R: Sample fiSAN

sample_fiSAN {SANple}

R Documentation

Sample fiSAN

Description

sample_fiSAN is used to perform posterior inference under the finite-infinite shared atoms nested (fiSAN) model with Gaussian likelihood. The model uses a Dirichlet process mixture prior at the distributional level, and Dirichlet mixture with an unknown number of components (following the specification of Frühwirth-Schnatter et al., 2021) at the observational level. The algorithm for the nonparametric component is based on the slice sampler for DPM of Kalli, Griffin and Walker (2011).

Usage

sample_fiSAN(nrep, y, group, 
             maxK = 50, maxL = 50, 
             m0 = 0, tau0 = 0.1, lambda0 = 3, gamma0 = 2,
             hyp_alpha1 = 1, hyp_alpha2 = 1,
             hyp_beta1 = 6, hyp_beta2 = 3, 
             eps_beta = NULL,
             alpha = NULL, beta = NULL,
             warmstart = TRUE, nclus_start = NULL,
             mu_start = NULL, sigma2_start = NULL,
             M_start = NULL, S_start = NULL,
             alpha_start = NULL, beta_start = NULL,
             progress = TRUE, seed = NULL)

Arguments

`nrep`	Number of MCMC iterations.
`y`	Vector of observations.
`group`	Vector of the same length of y indicating the group membership (numeric).
`maxK`	Maximum number of distributional clusters `K` (default = 50).
`maxL`	Maximum number of observational clusters `L` (default = 50).
`m0`, `tau0`, `lambda0`, `gamma0`	Hyperparameters on `(\mu, \sigma^2) \sim NIG(m_0, \tau_0, \lambda_0,\gamma_0)`. Default is (0, 0.1, 3, 2).
`hyp_alpha1`, `hyp_alpha2`	If a random `\alpha` is used, (`hyp_alpha1`,`hyp_alpha2`) specify the hyperparameters (default = (1,1)). The prior is `\alpha` ~ Gamma(`hyp_alpha1`, `hyp_alpha2`).
`hyp_beta1`, `hyp_beta2`, `eps_beta`	If a random `\beta` is used, (`hyp_beta1`,`hyp_beta2`) specifies the hyperparameter (default = (6,3)). The prior is `\beta\sim Gamma`(`hyp_beta1`, `hyp_beta2`). In this case, `eps_beta` is the tuning parameter of the MH step.
`alpha`	Distributional DP parameter if fixed (optional). The distribution is `\pi\sim GEM (\alpha)`.
`beta`	Observational Dirichlet parameter if fixed (optional). The distribution is Dirichlet( `rep(beta/maxL, maxL)` ).
`warmstart`, `nclus_start`	Initialization of the observational clustering. `warmstart` is logical parameter (default = `TRUE`) of whether a kmeans clustering should be used to initialize the chains. An initial guess of the number of observational clusters can be passed via the `nclus_start` parameter (optional)
`mu_start`, `sigma2_start`, `M_start`, `S_start`, `alpha_start`, `beta_start`	Starting points of the MCMC chains (optional). Default is `nclus_start = min(c(maxL, 30))`. `mu_start, sigma2_start` are vectors of length `maxL`. `M_start` is a vector of observational cluster allocation of length N. `S_start` is a vector of observational cluster allocation of length J. `alpha_start, alpha_start` are numeric.
`progress`	show a progress bar? (logical, default TRUE).
`seed`	set a fixed seed.

Details

Data structure

The finite-infinite common atoms mixture model is used to perform inference in nested settings, where the data are organized into J groups. The data should be continuous observations (Y_1,\dots,Y_J), where each Y_j = (y_{1,j},\dots,y_{n_j,j}) contains the n_j observations from group j, for j=1,\dots,J. The function takes as input the data as a numeric vector y in this concatenated form. Hence y should be a vector of length n_1+\dots+n_J. The group parameter is a numeric vector of the same size as y indicating the group membership for each individual observation. Notice that with this specification the observations in the same group need not be contiguous as long as the correspondence between the variables y and group is maintained.

Model

The data are modeled using a univariate Gaussian likelihood, where both the mean and the variance are observational-cluster-specific, i.e.,

y_{i,j}\mid M_{i,j} = l \sim N(\mu_l,\sigma^2_l)

where M_{i,j} \in \{1,\dots,L \} is the observational cluster indicator of observation i in group j. The prior on the model parameters is a Normal-Inverse-Gamma distribution (\mu_l,\sigma^2_l)\sim NIG (m_0,\tau_0,\lambda_0,\gamma_0), i.e., \mu_l\mid\sigma^2_l \sim N(m_0, \sigma^2_l / \tau_0), 1/\sigma^2_l \sim Gamma(\lambda_0, \gamma_0) (shape, rate).

