testcode/copyFitmarkovSame_v2.R

In lvhoskovec/psbpHMM: Covariate-dependent iHMMs for multiple time series

fitMarkovSame <- function(niter, nburn, y, ycomplete=NULL, X,
                          priors=NULL, K.start=NULL, z.true=NULL, lod=NULL,
                          mu.true=NULL, missing = FALSE, 
                          tau2 = 0.1, a.tune = 10, b.tune = 1,
                          resK = FALSE, eta.star = NULL, len.imp = 1){
  
  
  # change updates based on missing vs. complete data 
  if(missing){
    algorithm = "MH"
    SigmaPrior = "wishart"
  }else{
    algorithm = "Gibbs"
    SigmaPrior = "non-informative"
  }
  
  #####################
  ### Initial Setup ###
  #####################
  
  # how many time series and exposures 
  if(class(y)=="list"){
    p <- ncol(y[[1]]) # number of exposures 
    n <- length(y) # number of time series
    t.max <- nrow(y[[1]]) # number of time points 
  }else if(class(y)=="matrix"){
    p <- ncol(y)
    n <- 1
    t.max <- nrow(y)
    y <- list(y) # make y into a list 
    ycomplete <- list(ycomplete)
    z.true <- list(z.true)
  }else if(class(y)=="numeric"){
    p <- 1
    n <- 1
    t.max <- length(y)
    y <- list(matrix(y, ncol = 1))
    ycomplete <- list(matrix(ycomplete, ncol= 1))
    z.true <- list(z.true)
  }
  
  # let X go in as a matrix # 
  q <- ncol(X)
  
  #################
  ### Functions ###
  #################
  
  
  ## these are the functions you want to write in C++
  
  # go into K states, then sum to 1 for K+1 
  fun1 <- function(){
    first1 = sapply(1:K, FUN = function(k) pnorm(alpha.0k[k] + crossprod(X[1,],beta.k[[k]])))
    second1 = sapply(1:(K-1), FUN = function(k) 1-pnorm(alpha.0k[k] + crossprod(X[1,],beta.k[[k]])))
    prod1 = c(1, cumprod(second1))
    return(c(first1*prod1, 1 - sum(first1*prod1)))
  }
  
  # K only 
  fun2 <- function(t){
    t(sapply(1:(K), FUN = function(j){
      first = sapply(1:K, FUN = function(k) pnorm(alpha.jk[[j]][k] + crossprod(X[t,],beta.k[[k]])))
      second = sapply(1:(K-1), FUN = function(k) 1-pnorm(alpha.jk[[j]][k] + crossprod(X[t,],beta.k[[k]])))
      prod2 = c(1, cumprod(second))
      return(c(first*prod2, 1 - sum(first*prod2))) 
    }))
  }
  
  uppi <- function(t){
    if(t == 1) return(fun1())
    else return(fun2(t))
  }
  
  ##############
  ### Priors ###
  ##############
  
  if(missing(priors)) priors <- NULL
  # mu
  if(is.null(priors$mu0)) priors$mu0 <- matrix(0, p, 1) # mu_k|Sigma_k ~ N(mu0, 1/lambda*Sigma_k) prior mean on p exposures
  # alpha
  if(is.null(priors$mu.alpha)) priors$mu.alpha <- 0 # fixed mean parameter prior on intercepts alpha.jk
  if(is.null(priors$m0)) priors$m0 <- 0 # fixed mean on m.alpha, prior mean for alpha.jj # higher so self-transition prob is higher
  if(is.null(priors$v0)) priors$v0 <- 1 # fixed variance on m.alpha, prior mean for alpha.jj
  # beta
  if(is.null(priors$mu.beta)) priors$mu.beta <- rep(0, q) # basis weight prior mean 
  if(is.null(priors$Sigma.beta)) priors$Sigma.beta <- diag(q) # basis weight prior variance
  if(!is.null(X)) priors$SigInv.beta <- chol2inv(chol(priors$Sigma.beta)) # inverse of Sigma.beta, precision
  # hyperiors on alpha
  if(is.null(priors$a1)) priors$a1 <- 1 # shape for sig2inv.alpha
  if(is.null(priors$b1)) priors$b1 <- 1 # rate for sig2inv.alpha
  if(is.null(priors$a2)) priors$a2 <- 1 # shape for vinv.alpha
  if(is.null(priors$b2)) priors$b2 <- 1 # rate for vinv.alpha
  
  if(SigmaPrior == "wishart"){
    # missing data case (MH udpates)
    if(is.null(priors$R)) priors$R <- diag(p) # Sigma_k ~ Inv.Wish(nu, R) hyperparameter for Sigma_k
    if(is.null(priors$nu)) priors$nu <- p+2 # Sigma_k ~ Inv.Wish(nu, R) hyperparameter for Sigma_k, nu > p+1
    nu.df <- priors$nu # 
    R.mat <- priors$R 
  }else if(SigmaPrior == "non-informative"){
    # complete data case (Gibbs updates)
    # if SigmaPrior = "ni", the half-t prior on Sigma_k
    if(is.null(priors$bj)) priors$bj <- rep(1, p) # Huang and Wand advise 10e5
    if(is.null(priors$nu)) priors$nu <- 2 # Huang and Wand advise 2, p+4 for so the variance exists
    nu.df <- priors$nu + p - 1 # Huang and Wand, prior df 
    aj.inv <- rgamma(p, shape = 1/2, rate = 1/(priors$bj^2)) # these are 1/aj 
    # starting value for R.mat, the matrix parameter on Sigma.Inv
    R.mat <- 2*priors$nu*diag(aj.inv) 
    # we model aj.inv with gamma(shape = 1/2, rate = 1/(b_j^2))
  }
  
  if(is.null(priors$lambda)) priors$lambda <- 1 
  
  #############################
  ### Indicate Missing Type ###
  #############################
  
