prior_emiss_cont: Specifying informative hyper-prior on the continuous emission...

In mHMMbayes: Multilevel Hidden Markov Models Using Bayesian Estimation

Specifying informative hyper-prior on the continuous emission distribution(s) of the multilevel hidden Markov model

Description

prior_emiss_cont provides a framework to manually specify an informative hyper-prior on the Normal (i.e., Gaussian) emission distributions. prior_emiss_cont creates an object of class mHMM_prior_emiss used by the function mHMM, and additionally attaches the class cont to signal use for continuous observations. The set of hyper-prior distributions consists of a Normal-Inverse-Gamma distribution (i.e., assuming both unknown population mean and variance between subject level means) on the vector of means (i.e., intercepts and regression coefficients), and an Inverse gamma distribution (i.e., assuming a known mean) on each of the (fixed over subjects) emission variances.

Usage

prior_emiss_cont(
  gen,
  emiss_mu0,
  emiss_K0,
  emiss_V,
  emiss_nu,
  emiss_a0,
  emiss_b0,
  n_xx_emiss = NULL
)

Arguments

gen	List containing the following elements denoting the general model properties: m: numeric vector with length 1 denoting the number of hidden states n_dep: numeric vector with length 1 denoting the number of dependent variables q_emiss: only to be specified if the data represents categorical data. Numeric vector with length n_dep denoting the number of observed categories for the categorical emission distribution for each of the dependent variables.
emiss_mu0	A list containing n_dep matrices, i.e., one list for each dependent variable k. Each matrix contains the hypothesized hyper-prior means of the Normal emission distributions in each of the states. Hence, each matrix consists of one row (when not including covariates in the model) and m columns. If covariates are used, the number of rows in each matrix in the list is equal to 1 + n_xx (i.e., the first row corresponds to the hyper-prior means, the subsequent rows correspond to the hyper-prior values of the regression coefficients connected to each of the covariates).
emiss_K0	A list containing n_dep elements corresponding to each dependent variable k. Each element k is a numeric vector with length 1 (when no covariates are used) denoting the number of hypothetical prior subjects on which the set of hyper-prior means specified in emiss_mu0 are based. When covariates are used: each element is a numeric vector with length 1 + n_xx denoting the number of hypothetical prior subjects on which the set of means (first value) and set of regression coefficients (subsequent values) are based.
emiss_V	A list containing n_dep elements corresponding to each of the dependent variables k, where each element k is a vector with length m containing the hypothesized variance between the subject (emission distribution) means, which are assumed to follow a Inverse Gamma hyper-prior distribution (note: here, the Inverse Gamma hyper-prior distribution is parametrized as a scaled inverse chi-squared distribution).
emiss_nu	A list containing n_dep elements corresponding to each dependent variable k. Each element k is a numeric vector with length 1 denoting the degrees of freedom of the Inverse Gamma hyper-prior distribution on the between subject variance of the emission distribution means (note: here, the Inverse Gamma hyper-prior distribution is parametrized as a scaled inverse chi-squared distribution).
emiss_a0	A list containing n_dep elements corresponding to each of the dependent variables k, where each element k is a vector with length m containing the shape values of the Inverse Gamma hyper-prior on each of the (fixed over subjects) emission standard deviation^2 of the Normal emission distributions (note: here the standard Inverse Gamma parametrization is used).
emiss_b0	A list containing n_dep elements corresponding to each of the dependent variables k, where each element k is a vector with length m containing the scale values of the Inverse Gamma hyper-prior on each of the (fixed over subjects) emission standard deviation^2 of the Normal emission distributions (note: here the standard Inverse Gamma parametrization is used).
n_xx_emiss	Optional numeric vector with length n_dep denoting the number of (level 2) covariates used to predict the emission distribution of each of the dependent variables k. When omitted, the model assumes no covariates are used to predict the emission distribution(s).

Details

Estimation of the mHMM proceeds within a Bayesian context, hence a hyper-prior distribution has to be defined for the group level parameters. To avoid problems with 'label switching' when dealing with continuous emission distribution(s) (i.e., switching of the labels of the hidden states while sampling from the MCMC), the user is forced to specify hyper-prior parameter values when using continuous emission distributions (i.e., default, non-informative priors are not available for continuous emission distributions).

