R/logLik_Pajor.R

In glmmPen: High Dimensional Penalized Generalized Linear Mixed Models (pGLMM)

# Pajor Methods for logLik calcuations

indicator = function(bounds, samples){
  inbounds = function(x){mean(x > bounds[1,] & x < bounds[2,]) == 1}
  apply(samples, 1, inbounds)
}

indicator_1 = function(bounds, samples){
  inbounds = function(x){mean(x > bounds[1,] & x < bounds[2,]) == 1}
  apply(samples, 1, inbounds)
}

indicator_2 = function(bounds, samples){
  inbounds = function(x){mean(x > bounds[1] & x < bounds[2]) == 1}
  apply(samples, 1, inbounds)
}

# Pajor IS CAME Method for logLik calcuations
# article{pajor2017estimating,
#   title={Estimating the marginal likelihood using the arithmetic mean identity},
#   author={Pajor, Anna and others},
#   journal={Bayesian Analysis},
#   volume={12},
#   number={1},
#   pages={261--287},
#   year={2017},
#   publisher={International Society for Bayesian Analysis}
# }
#' @importFrom mvtnorm rmvnorm dmvnorm
#' @importFrom stats acf var dbinom dpois dnorm
CAME_IS = function(posterior, y, X, Z, group, coef, sigma, family, M, 
                   gaus_sig = NULL, trace, progress){
  
  if(progress == TRUE) message("Pajor Log-Likelihood Calculation")
  
  family_info = family_export(family)
  fam_fun = family_info$family_fun
  link = family_info$link
  link_int = family_info$link_int # Recoded link as integer, see "family_export.R" for details
  family = family_info$family
  
  # Define variables
  d = nlevels(group)
  num_var = ncol(Z) / d
  Gam = t(chol(sigma[which(diag(sigma)!=0),which(diag(sigma)!=0)]))
  eta_fef = X %*% coef[1:ncol(X)]
  post_mean = colMeans(posterior[,])
  
  # Calculate bounds of the posterior for each dimension
  bounds = apply(posterior[,], 2, range)
  
  # Thin posterior draws based on acf results
  grp_names = as.numeric(levels(group))
  d = nlevels(group)
  var_names = 1:ncol(X) - 1  # 0 = intercept, 1 to p for p covariates
  var_num = length(var_names)
  grp_index = rep(grp_names, times = var_num)
  var_index = rep(var_names, each = d)
  
  for(j in 1:ncol(posterior)){
    ACF = acf(posterior[,j], plot=F, lag.max = 30)
    ACF_df = with(ACF, data.frame(lag,acf))
    ACF_df$grp_names = grp_index[j]
    ACF_df$var_names = var_index[j]
    if(j == 1){
      ACF_all = ACF_df
    }else{
      ACF_all = rbind(ACF_all, ACF_df)
    }
  }
  
  # Want lag where acf < 0.1
  df_lag = subset(ACF_all, acf > 0.01)
  max_lag = max(max(df_lag$lag) + 1, 10) # At minimum, thin by 10
  thin = seq(from = 1, to = nrow(posterior), by = max_lag)
  post_thin = posterior[thin,]
  
  # Initiate logLik
  ll = 0
  
  for(k in 1:d){
    # Define group-specific individuals
    ids = (group == k)
    y_k = y[ids]
    n_k = sum(ids)
    
    # Identify columns for variables corresponding to group k
    cols_all = seq(from = k, by = d, length.out = num_var)
    # Restrict above colums to those belonging to variables with non-zero random effect variance
    cols = cols_all[which(diag(sigma) != 0)]
    Z_k = Z[ids,cols,drop=F]
    # if((length(cols) == 1) | (length(ids) == 1)){
    #   Z_k = matrix(Z_k, nrow = length(ids), ncol = length(cols))
    # }
    
    # Sample M samples from the importance function
    ## Importance function: multivariate normal (multivariate t?)
    if(length(cols) > 1){
      post_cov = var(post_thin[,cols])
    }else{
      post_cov = matrix(var(post_thin[,cols]), nrow = 1, ncol = 1)
    }
    
    imp_samp = rmvnorm(M, mean = post_mean[cols], sigma = post_cov)
    # imp_samp = rmvnorm(M, mean = post_mean[cols], sigma = sigma)
    # imp_samp = rmvt(M, df = 10, delta = post_mean[cols], sigma = sigma)
    
    # Determine Importance Weights - f/g
    prior = dmvnorm(imp_samp) # Mean 0, sigma I
    imp_dens = dmvnorm(imp_samp, post_mean[cols], sigma = post_cov)
    # imp_dens = dmvt(imp_samp, delta = post_mean[cols], sigma = sigma, df = 10, log = F)
    wt = prior / imp_dens
    
    # Calculate density given importance samples
    
    ## Fixed-effects component of eta (linear predictor)
    eta_fef_k = matrix(eta_fef[ids], nrow = 1) # 1 X n_k
    ## Random-effects component of eta (linear predictor)
    ## switched Gam to be transposed
    eta_ref = imp_samp %*% t(Gam) %*% t(Z_k) # M x n_k
    ## Full eta - fixed + random effects component
    eta = eta_fef_k[rep(1, M),] + eta_ref # M x n_k
    