  # indicate missing data: obs = 0, mar = 1, lod = 2
  mismat <- list()
  for(i in 1:n){
    mismat[[i]] <- matrix(sapply(y[[i]], ismissing), ncol = p)
  }
  
  mism <- numeric()
  for(i in 1:n){
    mism <- rbind(mism, mismat[[i]])
  }
  
  # for each i, which time points have any missing data? 
  missingTimes <- lapply(1:n, FUN = function(i){
    which(apply(mismat[[i]],1,sum)>0)
  })
  
  # for each i, which times points are observed? 
  observedTimes <- lapply(1:n, FUN = function(i){
    which(apply(mismat[[i]], 1, sum)==0)
  })
  
  ###############################################
  ### Impute Starting Values for Missing Data ###
  ###############################################
  
  for(i in 1:n){
    if(any(mismat[[i]]==2)){ # lod 
      expLod <- exp(lod)
      numlod <- apply(mismat[[i]], 1, FUN = function(x) length(which(x==2))) # how many LOD
      for(t in which(numlod>0)){ # loop thru LOD data
        whichlod <- which(mismat[[i]][t,]==2) # which exposures are below LOD 
        y[[i]][t,whichlod] <- log(expLod[whichlod]/sqrt(2)) # impute the LOD with the log(LOD/sqrt(2))
      }
    }
    if(any(mismat[[i]]==1)){ # mar
      nummis <- apply(mismat[[i]], 1, FUN = function(x) length(which(x==1))) # how many missing at each time point 
      for(t in which(nummis>0)){ # only loop thru time points with MAR
        whichmis <- which(mismat[[i]][t,]==1) # which exposures are missing at random 
        if(t == 1){
          for(ws in whichmis){
            lastT = max(which(!is.na(y[[i]][,ws])))
            y[[i]][t,ws] <- y[[i]][lastT, ws] # fill in with the last observed value from the end of the time series
          }
        }else{
          y[[i]][t,whichmis] <- y[[i]][t-1, whichmis] # fill in the missing by LVCF
        }
      }
    }
  } 
  
  #####################
  ## Starting Values ##  
  #####################
  
  z <- list()
  for(i in 1:n){
    K <- K.start
    if(is.null(K)) K <- 12
    z[[i]] <- sample(1:K, t.max, replace = TRUE)
  }
  
  mu <- list()
  Sigma <- list()
  D <- list()
  L <- list()
  lams <- list()
  al <- list()
  
  ymatrix <- NULL
  for(i in 1:n){
    ymatrix <- rbind(ymatrix, y[[i]])
  }
  
  for(k in 1:K){
    if(algorithm == "MH"){
      # we reparameterize Sigma and model L and D instead 
      vj0 <- sapply(1:p, FUN = function(j) priors$nu + j - p); vj0 # fixed for each k 
      deltaj0 <- rep(1,p); deltaj0 # fixed for each k 
      lams[[k]] <- 1/rgamma(3, vj0, rate = deltaj0)
      D[[k]] <- diag(lams[[k]])
      al.list <- list()
      for(j in 2:p){
        al.list[[j-1]] <- rnorm(j-1, 0, lams[[k]][j])
      }
      al[[k]] <- unlist(al.list) # for j = 2 to p
      which.lams <- unlist(sapply(2:p, FUN = function(j) rep(j,j-1))) # which lams to use for each al 
      L[[k]] <- diag(p)
      lowerTriangle(L[[k]]) <- al[[k]]
      Sigma[[k]] <- solve(L[[k]])%*%D[[k]]%*%t(solve(L[[k]])) # Sigma[[k]]
      mu[[k]] <- rmvn(1, priors$mu0, (1/priors$lambda)*Sigma[[k]])
    }else{
      Sigma[[k]] <- chol2inv(chol(matrix(rWishart(1, df = nu.df, Sigma = solve(R.mat)),p,p)))
      mu[[k]] <- rmvn(1, priors$mu0, (1/priors$lambda)*Sigma[[k]])
    }
  }
  
  # K only 
  alpha.0k <- rep(0, K) # initial state intercept
  alpha.jk <- list() # state intercepts
  m.alpha <- priors$m0 # mean on alpha.jj
  sig2inv.alpha <- priors$a1/priors$a2 # precision on alpha.jk
  vinv.alpha <- priors$b1/priors$b2 # precision on alpha.jj
  
  # K only 
  beta.k <- list() # regression coefficients
  for(k in 1:(K)){
    alpha.jk[[k]] <- rnorm(K, priors$mu.alpha, sqrt(1/sig2inv.alpha))
    alpha.jk[[k]][k] <- priors$m0
    beta.k[[k]] <- matrix(rmvn(1, priors$mu.beta, priors$Sigma.beta), nrow = q, ncol = 1) # length q
  }
  
  # transition probability of going from state j to state k in the current trajectory at time t 
  pi.z <- mclapply(1:t.max, FUN = uppi)
  
