Model II: Dirichlet Process Gaussian mixture model (DP-GMM)

Model I is a fast and efficient genotyping model for SNPs having large values of MAF. In real experiments, many SNPs with low MAF may result in the disappearance of one or two genotype clusters. Also even though some SNPs with low MAF display three genotype groups, some clusters may lack sufficient data to support and recognize. In this case, Model II, DP Gaussian Mixture Model, is motivated by the need to carry out the model selection for SNPs with an uncertain number of genotype clusters [24]. Generally speaking, this is a nonparametric Bayesian method that potentially allows a flexible number of mixture components and also provides estimates for the mixture component parameters and the relevant mixing proportions.

A DP Gaussian Mixture Model [24] fits the pair of raw intensity x_is into K-component Gaussian Mixture Model with K approaching a large number. The model is expressed as,

where K is the total number of clusters. Θ_s=(π_s, μ_s, Σ_s) denotes the unknown parameters at the sth SNP where π_s=(π_1s,..., π_Ks), μ_s=(μ_1s,..., μ_Ks), and Σ_s=(Σ_1s,..., Σ_Ks). Generally, the number of observations within the sth SNP (n_s) are partitioned into K components (n_1s, n_2s,..., n_Ks) with relevant mixing proportions (π_1s, π_2s,..., π_Ks). The distribution of n_1s, n_2s,..., n_Ks follows a multinomial distribution and its probability mass function is written by,

where n_s = ∑k=1Knks denotes the total number of individuals at the sth SNP. Then each pair of raw intensity for the sth SNP x_is has its own indicator z_is (i = 1,..., n_s), and the distribution of indicator variables is expressed as,

The model can then be expressed as:

where α is the DP concentration parameter and can be thought as the inverse variance of DP. The distribution of the reciprocal of α follows a Gamma distribution with 1 degree freedom and mean 1. K is the maximum number of clusters, then π_s is distributed with a symmetric Dirichlet distribution with parameter αK. m and r are hyperparameters being the mean and relative precision of μ_ks, and the hyperparameters ν and S⁻¹ are degrees of freedom and inverse mean of R_ks where R_ks follows a Wishart distribution with parameters ν and S⁻¹, respectively.

The inference on Model II relies on the posterior distribution of each parameter conditional on all other parameters, then the parameters, hyperparameters and indicator variables are repeatedly sampled from their posterior distributions. In particular, the conditional posterior probabilities are proportional to the likelihood function multiplying priors. Then the posterior probabilities of the cluster indicator variable z_is conditional on all other variables are expressed as:

Note that p(x_is|μ_ks,R_ks) and p(μ_ks,R_ks|m,r,ν,S) are the likelihood function and the joint function of parameters (μ_ks and R_ks), respectively. Once the optimal genotype clusters and their relevant component parameters are obtained, two measurements Posterior Rate (PR) and the Average Posterior Rate (APR) measuring the quality of the sth SNP can be calculated in the similar way.