Hierarchical Bayesian mixture modelling for antigen-specific T-cell subtyping in combinatorially encoded flow cytometry studies

7.3.2.1 Update phenotypic marker mixture weights and hyperparameter

Sampling mixture probabilities π_1:J and the hyperparameter of the DP model use a Metropolis-Hastings extension of the standard component distributions (Ishwaran and James, 2001; Ji et al., 2009), as follows. The mixture probabilities π_j are obtained from underlying beta variates ν_1:J–₁ as detailed in Appendix 7.1. Hence new π_1:J samples are computed directly from resampled ν_1:J–₁ samples. For the latter, we have

where for each j=1:J–1. The complications here are that, since the π_j are functions of the ν_j, then ν_j is implicitly involved in both numerator and denominator terms of the product expression multiplying the base beta distributions. Hence we use a Metropolis-Hastings sampler for this step, based on a customized proposal distribution

with

This proposal distribution is an approximation of p(ν_j|…) by taking off the denominator f(b_i|Θ)^–1 in the product expression and assuming that dominates the rest of the component values. Our experience with examples and the data analysis reported is that this generates acceptable convergence with acceptance rates for these components of the MCMC around 10–50%.

The weights π_1:J are then evaluated by the formula in Appendix 7.1. Next, the hyperparameter α_b is resampled from

7.3.2.2 Update phenotypic marker component means and variance matrices

For each j=1:J the mean μ_b,j has conditional posterior

with

where c_j=λ/(1+λa_1,j). Again we need a Metropolis-Hastings sampler for this step as the base normal distribution here is multiplied by a term that depends in complicated ways on μ_b,j. We use the customized proposal distribution where

with

A similar structure and MCMC strategy arises for each of the j=1:J variance matrices Σ_b,j; the conditional posteriors are

where

We use the customized proposal distribution

with

We update the pair (μ_b,j, Σ_b,j) together each iterate. We achieve acceptable convergence with acceptance rates for these components of the MCMC around 20–45%.

7.3.3.1 Update multimer mixture weights and hyperparameter

With the definitions and notation of the multimer mixture model parameters of Appendix 7.2, the logic and details of the MCMC steps are as follows.

For each k=1:K ϕ_k has conditional posterior

To choose a proposal distribution, first, for each i=1:n and independently over i, generate a set of auxiliary indicator variables q_i from conditional multinomials on k=1:K cells with number of trials=1 and

(q_i=k)∝η_kN(t_i; μ_t,k, Σ_t,k),  k=1:K.

Given these sampled values, generate

where for each r=1:K. We achieve acceptable convergence with acceptance rates for these components of the MCMC around 10–40%.

The sets of weights ω_j,k and the η_1:K probabilities are then evaluated by the formulæ given in Appendix 7.2. Further, the hyper-parameter γ_t is resampled from

Next, for each j=1:J and k=1:K, the latent probabilities ϕ_j,k of Appendix 7.2 have conditional posterior

We use the customized proposal distribution

where We achieve acceptable convergence with acceptance rates for these components of the MCMC around 5–50%.

7.3.3.2 Update multimer component means and variance matrices

For each k=1:K the mean μ_t,k is sampled using an additional auxiliary random quantity that allocates the multimer to one of the K anchor regions based on current parameters and indicators. That is, for each k independently, draw an auxiliary indicator τ_k from the multinomial with one trial and probabilities on k=1:K given by

Then draw (μ_t,k|C_k=r, …)~N (μ_t,k|m_t,k, M_t,k) where

with

Finally, resample the variance matrices from