  # fixed values for new state probabilities 
  gampp <- mgamma(nu = nu.df, p = p)
  # fixed for "wish", starting value for "ni" because R.mat will change with each iteration as aj.inv changes
  # if "ni" then need to update detR.star and log.stuff each iteration
  detR.star <- mclapply(1:n, FUN = function(i){
    sapply(1:t.max, FUN = function(t){
      x <- R.mat + priors$lambda*tcrossprod(priors$mu0) + tcrossprod(y[[i]][t,]) - 
        (1/(1+priors$lambda))*tcrossprod(priors$lambda*priors$mu0+y[[i]][t,])
      return(det(x))})
  })
  log.stuff <- (p/2)*log(priors$lambda/(pi*(priors$lambda+1)))+log(gampp)+(nu.df/2)*log(det(R.mat)); log.stuff
  
  ########################
  ### For MH algorithm ###
  ########################
  
  # calculate before loop for MH 
  ymatrix <- numeric()
  for(i in 1:n){
    ymatrix <- rbind(ymatrix, y[[i]])
  }
  ycomp <- ymatrix[which(rowSums(mism)==0),]
  
  ##############################
  ### MCMC Storage and Setup ###
  ##############################
  
  z.save <- list()
  mu.save <- list()
  Sigma.save <- list()
  K.save <- rep(NA, niter)
  ham <- rep(NA, niter)
  mu.mse <- rep(NA, niter)
  mu.sse <- rep(NA, niter)
  MH.a <- 0 # for MH update a
  MH.lam <- 0 # for MH update lams
  
  # missing data sets
  if(!is.null(len.imp)){
    imputes <- ceiling(seq.int(nburn, niter, length.out = len.imp))
    y.mar.save <- matrix(NA, len.imp, length(which(unlist(mismat)==1)))
    y.lod.save <- matrix(NA, len.imp, length(which(unlist(mismat)==2)))
    mar.mse <- rep(NA, len.imp)
    lod.mse <- rep(NA, len.imp)
    mar.sse <- rep(NA, len.imp)
    lod.sse <- rep(NA, len.imp)
    miss.mse <- rep(NA, len.imp)
    mar.bias <- rep(NA, len.imp)
    lod.bias <- rep(NA, len.imp)
    mar.sum.bias <- rep(NA, len.imp)
    lod.sum.bias <- rep(NA, len.imp)
    s.imp <- 1
  }else{
    imputes = 0
    y.mar.save <- NULL
    y.lod.save <- NULL
    mar.mse <- NULL
    lod.mse <- NULL
    mar.sse <- NULL
    lod.sse <- NULL
    miss.mse <- NULL
    mar.bias <- NULL
    lod.bias <- NULL
    mar.sum.bias <- NULL
    lod.sum.bias <- NULL
    s.imp <- NULL
  }
  
  beta.save <- list()
  
  ###############
  ### Sampler ###
  ###############
  
  for(s in 1:niter){
    
    #################
    ### debugging ### 
    #################
    
    par(mfrow = c(2,2))
    plot(1:t.max, y[[1]][,1], type = "p", pch = 19, col = z[[1]])
    abline(h = lod[1])
    plot(1:t.max, ycomplete[[1]][,1], type = "p", pch = 19, col = z.true[[1]])
    abline(h = lod[1])
    plot(1:t.max, y[[2]][,1], type = "p", pch = 19, col = z[[2]])
    abline(h = lod[1])
    plot(1:t.max, ycomplete[[2]][,1], type = "p", pch = 19, col = z.true[[2]])
    abline(h = lod[1])
    
    #####################
    ### initial stuff ### ### done 
    #####################
    
    
    print(paste("iteration", s))
    #start.time1 <- Sys.time()
    z.prev <- list()
    z.prev <- mclapply(1:n, FUN=function(i) return(z[[i]]))
    
    
    ################
    ### update u ### ### done 
    ################
    
    u <- list()
    for(i in 1:n){
      u[[i]] <- unlist(lapply(1:t.max, FUN = function(t){
        if(t==1){
          return(runif(1, 0, pi.z[[1]][z[[i]][1]]))
        }else{
          return(runif(1, 0, pi.z[[t]][z[[i]][t-1], z[[i]][t]]))
        }
      }))
    }
    
    #############################################
    ### update the possible states for each t ###  ### done
    #############################################
    
    state.list <- list()
    # state.list is a list of states that belong to some possible trajectory for each time point
    for(i in 1:n){
      # forward condition: considering the previous possible states, what are the current possible states?
      for.list <- list()
      for.list[[1]] <- which(pi.z[[1]] >= u[[i]][1])
      for(t in 2:t.max){
        
        for.list[[t]] <- sort(unique(unlist(lapply(intersect(for.list[[t-1]], 1:K),
                                                   FUN = function(k) which(pi.z[[t]][k,] >= u[[i]][t])))))
      }
      # backward condition
      back.list <- list()
      back.list[[t.max]] <- for.list[[t.max]]
      for(t in (t.max-1):1){
        back.list[[t]] <- sort(unique(unlist(lapply(intersect(back.list[[t+1]], 1:K),
                                                    FUN = function(k) which(pi.z[[t+1]][,k] >= u[[i]][t+1])))))
      }
      state.list[[i]] <- lapply(1:t.max, FUN = function(t) {
        return(intersect(for.list[[t]], back.list[[t]]))
      })
    }
    