    # Calculate indicator function
    if(length(cols) > 1){
      indic = indicator_1(bounds[,cols], imp_samp)
    }else{
      indic = indicator_2(bounds[,cols], imp_samp)
    }
    if(trace >= 1){
      cat("proportion of importance samples inside bounds:", mean(indic), "\n")
    }
    
    dens_k = rep(1, times = M)
    
    # Estimate fitted values (see "utility_glm.cpp" for invlink() function details)
    # Columns of mu correspond to individuals
    mu = matrix(0, nrow = nrow(eta), ncol = ncol(eta))
    for(i in 1:ncol(eta)){
      mu[,i] = invlink(link_int, eta[,i])
    }
    
    if(family == "binomial"){
      for(i in 1:n_k){
        dens_i = dbinom(y_k[i], size = 1, prob = mu[,i], log = FALSE)
        # Multiply together all elements belonging to same alpha_m
        dens_k = dens_k * dens_i
      } # End i loop
    }else if(family == "poisson"){
      for(i in 1:n_k){
        dens_i = dpois(y_k[i], lambda = mu[,i], log = FALSE)
        # Multiply together all elements belonging to same alpha_m
        dens_k = dens_k * dens_i
      } # End i loop
    }else if(family == "gaussian"){
      if(is.null(gaus_sig)){
        stop("gaus_sig must be specified for gaussian family")
      }
      for(i in 1:n_k){
        dens_i = dnorm(y_k[i], mean = mu[,i], sd = gaus_sig, log = FALSE)
        # Multiply together all elements belonging to same alpha_m
        dens_k = dens_k * dens_i
      } # End i loop
    }else{
      stop("specified family not currently available")
    }
    # else if(family == "negbin"){
    #   for(i in 1:n_k){
    #     dens_i = dnbinom(y_k[i], size = 1/phi, mu = mu[,i], log = FALSE)
    #     # Multiply together all elements belonging to same alpha_m
    #     dens_k = dens_k * dens_i
    #   } # End i loop
    
    lik_k = mean(dens_k*indic*wt)
    ll_k = log(lik_k)
    ll = ll + ll_k
    
  } # End k loop
  
  return(ll)
} # End CAME_IS function



# Pajor regular CAME Method for logLik calculations

# Sample from prior, no importance sampling
# CAME = function(posterior, y, X, Z, group, coef, sigma, family, M){
#   
#   # Define variables
#   d = nlevels(group)
#   num_var = ncol(Z) / d
#   Gam = t(chol(sigma[which(diag(sigma)!=0),which(diag(sigma)!=0)]))
#   eta_fef = X %*% coef[1:ncol(X)]
#   # post_mean = colMeans(posterior)
#   # post_cov = sigma
#   
#   # Calculate bounds of the posterior for each dimension
#   bounds = apply(posterior[,], 2, range)
#   
#   ll = 0
#   
#   for(k in 1:d){
#     # Define group-specific individuals
#     ids = (group == k)
#     y_k = y[ids]
#     n_k = sum(ids)
#     
#     # Identify columns corresponding to group k
#     cols_all = seq(from = k, by = d, length.out = num_var)
#     # Restrict above colums to those belonging to variables with non-zero random effect variance
#     cols = cols_all[which(diag(sigma) != 0)]
#     Z_k = Z[ids,cols]
#     
#     # Sample M samples from the prior function
#     zero = rep(0, num_var)
#     I = diag(x = 1, nrow = num_var)
#     prior = rmvnorm(M, zero, I) # Mean 0, sigma I
#     
#     # Calculate density given importance samples
#     
#     ## Fixed-effects component of eta (linear predictor)
#     eta_fef_k = matrix(eta_fef[ids], nrow = 1)
#     ## Random-effects component of eta (linear predictor)
#     eta_ref = prior %*% t(Gam) %*% t(Z_k)
#     ## Full eta - fixed + random effects component
#     eta = eta_fef_k[rep(1, M),] + eta_ref
#     
#     # Calculate indicator function
#     indic = indicator(bounds[,cols], prior)
#     cat("proportion of samples inside bounds:", mean(indic), "\n")
#     
#     dens_k = rep(1, times = M)
#     
#     if(family == "binomial"){
#       prob_mat = exp(eta) / (1+exp(eta))
#       # Columns of prob_mat correspond to individuals
#       
#       for(i in 1:n_k){
#         dens_i = dbinom(y_k[i], size = 1, prob = prob_mat[,i], log = F)
#         # Multiply together all elements belonging to same alpha_m
#         dens_k = dens_k * dens_i
#       } # End i loop
#     }else{
#       stop("specified family not currently available")
#     }
#     
#     lik_k = mean(dens_k*indic)
#     ll_k = log(lik_k)
#     ll = ll + ll_k
#     
#   } # End k loop
#   
#   return(ll)
# } # End CAME function