prior_gamma: Specifying informative hyper-prior on the transition probability matrix gamma of the multilevel hidden Markov model

Description

prior_gamma provides a framework to manually specify an informative hyper-prior on the transition probability matrix gamma, and creates an object of class mHMM_prior_gamma used by the function mHMM. Note that the hyper-prior distribution on the transition probabilities are on the intercepts (and, if subject level covariates are used, regression coefficients) of the Multinomial logit model used to accommodate the multilevel framework of the data, instead of on the probabilities directly. The set of hyper-prior distributions consists of a multivariate Normal hyper-prior distribution on the vector of means (i.e., intercepts and regression coefficients), and an Inverse Wishart hyper-prior distribution on the covariance matrix.

Usage

prior_gamma(
  m,
  gamma_mu0,
  gamma_K0 = NULL,
  gamma_nu = NULL,
  gamma_V = NULL,
  n_xx_gamma = NULL
)

Value

prior_gamma returns an object of class mHMM_prior_gamma, containing informative hyper-prior values for the transition probability matrix gamma 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:

m: Numeric vector denoting the number of hidden states, used for checking equivalent general model properties specified under prior_gamma and mHMM.
gamma_mu0: A list containing the hypothesized hyper-prior mean values of the intercepts of the Multinomial logit model on the transition probability matrix gamma.
gamma_K0: A numeric vector denoting the number of hypothetical prior subjects on which the set of hyper-prior mean intercepts specified in gamma_mu0 are based.
gamma_nu: A numeric vector denoting the degrees of freedom of the hyper-prior Inverse Wishart distribution on the covariance of the Multinomial logit intercepts.
gamma_V: A matrix containing the variance-covariance of the hyper-prior Inverse Wishart distribution on the covariance of the Multinomial logit intercepts.
n_xx_gamma: A numeric vector denoting the number of (level 2) covariates used to predict the transition probability matrix gamma. When no covariates are used, n_xx_gamma equals NULL.

Arguments

m: Numeric vector with length 1 denoting the number of hidden states.
gamma_mu0: A list containing m matrices; one matrix for each row of the transition probability matrix gamma. Each matrix contains the hypothesized hyper-prior mean values of the intercepts of the Multinomial logit model on the transition probabilities gamma. Hence, each matrix consists of one row (when not including covariates in the model) and m - 1 columns. If covariates are used, the number of rows in each matrix in the list is equal to 1 + n_xx_gamma (i.e., the first row corresponds to the hyper-prior mean values of the intercepts, the subsequent rows correspond to the hyper-prior mean values of the regression coefficients connected to each of the covariates).
gamma_K0: Optional numeric vector with length 1 (when no covariates are used) denoting the number of hypothetical prior subjects on which the set of hyper-prior mean intercepts specified in gamma_mu0 are based. When covariates are used: Numeric vector with length 1 + n_xx_gamma denoting the number of hypothetical prior subjects on which the set of intercepts (first value) and set of regression coefficients (subsequent values) are based.
gamma_nu: Optional numeric vector with length 1 denoting the degrees of freedom of the hyper-prior Inverse Wishart distribution on the covariance of the Multinomial logit intercepts.
gamma_V: Optional matrix of m - 1 by m - 1 containing the variance-covariance matrix of the hyper-prior Inverse Wishart distribution on the covariance of the Multinomial logit intercepts.
n_xx_gamma: Optional numeric vector with length 1 denoting the number of (level 2) covariates (excluding the intercept) used to predict the transition probability matrix gamma. When omitted, the model assumes no covariates are used to predict gamma.

Details

Estimation of the mHMM proceeds within a Bayesian context, hence a hyper-prior distribution has to be defined for the group level parameters. Default, non-informative priors are used unless specified otherwise by the user. Each row of the transition probability matrix has its own set of Multinomial logit intercepts, which are assumed to follow a multivariate normal distribution. Hence, the hyper-prior distributions for the intercepts consists of a multivariate Normal hyper-prior distribution on the vector of means, and an Inverse Wishart hyper-prior distribution on the covariance matrix. Note that only the number of states m and values of the hypothesized hyper-prior mean values of the Multinomial logit intercepts have to be specified by the user, default values are available for all other hyper-prior distribution parameters.

