in the sparse setting by using our decision-rule procedure with the Ledoit and Wolf estimator. Again, both empirical Bayes and graphical lasso estimates were calculated to compare methods.

Because of the poor performance of the EBIC with Glasso in the risk minimization setting in the previous chapter, we used the 5-fold cross-validation described in [ 8 ] to choose the regularization parameter for Glasso. The classification diagnostics are presented in Table 1. We also tested the EBIC as the penalization parameter selection criterion for Glasso but this resulted in high frequentist risk estimates results not shown compared to the ones presented in Table 1.

The performance of the Glasso improves substantially when using the 5-fold cross-validation. Now the Glasso is able to give comparable results and even outperform our empirical Bayes estimate. The Glasso seems to have some problems with consistency in this setting because the risk estimates appear not to decrease as the sample size increases.

Still our empirical Bayes method gives lower risk in some of the cases and seems to be consistent as the sample size increases. Again, the decision-rule always produced a positive definite estimate with no need to tamper with the final sparse estimate. The choice between EBIC and cross-validation is a trade off between two features. As mentioned by [ 21 ], the cross-validation tends to produce dense graphs with many false positive edges.

This is not desirable if the Glasso is used in the graph estimation in high dimensions. On the other hand, the empirical Bayes estimate produces way too sparse estimate with no sign of the structure of the true precision matrix with this data set. In this chapter, we extend previously examined estimates for the precision matrix to the graph estimation in a real data example. First we introduce some concepts needed for the graph construction. Denote that is a graphical model with node set and edge set in. One can parallel the nodes with the variables of interest.

We estimate the graph associated with the GGM by using the elements of the precision matrix to compose a symmetric adjacency matrix.

Multivariate Bayesian Statistics: Models for Source Separation and Signal Unmixing

The pair i , j is contained in the edge set if and only if the element Ad ij is non-zero. The diagonal of the adjacency matrix can be set as zero so that there are no pairs i , i in the edge set. Otherwise there is no edge between the nodes and we posit that the variable i is conditionally independent of the variable j given all remaining variables see for example [ 26 ]. For this reason, one only needs to examine either the upper or the lower diagonal elements of the adjacency matrix Ad.

For reasons of clarity, a flow cytometry dataset from [ 13 ] is analyzed. The results were compared to the undirected graph presented in [ 4 ]. A directed acyclic graph of the data can be found in [ 3 ], and an undirected graph similar to ours from [ 4 ]. Unlike [ 4 ], we used standardized data and did not precorrect the systematic effects from the data. The precorrection could reduce some edges between the nodes in the network with graphical lasso. Other analyses were done with R. We used this same value in the MB approximation.

The estimated graphs based on the precision matrix are shown in Fig 8. We also investigated how the methods performed in the estimation of the graphical structure compared to the network of Sachs. For this, we computed the specificity, sensitivity, fall-out, precision and Matthew Correlation Coefficients MCC which are defined as follows: 6 7 8 9 10 in which TN is the number of true negatives, FP is the number of false positives, TP is the number of true positives and FN is the number of false negatives.

The closer the values of specificity, sensitivity, precision and MCC are to one, the better the classification is. The results are presented in Table 2. Comparing graphs in the Fig 8 it appears that all of the Bayesian methods produce more sparse graphs than the Glasso and MB approximation. The diagnostics in the Table 2 indicate that the Bayesian approaches performed at least comparable manner to the frequentist competitors.

Multivariate Bayesian statistics : models for source separation and signal unmixing

We note, that the MCC is higher with the frequentist methods. On the other hand, these methods produce the highest fall-out and sensitivity, which is due to the dense graph estimates. From the practical point of view, the Bayesian methods produce graphs which are visually easier to examine. Overall the MAP-estimate with our decision-rule is able to detect false positive edges from the graph associated with this data set quite efficiently. We also tried the analysis with the 5-fold cross-validation but the results were identical to those with regularization parameter chosen by EBIC.

The inferred networks drawn from the estimated precision matrices may be dense because there are several edges between nodes which are redundant, such as the node between PKC and Erk. Based on [ 13 ], there are also some unmeasured variables which cause indirect connections. We have proposed improvements to the classic Bayesian estimates when using Wishart prior for the precision matrix by just increasing the degrees of freedom parameter of the Wishart prior.

By monitoring the condition number of the estimate, we can determine an estimate with lower risk, without loss of computational speed. Apart from graphical lasso, analytical precision estimates for the matrix elements are available and can be calculated without iterative methods or MCMC sampling. The simulations with several sparsity patterns of the precision matrix indicate that there is no happy compromise between sparse estimate for the precision matrix and low risk estimate when measured with the loss-functions we have used.

In the regression-based lasso approach, we know that the cross-validation does produce a model with a reliable prediction ability but this generally does not lend itself to a very sparse model [ 28 ] without some pre-modification of the data [ 4 ].

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From this, a contradiction arises in terms of classic Bayesian analysis. When there are more data points, it is natural that the data starts to dominate over the prior and the posterior estimate comes closer to the MLE. This is troublesome because it may cause overly dense graphs when the true precision or covariance matrix is sparse. Also methods such as stability approach to regularization selection StARS [ 21 ] could be used with the graphical lasso. With StARS, the Glasso derived network was very sparse just eight edges with the following diagnostics: Specificity 0.

Clearly the performance of the Glasso depends on the procedure used to choose the regularization parameter. It is possible to simulate independent posterior samples and obtain credible regions for the whole precision matrix because of the known analytic posterior distribution. In [ 4 ], credible region based thresholding is used to choose which off-diagonal elements should be set at zero; if the credible region contained a value of zero, the corresponding off-diagonal element was set as zero.