# Proposal distribution settings RW Metropolis sampler for mHMM Poisson-lognormal emission distribution(s) `pd_RW_emiss_count` provides a framework to manually specify the settings of the proposal distribution of the random walk (RW) Metropolis sampler of Poisson emission distribution(s) of the multilevel hidden Markov model, and creates on object of the class `mHMM_pdRW_emiss`. The RW metropolis sampler is used for sampling the subject level parameter estimates relating to the emission distributions of the dependent variables `k`, that is, the Poisson parameters lambda. ```r pd_RW_emiss_count(gen, emiss_scalar, emiss_w) ``` ## 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_scalar`: A list containing `n_dep` elements corresponding to each of the dependent variables, where each element is 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. - `emiss_w`: A list containing `n_dep` elements corresponding to each of the dependent variables, where each element is 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. ## Returns `pd_RW_emiss_count` returns an object of class `mHMM_pdRW_emiss`, containing settings of the proposal distribution of the random walk (RW) Metropolis sampler on the categorical emission distribution(s) 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: - **`gen`**: A list containing the elements `m` and `n_dep`, used for checking equivalent general model properties specified under `pd_RW_emiss_count` and `mHMM`. - **`emiss_scalar`**: A list containing `n_dep` elements denoting the scale factor `s` of the proposal distribution. - **`emiss_w`**: A list containing `n_dep` elements denoting denoting the weight for the overall log likelihood 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_emiss_count` is not used), `emiss_scalar` set to 2.38, and `emiss_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 emission distribution(s) can be tracked using the output parameter `emiss_naccept`. An acceptance rate of about 45% is considered optimal when a single parameter is being updated (Gelman, Carlin, Stern & Rubin, 2014). ## Examples ```r ###### Example using simulated data # specifying general model properties: n_t <- 200 # Number of observations on the dependent variable m <- 3 # Number of hidden states n_dep <- 2 # Number of dependent variables n_subj <- 30 # Number of subjects gamma <- matrix(c(0.9, 0.05, 0.05, 0.2, 0.7, 0.1, 0.2,0.3, 0.5), ncol = m, byrow = TRUE) emiss_distr <- list(matrix(c(20, 10, 5), nrow = m, byrow = TRUE), matrix(c(50, 3, 20), nrow = m, byrow = TRUE)) # Define between subject variance to use on the simulating function: # here, the variance is varied over states within the dependent variable. var_emiss <- list(matrix(c(5.0, 3.0, 1.5), nrow = m), matrix(c(5.0, 5.0, 5.0), nrow = m)) # Simulate count data: data_count <- sim_mHMM(n_t = n_t, n = n_subj, data_distr = "count", gen = list(m = m, n_dep = n_dep), gamma = gamma, emiss_distr = emiss_distr, var_gamma = 0.1, var_emiss = var_emiss, return_ind_par = TRUE) # Transition probabilities start_gamma <- diag(0.8, m) start_gamma[lower.tri(start_gamma) | upper.tri(start_gamma)] <- (1 - diag(start_gamma)) / (m - 1) # Emission distribution start_emiss <- list(matrix(c(20,10, 5), nrow = m, byrow = TRUE), matrix(c(50, 3,20), nrow = m, byrow = TRUE)) # Specify hyper-prior for the count emission distribution manual_prior_emiss <- prior_emiss_count( gen = list(m = m, n_dep = n_dep), emiss_mu0 = list(matrix(c(20, 10, 5), byrow = TRUE, ncol = m), matrix(c(50, 3, 20), byrow = TRUE, ncol = m)), emiss_K0 = rep(list(0.1),n_dep), emiss_nu = rep(list(0.1),n_dep), emiss_V = rep(list(rep(10, m)),n_dep) ) # Specify the desired values for the sampler manual_emiss_sampler <- pd_RW_emiss_count(gen = list(m = m, n_dep = n_dep), emiss_scalar = rep(list(2.38),n_dep), emiss_w = rep(list(0.1), n_dep)) # Run 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_count_RWemiss <- mHMM(s_data = data_count$obs, data_distr = 'count', gen = list(m = m, n_dep = n_dep), start_val = c(list(start_gamma), start_emiss), emiss_hyp_prior = manual_prior_emiss, emiss_sampler = manual_emiss_sampler, mcmc = list(J = 11, burn_in = 5), show_progress = TRUE) # Examine acceptance rates over dependent variable, individual, and states: lapply(out_3st_count_RWemiss$emiss_naccept, function(e) e/out_3st_count_RWemiss$input$J) # Finally, take the average acceptance rate by dependent variable and state: lapply(out_3st_count_RWemiss$emiss_naccept, function(e) colMeans(e/out_3st_count_RWemiss$input$J)) ``` ## References \insertRef gelman2014mHMMbayes \insertRef rossi2012mHMMbayes

pd_RW_emiss_count function