Given that the hyper-priors are specified on the intercepts of the Multinomial logit model intercepts instead of on the probabilities of the transition probability matrix gamma directly, specifying a hyper-prior can seem rather daunting. However, see the function prob_to_int and int_to_prob for translating probabilities to a set of Multinomial logit intercepts and vice versa.

Note that gamma_K0, gamma_nu and gamma_V are assumed equal over the states. When the hyper-prior values for gamma_K0, gamma_nu and gamma_V are not manually specified, the default values are as follows. gamma_K0 set to 1, gamma_nu set to 3 + m - 1, and the diagonal of gamma_V (i.e., the variance) set to 3 + m - 1 and the off-diagonal elements (i.e., the covariance) set to 0. In addition, when no manual values for the hyper-prior on gamma are specified at all (that is, the function prior_gamma is not used), all elements of the matrices contained in gamma_mu0 are set to 0 in the function mHMM.

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.

Examples

Run this code

###### Example using package example data, see ?nonverbal
# specifying general model properties:
m <- 3
# representing a prior belief that switching to state 3 does not occur often and
# state 3 has a relative short duration
prior_prob_gamma <- matrix(c(0.70, 0.25, 0.05,
                             0.25, 0.70, 0.05,
                             0.30, 0.30, 0.40), nrow = m, ncol = m, byrow = TRUE)

# using the function prob_to_int to obtain intercept values for the above specified
# transition probability matrix gamma
prior_int_gamma <- prob_to_int(prior_prob_gamma)
gamma_mu0 <- list(matrix(prior_int_gamma[1,], nrow = 1, ncol = m-1),
                  matrix(prior_int_gamma[2,], nrow = 1, ncol = m-1),
                  matrix(prior_int_gamma[3,], nrow = 1, ncol = m-1))

gamma_K0 <- 1
gamma_nu <- 5
gamma_V <- diag(5, m - 1)

manual_prior_gamma <- prior_gamma(m = m, gamma_mu0 = gamma_mu0,
                                  gamma_K0 = gamma_K0, gamma_nu = gamma_nu,
                                  gamma_V = gamma_V)


# using the informative hyper-prior in a model
n_dep <- 4
q_emiss <- c(3, 2, 3, 2)

# specifying starting values
start_TM <- diag(.7, m)
start_TM[lower.tri(start_TM) | upper.tri(start_TM)] <- .1
start_EM <- list(matrix(c(0.05, 0.90, 0.05,
                          0.90, 0.05, 0.05,
                          0.55, 0.45, 0.05), byrow = TRUE,
                        nrow = m, ncol = q_emiss[1]), # vocalizing patient
                 matrix(c(0.1, 0.9,
                          0.1, 0.9,
                          0.1, 0.9), byrow = TRUE, nrow = m,
                        ncol = q_emiss[2]), # looking patient
                 matrix(c(0.90, 0.05, 0.05,
                          0.05, 0.90, 0.05,
                          0.55, 0.45, 0.05), byrow = TRUE,
                        nrow = m, ncol = q_emiss[3]), # vocalizing therapist
                 matrix(c(0.1, 0.9,
                          0.1, 0.9,
                          0.1, 0.9), byrow = TRUE, nrow = m,
                        ncol = q_emiss[4])) # looking therapist

# 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.
# \donttest{
out_3st_infgamma <- mHMM(s_data = nonverbal,
                    gen = list(m = m, n_dep = n_dep, q_emiss = q_emiss),
                    start_val = c(list(start_TM), start_EM),
                    gamma_hyp_prior = manual_prior_gamma,
                    mcmc = list(J = 11, burn_in = 5))

out_3st_infgamma
summary(out_3st_infgamma)
# }
# \dontshow{
out_3st_infgamma <- mHMM(s_data = nonverbal,
                    gen = list(m = m, n_dep = n_dep, q_emiss = q_emiss),
                    start_val = c(list(start_TM), start_EM),
                    gamma_hyp_prior = manual_prior_gamma,
                    mcmc = list(J = 5, burn_in = 3))
# }