    ###########################
    ### Determine t minus 1 ### 
    ###########################
    
    detzAll <- mclapply(1:n, FUN = function(i){
      detZminus1(i = i, state.list.i = state.list[[i]], pi.z = pi.z, u.i = u[[i]], t.max = t.max)
    })
    
    ################
    ### Sample Z ### 
    ################
    
    cholSigma <- lapply(1:K, FUN = function(k) chol(Sigma[[k]]))
    K <- length(unique(unlist(z)))
    
    # only the joint NIW option 
    z <- mclapply(1:n, FUN = function(i) {
      updateZ(i=i, y.i=y[[i]], mu=mu, cholSigma=cholSigma, detR.stari=detR.star[[i]],
              nu.df=nu.df, K=K, log.stuff=log.stuff, t.max=t.max, 
              detz = detzAll[[i]], state.list.i = state.list[[i]])
    })
    
    #########################
    ### new state fillers ### 
    #########################
    
    if(any(unlist(z)>K)){
      
      if(algorithm == "MH"){
        lamsNew <- 1/rgamma(3, vj0, rate = deltaj0); lamsNew
        DNew <- diag(lamsNew); DNew
        al.listNew <- list()
        for(j in 2:p){
          al.listNew[[j-1]] <- rnorm(j-1, 0, lamsNew[j])
        }
        alNew <- unlist(al.listNew); alNew # for j = 2 to p
        LNew <- diag(p)
        lowerTriangle(LNew) <- alNew; LNew
        SigmaNew <- solve(LNew)%*%DNew%*%t(solve(LNew)) # Sigma[[k]]
        cholSigmaNew <- chol(SigmaNew)
        muNew <- rmvn(1, priors$mu0, (1/priors$lambda)*SigmaNew)
      }else if(algorithm == "Gibbs"){
        SigmaNew <- chol2inv(chol(matrix(rWishart(1, df = nu.df, Sigma = solve(R.mat)),p,p)))
        muNew <- rmvn(1, priors$mu0, (1/priors$lambda)*SigmaNew)
      }
      
      # sample starting values if we got a new state
      mu[[K+1]] <- muNew
      if(algorithm == "MH"){
        lams[[K+1]] <- lamsNew
        D[[K+1]] <- DNew
        al[[K+1]] <- alNew
        L[[K+1]] <- LNew
      }
      Sigma[[K+1]] <- SigmaNew
      alpha.0k[K+1] <- 0
      for(k in 1:K){
        alpha.jk[[k]][K+1] <- rnorm(1, priors$mu.alpha, sqrt(1/sig2inv.alpha))
      }
      alpha.jk[[K+1]] <- rnorm(K+1, priors$mu.alpha, sqrt(1/sig2inv.alpha))
      alpha.jk[[K+1]][K+1] <- priors$m0
      beta.k[[K+1]] <- matrix(rmvn(1, priors$mu.beta, priors$Sigma.beta), nrow = q, ncol = 1) # length q
      EnterNew <- TRUE # indicator that we entered into a new state this round 
    }else{
      EnterNew <- FALSE # didn't go into a new state 
    }
    
    #########################################
    ### Relabel the States and Parameters ###
    #########################################
    
    z.new <- recode_Z(unlist(z))
    splits <- seq(1,n*t.max, t.max)
    z.new <- lapply(1:length(splits), FUN = function(i) z.new[splits[i]:(splits[i]+t.max-1)])
    K <- length(sort(unique(unlist(z.new)))) # new K 
    zu <- sort(unique(unlist(z))) # z.unique (old labels for current states to grab from) 
    
    alpha.new <- list()
    beta.new <- list()
    mu.new <- list()
    Sigma.new <- list()
    al.new <- list()
    lams.new <- list()
    L.new <- list()
    D.new <- list()
    
    rj <- 1
    for(k in zu){ # 
      alpha.new[[rj]] <- alpha.jk[[k]][zu]
      beta.new[[rj]] <- beta.k[[k]]
      mu.new[[rj]] <- mu[[k]]
      Sigma.new[[rj]] <- Sigma[[k]]
      if(algorithm == "MH"){
        al.new[[rj]] <- al[[k]]
        lams.new[[rj]] <- lams[[k]]
        D.new[[rj]] <- D[[k]]
        L.new[[rj]] <- L[[k]]
      }
      rj <- rj + 1
    }
    
    alpha.jk <- alpha.new
    beta.k <- beta.new
    z <- z.new
    Sigma <- Sigma.new
    mu <- mu.new
    if(algorithm == "MH"){
      al <- al.new
      lams <- lams.new 
      D <- D.new
      L <- L.new
    }
    
    cholSigma <- lapply(1:K, FUN = function(k) chol(Sigma[[k]]))
    
