The DDPdensity function generates posterior density samples for a univariate Griffiths-Milne dependent Dirichlet process mixture model with Gaussian
kernel, for partially exchangeable data. The function implements the importance conditional sampler method.
DDPdensity(y, group, mcmc = list(), prior = list(), output = list())a vector or matrix giving the data based on which densities are to be estimated;
vector of length length(y) containing the group labels (integers)
for the elements of y;
list of MCMC arguments:
niter (mandatory), number of iterations.
nburn (mandatory), number of iterations to discard as burn-in.
nupd, argument controlling the number of iterations to be displayed on screen: the function reports
on standard output every time nupd new iterations have been carried out (default is niter/10).
print_message, control option. If equal to TRUE, the status is printed
to standard output every nupd iterations (default is TRUE).
m_imp, number of generated values for the importance sampling step of the
importance conditional sampler (default is 10). See details.
a list giving the prior information, which contains:
strength, the strength parameter, or total mass, of the marginal Dirichlet processes (default 1);
m0, mean of the normal base measure on the location parameter (default 0);
k0, scale factor appearing in the normal base measure on the location parameter (default 1);
a0, shape parameter of the inverse gamma base measure on the scale parameter (default 2);
b0, scale parameter of the inverse gamma base measure on the scale parameter (default 1);
wei, parameter controlling the strength of dependence across Dirichlet processes (default 1/2).
a list of arguments for generating posterior output. It contains:
grid, a grid of points at which to evaluate the estimated posterior mean densities (common for all the groups).
out_type, if out_type = "FULL", return the estimated partitions and the realizations of the posterior density for each iterations. If out_type = "MEAN", return
the estimated partitions and the mean of the densities sampled at each iterations. If out_type = "CLUST", return the estimated partitions. Default out_type = "FULL".
A BNPdensity class object containing the estimated densities for each iteration,
the allocations for each iteration; the grid used to evaluate the densities (for each group); the
densities sampled from the posterior distribution (for each group); the groups; the weights of the processes.
The function returns also informations regarding the estimation: the number of iterations, the number
of burn-in iterations and the execution time.
This function fits a Griffiths-Milne dependent Dirichlet process (GM-DDP) mixture
for density estimation for partially exchangeable data (Lijoi et al., 2014).
For each observation the group variable allows the observations to be gathered
into \(L\)=length(unique(group)) distinct groups.
The model assumes exchangeability within each group, with observations in the \(l\)th group marginally
modelled by a location-scale Dirichlet process mixtures, i.e.
$$\tilde f_l(y) = \int \phi(y; \mu, \sigma^2) \tilde p_l (d \mu, d \sigma^2)$$
where each \(\tilde p_l\) is a Dirichlet process with total mass strength and base measure \(P_0\).
The vector \(\tilde p = (\tilde p_1,\ldots,\tilde p_L)\) is assumed to be jointly distributed as a vector of
GM-DDP(strength, wei; \(P_0\)), where strength and
\(P_0\) are the total mass parameter and the base measure of each \(\tilde p_l\), and wei
controls the dependence across the components of
\(\tilde p\). Admissible values for wei are in \((0,1)\), with the two extremes of the range
corresponding to full exchangeability (wei\(\rightarrow 0\))
and independence across groups (wei\(\rightarrow 1\)).
\(P_0\) is a normal-inverse gamma base measure, i.e. $$P_0(d\mu,d\sigma^2) = N(d \mu; m_0, \sigma^2 / k_0) \times IGa(d \sigma^2; a_0, b_0).$$
Posterior sampling is obtained by implementing the importance conditional sampler (Canale et al., 2019).
Lijoi, A., Nipoti, B., and Pruenster, I. (2014). Bayesian inference with dependent normalized completely random measures. Bernoulli 20, 1260<U+2013>1291.
Canale, A., Corradin, R., & Nipoti, B. (2019). Importance conditional sampling for Bayesian nonparametric mixtures. arXiv preprint arXiv:1906.08147.
# NOT RUN {
data_toy <- c(rnorm(50, -4, 1), rnorm(100, 0, 1), rnorm(50, 4, 1))
group_toy <- c(rep(1,100), rep(2,100))
grid <- seq(-7, 7, length.out = 50)
est_model <- DDPdensity(y = data_toy, group = group_toy,
mcmc = list(niter = 200, nburn = 100, napprox_unif = 50),
output = list(grid = grid))
summary(est_model)
plot(est_model)
# }
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