Clustering

The model performs a clustering of both observations and groups. The clustering of groups (distributional clustering) is provided by the allocation variables S_j \in \{1,2,\dots\}, with

Pr(S_j = k \mid \dots ) = \pi_k \qquad \text{for } \: k = 1,2,\dots

The distribution of the probabilities is \{\pi_k\}_{k=1}^{\infty} \sim GEM(\alpha), where GEM is the Griffiths-Engen-McCloskey distribution of parameter \alpha, which characterizes the stick-breaking construction of the DP (Sethuraman, 1994).

The clustering of observations (observational clustering) is provided by the allocation variables M_{i,j} \in \{1,\dots,L\}, with

Pr(M_{i,j} = l \mid S_j = k, \dots ) = \omega_{l,k} \qquad \text{for } \: k = 1,2,\dots \, ; \: l = 1,\dots,L.

The distribution of the probabilities is (\omega_{1,k},\dots,\omega_{L,k})\sim Dirichlet_L(\beta/L,\dots,\beta/L) for all k = 1,2,\dots. Moreover, the dimension L is random (see Frühwirth-Schnatter et al., 2021).

Value

sample_fiSAN returns four objects:

model: name of the fitted model.
params: list containing the data and the parameters used in the simulation. Details below.
sim: list containing the simulated values (MCMC chains). Details below.
time: total computation time.

Data and parameters: params is a list with the following components:

nrep: Number of MCMC iterations.
y, group: Data and group vectors.
maxK, maxL: Maximum number of distributional and observational clusters.
m0, tau0, lambda0, gamma0: Model hyperparameters.
(hyp_alpha1,hyp_alpha2) or alpha: Either the hyperparameters on \alpha (if \alpha random), or the value for \alpha (if fixed).
(hyp_beta1, hyp_beta2, eps_beta) or beta: Either the hyperparameters on \beta and MH step size (if \beta random), or the value for \beta (if fixed).

Simulated values: sim is a list with the following components:

mu: Matrix of size (nrep, maxL). Each row is a posterior sample of the mean parameter for each observational cluster (\mu_1,\dots\mu_L).
sigma2: Matrix of size (nrep, maxL). Each row is a posterior sample of the variance parameter for each observational cluster (\sigma^2_1,\dots\sigma^2_L).
obs_cluster: Matrix of size (nrep, n), with n = length(y). Each row is a posterior sample of the observational cluster allocation variables (M_{1,1},\dots,M_{n_J,J}).
distr_cluster: Matrix of size (nrep, J), with J = length(unique(group)). Each row is a posterior sample of the distributional cluster allocation variables (S_1,\dots,S_J).
pi: Matrix of size (nrep, maxK). Each row is a posterior sample of the distributional cluster probabilities (\pi_1,\dots,\pi_{maxK}).
omega: 3-d array of size (maxL, maxK, nrep). Each slice is a posterior sample of the observational cluster probabilities. In each slice, each column k is a vector (of length maxL) observational cluster probabilities (\omega_{1,k},\dots,\omega_{L,k}) for distributional cluster k.
alpha: Vector of length nrep of posterior samples of the parameter \alpha.
beta: Vector of length nrep of posterior samples of the parameter \beta.
maxK: Vector of length nrep of the number of distributional DP components used by the slice sampler.
L_plus: Vector of length nrep of posterior samples of the number of observational clusters.
L: Vector of length nrep of posterior samples of the observational Dirichlet dimension.

References

Frühwirth-Schnatter, S., Malsiner-Walli, G. and Grün, B. (2021). Generalized mixtures of finite mixtures and telescoping sampling. Bayesian Analysis, 16(4), 1279–1307. <doi:10.1214/21-BA1294>

Kalli, M., Griffin, J.E., and Walker, S.G. (2011). Slice Sampling Mixture Models, Statistics and Computing, 21, 93–105. <doi:10.1007/s11222-009-9150-y>

Sethuraman, A.J. (1994). A Constructive Definition of Dirichlet Priors, Statistica Sinica, 4, 639–650.

Examples

set.seed(123)
y <- c(rnorm(40,0,0.3), rnorm(20,5,0.3))
g <- c(rep(1,30), rep(2, 30))
plot(density(y[g==1]), xlim = c(-5,10))
lines(density(y[g==2]), col = 2)
out <- sample_fiSAN(nrep = 500, y = y, group = g, 
                    nclus_start = 2,
                    maxK = 20, maxL = 20,
                    beta = 1)
out

[Package SANple version 0.1.0 Index]