    ######################
    ### update theta_k ###
    ######################
    
    # first update mu and Sigma 
    for(k in 1:K){
      itimes <- lapply(1:n, FUN = function(i)  which(z[[i]] == k))
      nkk.tilde  <- length(unlist(itimes)) # number in state k 
      y.list <- lapply(1:n, FUN = function(i) matrix(y[[i]][itimes[[i]],], ncol = p))
      yk <- numeric()
      for(i in 1:n){
        yk <- rbind(yk, y.list[[i]])
      }
      ybark <- matrix(apply(yk, 2, mean),p,1)
      nu_nk <- nu.df + nkk.tilde 
      
      if(algorithm == "Gibbs"){
        
        mu_nk <- (priors$lambda*priors$mu0 + nkk.tilde*ybark)/(priors$lambda+nkk.tilde) 
        lambda_nk <- priors$lambda + nkk.tilde 
        if(nkk.tilde == 1){
          M <- R.mat
        }else{
          M <- R.mat + (nkk.tilde-1)*cov(yk) 
        }
        Sigma_nk <- M + (priors$lambda*nkk.tilde)/(nkk.tilde + priors$lambda)*tcrossprod(ybark - priors$mu0) 
        Sigma[[k]] <- chol2inv(chol(matrix(rWishart(1,df=nu_nk, Sigma=chol2inv(chol(Sigma_nk))),p,p)))
        mu[[k]] <- rmvn(n=1, mu=mu_nk, sigma=chol((1/lambda_nk)*as.matrix(Sigma[[k]], p, p)), isChol = TRUE)  
        
      }else if(algorithm == "MH"){
        # update a
        for(j in 1:length(al[[k]])){
          
          if(resK){
            eta <- rgeom(1, (1/eta.star)) + 1
          }else eta <- 1
          
          if(eta>0){
            for(m in 1:eta){
              al.star <- al[[k]]
              L.star <- L[[k]]
              
              #a.star <- rnorm(1, al[[k]][j], sqrt(tau2)); a.star # proposed value 
              a.star <- rnorm(1, 0, sqrt(tau2)); a.star # proposed value 
              
              al.star[j] <- a.star; al.star
              lowerTriangle(L.star) <- al.star; L.star
              SigmaStar <- solve(L.star)%*%D[[k]]%*%t(solve(L.star)); SigmaStar # function of a.star
              
              # likelihoods
              da.curr <- sum(dmvn(yk, mu = mu[[k]], sigma = cholSigma[[k]], log = TRUE, isChol = TRUE)) +
                dmvn(mu[[k]], mu = priors$mu0, sigma = chol((1/priors$lambda)*Sigma[[k]]), log = TRUE, isChol = TRUE)
              da.star <- sum(dmvn(yk, mu = mu[[k]], sigma = chol(SigmaStar), log = TRUE, isChol = TRUE)) +
                dmvn(mu[[k]], mu = priors$mu0, sigma = chol((1/priors$lambda)*SigmaStar), log = TRUE, isChol = TRUE)
              
              # priors 
              pa.curr <- dnorm(al[[k]][j], 0, sqrt(lams[[k]][which.lams[j]]), log = TRUE)
              pa.star <- dnorm(a.star, 0, sqrt(lams[[k]][which.lams[j]]), log = TRUE)
              
              # proposals
              qa.curr <- dnorm(al[[k]][j], 0, sqrt(tau2), log = TRUE)
              qa.star <- dnorm(a.star, 0, sqrt(tau2), log = TRUE)
              
              mh1 <- pa.star + da.star + qa.curr; mh1
              mh2 <- pa.curr + da.curr + qa.star
              # catch error on da.curr
              
              ar <- mh1-mh2
              
              if(runif(1) < exp(ar)){
                al[[k]][j] <- a.star
                L[[k]] <- L.star # update this too, fxn of a.star
                Sigma[[k]] <- SigmaStar # update this too, fxn of a.star
                MH.a <- MH.a + 1 
              }
            }
          }
        }
        # update lams
        for(j in 1:p){
          D.star <- D[[k]]
          lam.star <- 1/rgamma(1, a.tune, rate = b.tune); lam.star # proposed value 
          D.star[j,j] <- lam.star
          SigmaStar <- solve(L[[k]])%*%D.star%*%t(solve(L[[k]]))
          
          # likelihoods
          dlam.curr <- sum(dmvn(yk, mu = mu[[k]], sigma = cholSigma[[k]], log = TRUE, isChol = TRUE)) + 
            dmvn(mu[[k]], mu = priors$mu0, sigma = chol((1/priors$lambda)*Sigma[[k]]), log = TRUE, isChol = TRUE)
          dlam.star <- sum(dmvn(yk, mu = mu[[k]], sigma = chol(SigmaStar), log = TRUE, isChol = TRUE)) +
            dmvn(mu[[k]], mu = priors$mu0, sigma = chol((1/priors$lambda)*SigmaStar), log = TRUE, isChol = TRUE)
          
          # priors
          plam.curr <- dinvgamma(lams[[k]][j], vj0[j]/2, rate = deltaj0[j]/2, log = TRUE)
          plam.star <- dinvgamma(lam.star, vj0[j]/2, rate = deltaj0[j]/2, log = TRUE)
          
          # proposal 
          qlam.curr <- dinvgamma(lams[[k]][j], a.tune, b.tune, log = TRUE)
          qlam.star <- dinvgamma(lam.star, a.tune, b.tune, log = TRUE)
          
          mh1 <- plam.star + dlam.star + qlam.curr; mh1
          mh2 <- plam.curr + dlam.curr + qlam.star; mh2
          ar <- mh1-mh2
          
          if(runif(1) < exp(ar)){
            lams[[k]][j] <- lam.star
            D[[k]] <- D.star # fxn of lam.star
            Sigma[[k]] <- SigmaStar # update this too, fxn of lam.star
            MH.lam <- MH.lam + 1 
          }
        }
        
