Computation Times in Simulation Studies.

<p>The computation times (unit in second) averaging over scenarios having with the same numbers of components (<i>K</i>) and sample size (<i>N</i>).</p><p>Computation Times in Simulation Studies.</p

Allocation Variable-Based Probabilistic Algorithm to Deal with Label Switching Problem in Bayesian Mixture Models

<div><p>The label switching problem occurs as a result of the nonidentifiability of posterior distribution over various permutations of component labels when using Bayesian approach to estimate parameters in mixture models. In the cases where the number of components is fixed and known, we propose a relabelling algorithm, an allocation variable-based (denoted by AVP) probabilistic relabelling approach, to deal with label switching problem. We establish a model for the posterior distribution of allocation variables with label switching phenomenon. The AVP algorithm stochastically relabel the posterior samples according to the posterior probabilities of the established model. Some existing deterministic and other probabilistic algorithms are compared with AVP algorithm in simulation studies, and the success of the proposed approach is demonstrated in simulation studies and a real dataset.</p></div

The Performance of AVP, ECR, SJW, HPD and KL in Poisson Mixture Models with Fixed Component Weights under Scenarios (1)–(4).

<p>This table summaries averages (avg) and standard deviations (sd) of posterior means over 100 replications for OC, AVP, ECR, SJW, OC on <b><i>θ</i></b>, HPD and KL, where OC stands for ordering constraints on <b><i>η</i></b>, and OC on <b><i>θ</i></b> represents ordering constraints on <b><i>θ</i></b>.</p><p>The Performance of AVP, ECR, SJW, HPD and KL in Poisson Mixture Models with Fixed Component Weights under Scenarios (1)–(4).</p

Plot (a) is the 3-dimensional scatter plot of unconstrained sample with (<i>γ</i>₂₁₁; <i>γ</i>₂₁₂; <i>γ</i>₂₁₃).

<p>The six colors represent the 3! sets of labels before relabelling. The relabelled samples applied by AVP algorithm are shown in Plot (b). Plot (c) is the trace plots of <i>γ</i>₈₁₁, <i>γ</i>₈₁₂ and <i>γ</i>₈₁₃.</p

Plots (a)–(f) are scatter plots of posterior samples of (<i>ϕ</i>₁, <i>ϕ</i>₂) for Scenario (3) (n = 10, <i>ϕ</i>₁ = 5, <i>ϕ</i>₂ = 5.5, η₁ = 0.3 and η₂ = 0.7).

<p>Plot (a) is the posterior samples with correct labels. Plots (b)–(f) are the relabelled samples under various relabelling algorithms.</p

The relationship between underlying subgroups and covariates from hierarchical LCA.

<p>^a OR: odds ratio</p><p>^b CI: 95% credible interval of OR</p><p>* Asterisk is added if value is significantly different from 1, judged by CI not covering 1.</p><p>The relationship between underlying subgroups and covariates from hierarchical LCA.</p

The association between the PANSS symptoms’ probability and covariates from hierarchical RLCA.

<p>^a OR: odds ratio</p><p>^b CI: 95% credible interval of OR</p><p>* Asterisk is added if value is significantly different from 1, judged by CI not covering 1.</p><p>The association between the PANSS symptoms’ probability and covariates from hierarchical RLCA.</p

The density plots of relabelling samples from various relabelling methods in Scenarios (3) and (4).

<p>The black dashed line represents the density plot of the true posterior distributions. The grey, blue, purple, blue and red lines represent the density plots of KL, ECR, HPD, SJW and AVP, respectively. Plots (a) and (b) are the density plots of <i>ϕ</i>₁ and <i>ϕ</i>₂ for Scenario (3), respectively. Plots (c) and (d) are the density plots of <i>ϕ</i>₁ and <i>ϕ</i>₂ for Scenario (4), respectively.</p

Plots (a)–(f) are scatter plots of posterior samples of (<i>ϕ</i>₁, <i>ϕ</i>₂) for Scenario (2) (n = 100, <i>ϕ</i>₁ = 5, <i>ϕ</i>₂ = 7, η₁ = η₂ = 0.5).

<p>Plot (a) is the unconstrained samples. Plots (b)–(f) are the relabelled samples under various relabelling algorithms.</p

Genome-wide nonparametric LOD scores of two indices attaining genome-wide significance in the nested OSA linkage analyses over 22 autosomal chromosomes.

<p>Genome-wide nonparametric LOD scores of two indices attaining genome-wide significance in the nested OSA linkage analyses over 22 autosomal chromosomes.</p