pd_RW_gamma {mHMMbayes}

Proposal distribution settings RW Metropolis sampler for mHMM transition probability matrix gamma

Description

pd_RW_gamma provides a framework to manually specify the settings of the proposal distribution of the random walk (RW) Metropolis sampler of the transition probability matrix gamma of the multilevel hidden Markov model, and creates on object of the class mHMM_pdRW_gamma. The RW metropolis sampler is used for sampling the subject level parameter estimates relating to the transition probability matrix gamma, that is, the Multinomial logistic regression intercepts.

Usage

pd_RW_gamma(m, gamma_int_mle0, gamma_scalar, gamma_w)

Arguments

m

Numeric vector with length 1 denoting the number of hidden states.

gamma_int_mle0

A matrix with m rows and m - 1 columns denoting the starting values for the maximum likelihood (ML) estimates of the Multinomial logit regression intercepts of the transition probability matrix gamma. ML parameters to be estimated are based on the pooled data (data over all subjects).

gamma_scalar

A numeric vector with length 1 denoting the scale factor s. That is, the scale of the proposal distribution is composed of a covariance matrix Sigma, which is then tuned by multiplying it by a scaling factor s^2.

gamma_w

A numeric vector with length 1 denoting the weight for the overall log likelihood (i.e., log likelihood based on the pooled data over all subjects) in the fractional likelihood.

Details

When no manual values for the settings of the proposal distribution of the random walk (RW) Metropolis sampler are specified at all (that is, the function pd_RW_gamma is not used), all elements in gamma_int_mle0 set to 0, gamma_scalar set to 2.93 / sqrt(m - 1), and gamma_w set to 0.1. See the section Scaling the proposal distribution of the RW Metropolis sampler in vignette("estimation-mhmm") for details.

Within the function mHMM, the acceptance rate of the RW metropolis sampler relating to the transition probability matrix gamma can be tracked using the output parameter gamma_naccept. An acceptance rate of about 23% is considered optimal when many parameters are being updated at once (Gelman, Carlin, Stern & Rubin, 2014).

Value

pd_RW_gamma returns an object of class mHMM_pdRW_gamma, containing settings of the proposal distribution of the random walk (RW) Metropolis sampler on 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 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 pd_RW_gamma and mHMM.

gamma_int_mle0

A matrix containing the starting values for the maximum likelihood (ML) estimates of the Multinomial logit regression intercepts of the transition probability matrix gamma.

gamma_scalar

A numeric vector with length 1 denoting the scale factor s of the proposal distribution.

gamma_w

A numeric vector with length 1 denoting denoting the weight for the overall log likelihood in the fractional likelihood.

References

Gelman A, Carlin JB, Stern HS, Rubin DB (2014). Bayesian Data Analysis vol. 2. Taylor & Francis.

Rossi PE, Allenby GM, McCulloch R (2012). Bayesian statistics and marketing. John Wiley & Sons.

Examples

###### Example using package example data, see ?nonverbal
# specifying general model properties:
m <- 3

# specifying manual values for RW metropolis sampler on gamma
gamma_int_mle0 <- matrix(c( -2, -2,
                             2,  0,
                             0,  3), byrow = TRUE, nrow = m, ncol = m - 1)
gamma_scalar <- c(2)
gamma_w <- c(0.2)
manual_gamma_sampler <- pd_RW_gamma(m = m, gamma_int_mle0 = gamma_int_mle0,
                                    gamma_scalar = gamma_scalar,
                                    gamma_w = gamma_w)

# specifying starting values
n_dep <- 4
q_emiss <- c(3, 2, 3, 2)

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.

out_3st_RWgamma <- 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_sampler = manual_gamma_sampler,
                         mcmc = list(J = 11, burn_in = 5))
out_3st_RWgamma
summary(out_3st_RWgamma)

# checking acceptance rate (for illustrative purposes, in the example,
# J is too low for getting a fair indication)
J_it <- 11 - 1 # accept/reject starts at iteration 2 of MCMC algorithm
out_3st_RWgamma$gamma_naccept / J_it
# average acceptance rate over all subjects per parameter
apply(out_3st_RWgamma$gamma_naccept / J_it, 2, mean)