        # update mu by Gibbs
        mu_nk <- (priors$lambda*priors$mu0 + nkk.tilde*ybark)/(priors$lambda+nkk.tilde) 
        lambda_nk <- priors$lambda + nkk.tilde 
        mu[[k]] <- rmvn(n=1, mu=mu_nk, sigma=chol((1/lambda_nk)*as.matrix(Sigma[[k]], p, p)), isChol = TRUE)  
      } # end if MH 
    } # end sample theta  
    
    ######################################################################
    ### if SigmaPrior == "NI": Update aj.inv, detR.star and log.stuff  ###
    ######################################################################
    
    if(SigmaPrior == "non-informative"){
      # then update aj.inv for j = 1 to p 
      shape.aj <- (K*(priors$nu+p-1)+1)/2
      sigmajj <- lapply(1:K, FUN = function(k){
        diag(chol2inv(chol(Sigma[[k]])))
      }) # diagonal elements of each Sigma.Inv_k
      diags <- numeric()
      for(k in 1:K){
        diags <- rbind(diags, sigmajj[[k]])
      }
      sumdiags <- apply(diags, 2, sum) # sum of diagonals of Sigma.Inv for k = 1 to K 
      rate.aj <- 1/(priors$bj^2) + priors$nu*sumdiags
      aj.inv <- rgamma(p, shape = shape.aj, rate = rate.aj) # these are 1/aj 
      R.mat <- 2*priors$nu*diag(aj.inv)
      
      detR.star <- mclapply(1:n, FUN = function(i){
        sapply(1:t.max, FUN = function(t){
          x <- R.mat + priors$lambda*tcrossprod(priors$mu0) + tcrossprod(y[[i]][t,]) - 
            (1/(1+priors$lambda))*tcrossprod(priors$lambda*priors$mu0+y[[i]][t,])
          return(det(x))
        })})
      
      log.stuff <- (p/2)*log(priors$lambda/(pi*(priors$lambda+1)))+log(gampp)+(nu.df/2)*log(det(R.mat))
    }
    
    ################
    ### update W ### 
    ################
    
    # for each t, w.z gives me a VECTOR based on the previous time point and values up to the current time point
    # so each w.z[[t]] should be a VECTOR of length z_t
    
    w.z <- list()
    for(i in 1:n){
      w.z[[i]] <- list()
      for(t in 1:t.max){
        w.z[[i]][[t]] <- sapply(1:z[[i]][t], FUN = function(l) upWsame(t=t, l=l, i=i, alpha.0k=alpha.0k, X=X, 
                                                                       beta.k=beta.k, alpha.jk=alpha.jk, z=z))
      }
    }
    
    #######################
    ### update alpha.0k ### ### done 
    #######################
    
    # number of i's s.t z_i1 >= k for each k 
    itime0 <- sapply(1:K, FUN = function(k) sum(sapply(1:n, FUN = function(i) z[[i]][1] >= k)))
    
    # sum over the i's s.t. z_i1 >= k for each k
    # which.i for each k
    which.i <- sapply(1:K, FUN = function(k) {
      if(any(lapply(1:n, FUN = function(i) z[[i]][1] >= k) == TRUE)){
        return(which(lapply(1:n, FUN = function(i) z[[i]][1] >= k) == TRUE))
      }else{
        return(0)
      }
    })
    # for each k, tells me which i's to sum over, if any
    # sum over w-beta for the i's s.t z_i1 >= k for each k 
    wminusbeta <- sapply(1:K, FUN = function(k) {
      if(any(which.i[[k]] != 0)){
        return(sum(sapply(which.i[[k]], FUN = function(i) wMinusb(i=i, t=1, k=k, w.z=w.z, beta.k=beta.k, X=X))))
      }else{
        return(0)
      }
    })
    
    
    # update alpha.0k
    v0k <- 1/(sig2inv.alpha + itime0)
    m0k <- v0k*(priors$mu.alpha*sig2inv.alpha + wminusbeta)
    alpha.0k <- rnorm(K, m0k, sqrt(v0k))
    
    #######################
    ### update alpha.jk ### 
    #######################
    
    alpha.jk <- lapply(1:(K), FUN = function(j){
      unlist(lapply(1:(K), FUN = function(k){
        updateAlphaJK(j=j, k=k, n=n, t.max=t.max, z=z, vinv.alpha=vinv.alpha,
                      sig2inv.alpha = sig2inv.alpha, w.z = w.z, X = X, beta.k = beta.k,
                      m.alpha = m.alpha, mu.alpha = priors$mu.alpha)
      }))
    })
    
    
    if(anyNA(unlist(alpha.jk))) {
      stop("NA's in alpha.jk")
    }
    
    ######################
    ### update m.alpha ###
    ######################
    
    alpha.jj <- list()
    for(k in 1:(K)){
      alpha.jj[[k]] <- alpha.jk[[k]][k]
    }
    sumjj <- sum(unlist(alpha.jj))
    v.star <- 1/(K*vinv.alpha + 1/priors$v0) 
    m.star <- v.star*(sumjj*vinv.alpha + priors$m0/priors$v0)
    m.alpha <- rnorm(1, m.star, sqrt(v.star))
    
    ############################
    ### update sig2inv.alpha ###
    ############################
    
    alpha.jnotk <- list()
    for(k in 1:K){
      alpha.jnotk[[k]] <- alpha.jk[[k]][-k]
    }
    sumAlphajk <- sum((unlist(alpha.jnotk) - priors$mu.alpha)^2)
    sig2inv.alpha <- rgamma(1, priors$a1 + K*(K-1)/2, priors$b1 + .5*sumAlphajk) 
    