Note that emiss_K0 and emiss_nu are assumed equal over the states. Also note that in case covariates are specified, the hyper-prior parameter values of the inverse Wishart distribution on the covariance matrix remain unchanged, as the estimates of the regression coefficients for the covariates are fixed over subjects.

Value

prior_emiss_cont returns an object of class mHMM_prior_emiss, containing informative hyper-prior values for the continuous emission distribution(s) of the multilevel hidden Markov model. The object is specifically created and formatted for use by the function mHMM, and thoroughly checked for correct input dimensions. The object contains the following components:

gen

A list containing the elements m, and n_dep, used for checking equivalent general model properties specified under prior_emiss_cont and mHMM.

emiss_mu0

A lists containing the hypothesized hyper-prior means of the Normal distribution on the continuous emission probabilities.

emiss_K0

A list containing n_dep elements denoting the number of hypothetical prior subjects on which the set of hyper-prior means specified in emiss_mu0 are based.

emiss_V

A list containing n_dep elements containing the variance of the Inverse Gamma hyper-prior distribution on the between subject variance of the emission distribution means.

emiss_nu

A list containing n_dep elements denoting the degrees of freedom of the Inverse Gamma hyper-prior distribution on the between subject variance of the emission distribution means.

emiss_a0

A list containing n_dep elements denoting the shape values of the Inverse Gamma hyper-prior on each of the (fixed over subjects) emission standard deviation^2 of the Normal emission distributions.

emiss_b0

A list containing n_dep elements denoting the scale values of the Inverse Gamma hyper-prior on each of the (fixed over subjects) emission standard deviation^2 of the Normal emission distributions.

n_xx_emiss

A numeric vector denoting the number of (level 2) covariates used to predict the emission distribution of each of the dependent variables. When no covariates are used, n_xx_emiss equals NULL.

See Also

prior_gamma for manually specifying an informative hyper-prior on the transition probability matrix gamma, and mHMM for fitting a multilevel hidden Markov model.

Examples

###### Example using simulated data
# specifying general model properties:
m <- 3
n_dep <- 2

# hypothesized hyper-prior values for the continuous emission distribution
manual_prior_emiss <- prior_emiss_cont(
                        gen = list(m = m, n_dep = n_dep),
                        emiss_mu0 = list(matrix(c(30, 70, 170), nrow = 1),
                                         matrix(c(7, 8, 18), nrow = 1)),
                        emiss_K0 = list(1, 1),
                        emiss_V =  list(rep(100, m), rep(25, m)),
                        emiss_nu = list(1, 1),
                        emiss_a0 = list(rep(1, m), rep(1, m)),
                        emiss_b0 = list(rep(1, m), rep(1, m)))

# to use the informative priors in a model, simulate multivariate continuous data
n_t     <- 100
n       <- 10

gamma   <- matrix(c(0.8, 0.1, 0.1,
                    0.2, 0.7, 0.1,
                    0.2, 0.2, 0.6), ncol = m, byrow = TRUE)

emiss_distr <- list(matrix(c( 50, 10,
                              100, 10,
                              150, 10), nrow = m, byrow = TRUE),
                    matrix(c(5, 2,
                             10, 5,
                             20, 3), nrow = m, byrow = TRUE))

data_cont <- sim_mHMM(n_t = n_t, n = n, data_distr = 'continuous', gen = list(m = m, n_dep = n_dep),
                  gamma = gamma, emiss_distr = emiss_distr, var_gamma = .1, var_emiss = c(.5, 0.01))

# using the informative hyper-prior in a model
# Note that for reasons of running time, J is set at a ridiculous low value.
# One would typically use a number of iterations J of at least 1000,
# and a burn_in of 200.
out_3st_cont_sim_infemiss <- mHMM(s_data = data_cont$obs,
                    data_distr = "continuous",
                    gen = list(m = m, n_dep = n_dep),
                    start_val = c(list(gamma), emiss_distr),
                    emiss_hyp_prior = manual_prior_emiss,
                    mcmc = list(J = 11, burn_in = 5))

out_3st_cont_sim_infemiss
summary(out_3st_cont_sim_infemiss)

mHMMbayes documentation built on Oct. 2, 2023, 5:06 p.m.