    #########################
    ### update vinv.alpha ###
    #########################
    
    sumjj2 <- sum((unlist(alpha.jj) - m.alpha)^2)
    vinv.alpha <- rgamma(1, priors$a2 + K/2, priors$b2 + sumjj2/2) 
    
    #####################
    ### update beta.k ### 
    #####################
    
    beta.k <- mclapply(1:K, FUN = function(k){
      itimes <- lapply(1:n, FUN = function(i)  which(z[[i]] >= k))
      nk.tilde  <- length(unlist(itimes)) # total number we're looking at the dimension of everything
      if(nk.tilde > 0){
        w.k <- unlist(lapply(1:n, FUN = function(i) {
          if(any(itimes[[i]]>0)){
            sapply(itimes[[i]], FUN = function(t) w.z[[i]][[t]][k])
          }
        }))
        X.k <- numeric()
        for(i in 1:n){
          if(any(itimes[[i]]>0)){
            X.k <- rbind(X.k, X[itimes[[i]],])
          }
        }
        alpha.k <- list()
        for(i in 1:n){
          alphaTHIS <- numeric()
          if(any(itimes[[i]]>0)){
            for(j in z[[i]][itimes[[i]]-1]){
              alphaTHIS <- c(alphaTHIS, alpha.jk[[j]][k])
            }
            if(1 %in% itimes[[i]]){
              alphaTHIS <- c(alpha.0k[k], alphaTHIS)
            }
          }
          alpha.k[[i]] <- alphaTHIS
        }
        alpha.k <- unlist(alpha.k)
        if(length(alpha.k) != nk.tilde | length(w.k) != nk.tilde | nrow(X.k) != nk.tilde){
          stop("dimension of alpha, w, or X is wrong")
        }
        # update beta.k
        V.k <- chol2inv(chol(priors$SigInv.beta + crossprod(X.k, X.k)))
        m.k <- crossprod(V.k,crossprod(priors$SigInv.beta,priors$mu.beta) + crossprod(X.k,w.k - alpha.k))
        return(matrix(rmvn(n = 1, m.k, V.k), nrow = q))
      }else{ 
        # update from prior
        return(matrix(rmvn(n = 1, mu = priors$mu.beta, sigma = priors$Sigma.beta), nrow = q)) 
      }
    })
    if(anyNA(unlist(beta.k))) {
      stop("NA's in beta.k")
    }
    
    
    ###################
    ### update pi.z ### ### done 
    ###################
    
    pi.z <- mclapply(1:t.max, FUN = uppi)
    
    #################################
    ### Sample New Missing Values ###
    #################################
    
    # Sample new MAR values conditional on observed data and imputed LOD data ###
    for(i in 1:n){
      if(any(mismat[[i]]==1)){ # MAR = 1 
        nummis <- apply(mismat[[i]], 1, FUN = function(x) length(which(x==1))) # how many missing at each time point 
        for(t in which(nummis>0)){ # only loop through time points with missing data
          whichmis <- which(mismat[[i]][t,]==1) # which ones are missing 
          if(length(whichmis)==p){
            y[[i]][t,] <- rmvn(1, mu[[z[[i]][t]]], chol(Sigma[[z[[i]][t]]]), isChol = TRUE)
          }else{
            y.obs <- y[[i]][t,-whichmis]
            mu.obs <- mu[[z[[i]][t]]][,-whichmis]
            mu.miss <- mu[[z[[i]][t]]][,whichmis]
            Sigma.obs <- matrix(Sigma[[z[[i]][t]]][-whichmis, -whichmis], p-length(whichmis), p-length(whichmis))
            Sigma.miss <- matrix(Sigma[[z[[i]][t]]][whichmis, whichmis], length(whichmis), length(whichmis))
            Sigma.obs.miss <-  matrix(Sigma[[z[[i]][t]]][-whichmis, whichmis], p-length(whichmis), length(whichmis))
            Sigma.miss.obs <- t(Sigma.obs.miss)
            Sigma.mis.obs.inv <- Sigma.miss.obs%*%solve(Sigma.obs)
            mu.mgo <- as.numeric(mu.miss + Sigma.mis.obs.inv%*%(y.obs - mu.obs))
            Sigma.mgo <- Sigma.miss + Sigma.mis.obs.inv%*%Sigma.obs.miss
            y[[i]][t,whichmis] <- rmvn(1, mu.mgo, chol(Sigma.mgo), isChol = TRUE)
          }
        }
      }
    }
    
    # Sample new LOD values conditional on observed data and imputed MAR data ###
    for(i in 1:n){
      if(any(mismat[[i]]==2)){ # LOD = 2
        numlod <- apply(mismat[[i]], 1, FUN = function(x) length(which(x==2))) # how many lod at each time point 
        for(t in which(numlod>0)){ # only loop through time points with missing data
          whichlod <- which(mismat[[i]][t,]==2) # which ones are below lod  
          if(length(whichlod)==p){ 
            y[[i]][t,] <- rtmvn(1, Mean = as.vector(mu[[z[[i]][t]]]), Sigma = Sigma[[z[[i]][t]]], lower = rep(-Inf, p),
                                upper = lod, int = y[[i]][t,], burn = 10, thin = 1)
          }else{
            y.obs <- y[[i]][t,-whichlod]
            mu.obs <- mu[[z[[i]][t]]][,-whichlod]
            mu.miss <- mu[[z[[i]][t]]][,whichlod]
            Sigma.obs <- matrix(Sigma[[z[[i]][t]]][-whichlod, -whichlod], p-length(whichlod), p-length(whichlod))
            Sigma.miss <- matrix(Sigma[[z[[i]][t]]][whichlod, whichlod], length(whichlod), length(whichlod))
            Sigma.obs.miss <-  matrix(Sigma[[z[[i]][t]]][-whichlod, whichlod], p-length(whichlod), length(whichlod))
            Sigma.miss.obs <- t(Sigma.obs.miss)
            Sigma.mis.obs.inv <- Sigma.miss.obs%*%chol2inv(chol(Sigma.obs))
            mu.mgo <- as.numeric(mu.miss + Sigma.mis.obs.inv%*%(y.obs - mu.obs))
            Sigma.mgo <- Sigma.miss + Sigma.mis.obs.inv%*%Sigma.obs.miss
            y[[i]][t,whichlod] <- rtmvn(1, Mean = mu.mgo, Sigma = Sigma.mgo, lower = rep(-Inf, length(whichlod)),
                                        upper = lod[whichlod], int = y[[i]][t, whichlod], burn = 10, thin = 1)
            
          }
        }
      }
    }
    
    #####################
    ### Error Measure ###
    #####################
    
    ## Hamming distance ##
    if(!is.null(unlist(z.true))){
      ham.error <- hamdist(unlist(z.true), unlist(z)) 
      ham[s] <- ham.error/(n*t.max) # proportion of misplaced states
    }else{
      ham <- NULL
    }    
    
    ## MSE for mu ##
    if(!is.null(mu.true)){
      sse <- list()
      for(i in 1:n){
        sse[[i]] <- sapply(1:t.max, FUN = function(t){
          as.numeric(crossprod(unlist(mu[z[[i]][t]]) - mu.true[z.true[[i]][t],]))/p
        })
      }
      mu.sse[s] <- sum(unlist(sse))
      mu.mse[s] <- mean(unlist(sse)) # vector mse for mu, divide by # exposures 
    }else{
      mu.sse <- NULL
      mu.mse <- NULL
    }
    
    
    #####################
    ### Store Results ###
    #####################
    
    z.save[[s]] <- z
    K.save[s] <- K 
    beta.save[[s]] <- beta.k
    mu.save[[s]] <- mu
    
    if(s%in%imputes){
      # imputed values for complete data sets 
      y.mar.save[s.imp,] <- unlist(y)[which(unlist(mismat)==1)] # mar imputations
      y.lod.save[s.imp,] <- unlist(y)[which(unlist(mismat)==2)] # lod imputations
      
      if(!is.null(ycomplete)){
        # MSE
        mar.mse[s.imp] <- mean((unlist(ycomplete)[which(unlist(mismat)==1)] - unlist(y)[which(unlist(mismat)==1)])^2)
        lod.mse[s.imp] <- mean((unlist(ycomplete)[which(unlist(mismat)==2)] - unlist(y)[which(unlist(mismat)==2)])^2)
        
        # SSE
        mar.sse[s.imp] <- sum((unlist(ycomplete)[which(unlist(mismat)==1)] - unlist(y)[which(unlist(mismat)==1)])^2)
        lod.sse[s.imp] <- sum((unlist(ycomplete)[which(unlist(mismat)==2)] - unlist(y)[which(unlist(mismat)==2)])^2)
        
        # mean bias
        mar.bias[s.imp] <- mean((unlist(y)[which(unlist(mismat)==1)] - unlist(ycomplete)[which(unlist(mismat)==1)]))
        lod.bias[s.imp] <- mean((unlist(y)[which(unlist(mismat)==2)] - unlist(ycomplete)[which(unlist(mismat)==2)]))
        
        # sum bias
        mar.sum.bias[s.imp] <- sum((unlist(y)[which(unlist(mismat)==1)] - unlist(ycomplete)[which(unlist(mismat)==1)]))
        lod.sum.bias[s.imp] <- sum((unlist(y)[which(unlist(mismat)==2)] - unlist(ycomplete)[which(unlist(mismat)==2)]))
      }
      
      s.imp <- s.imp+1
    }
    
    #end.time1 <- Sys.time()
    #print(s)
    #if(s %% 10 == 0) print(K)     
    #if(s %% 10 == 0) print(min(unlist(y)))
    #print(mhacc)
    #print(end.time1 - start.time1)
    
    
  }
  
  
  list1 <- list(z.save = z.save[-(1:nburn)], K.save = K.save[-(1:nburn)],
                ymar = y.mar.save, ylod = y.lod.save,
                beta.save = beta.save[-(1:nburn)],
                mu.save = mu.save[-(1:nburn)],
                hamming = ham[-(1:nburn)], mu.mse = mu.mse[-(1:nburn)], 
                mu.sse = mu.sse[-(1:nburn)],
                mar.mse = mar.mse, lod.mse = lod.mse,
                mar.sse = mar.sse, lod.sse = lod.sse,
                mar.bias = mar.bias, lod.bias = lod.bias,
                mar.sum.bias = mar.sum.bias, lod.sum.bias = lod.sum.bias,
                mismat = mismat, ycomplete = ycomplete,
                MH.arate = MH.a/(length(al)*sum(K.save)),
                MH.lamrate = MH.lam/(p*sum(K.save)))
  
  class(list1) <- "ihmm"
  return(list1)
  
  
}