R/post_correction.R

In bssm: Bayesian Inference of Non-Linear and Non-Gaussian State Space Models

#' Get MAP estimate of theta
#' @param x Object of class \code{mcmc_output} or any other list style object 
#' which has matrix theta (where each row corresponds to one iteration) and 
#' vector \code{posterior},
#' @return A vector containing theta corresponding to maximum log-posterior 
#' value of the posterior sample.
#' @noRd
get_map <- function(x) {
  x$theta[which.max(x$posterior), ]
}

#' Suggest Number of Particles for \eqn{\psi}-APF Post-correction
#'
#' Function \code{estimate_N} estimates suitable number particles needed for 
#' accurate post-correction of approximate MCMC.
#' 
#' Function \code{suggest_N} estimates the standard deviation of the 
#' logarithm of the post-correction weights at approximate MAP of theta, 
#' using various particle sizes and suggest smallest number of particles 
#' which still leads standard deviation less than 1. Similar approach was 
#' suggested in the context of pseudo-marginal MCMC by Doucet et al. (2015),
#'  but see also Section 10.3 in Vihola et al (2020).
#' 
#' @param model Model of class \code{nongaussian} or \code{ssm_nlg}.
#' @param theta A vector of theta corresponding to the model, at which point 
#' the standard deviation of the log-likelihood is computed. Typically MAP 
#' estimate from the (approximate) MCMC run. Can also be an output from 
#' \code{run_mcmc} which is then used to compute the MAP 
#' estimate of theta.
#' @param candidates A vector of positive integers containing the candidate 
#' number of particles to test. Default is \code{seq(10, 100, by = 10)}. 
#' @param replications Positive integer, how many replications should be used 
#' for computing the standard deviations? Default is 100.
#' @param seed Seed for the C++ RNG  (positive integer).
#' @return List with suggested number of particles \code{N} and matrix 
#' containing estimated standard deviations of the log-weights and 
#' corresponding number of particles.
#' @references 
#' Doucet, A, Pitt, MK, Deligiannidis, G, Kohn, R (2015). 
#' Efficient implementation of Markov chain Monte Carlo when using an 
#' unbiased likelihood estimator, Biometrika, 102(2) p. 295-313,
#' https://doi.org/10.1093/biomet/asu075
#' 
#' Vihola, M, Helske, J, Franks, J (2020). Importance sampling type estimators 
#' based on approximate marginal Markov chain Monte Carlo. 
#' Scand J Statist. 1-38. https://doi.org/10.1111/sjos.12492
#' @export
#' @examples 
#' 
#' set.seed(1)
#' n <- 300
#' x1 <- sin((2 * pi / 12) * 1:n)
#' x2 <- cos((2 * pi / 12) * 1:n)
#' alpha <- numeric(n)
#' alpha[1] <- 0
#' rho <- 0.7
#' sigma <- 1.2
#' mu <- 1
#' for(i in 2:n) {
#'   alpha[i] <- rnorm(1, mu * (1 - rho) + rho * alpha[i-1], sigma)
#' }
#' u <- rpois(n, 50)
#' y <- rbinom(n, size = u, plogis(0.5 * x1 + x2 + alpha))
#' 
#' ts.plot(y / u)
#' 
#' model <- ar1_ng(y, distribution = "binomial", 
#'   rho = uniform(0.5, -1, 1), sigma = gamma_prior(1, 2, 0.001),
#'   mu = normal(0, 0, 10),
#'   xreg = cbind(x1,x2), beta = normal(c(0, 0), 0, 5),
#'   u = u)
#' 
#' # theta from earlier approximate MCMC run
#' # out_approx <- run_mcmc(model, mcmc_type = "approx", 
#' #   iter = 5000) 
#' # theta <- out_approx$theta[which.max(out_approx$posterior), ]
#'
#' theta <- c(rho = 0.64, sigma = 1.16, mu = 1.1, x1 = 0.56, x2 = 1.28)
#'
#' estN <- suggest_N(model, theta, candidates = seq(10, 50, by = 10),
#'   replications = 50, seed = 1)
#' plot(x = estN$results$N, y = estN$results$sd, type = "b")
#' estN$N
#' 
suggest_N <- function(model, theta, 
  candidates = seq(10, 100, by = 10), replications = 100, 
  seed = sample(.Machine$integer.max, size = 1)) {
  
  check_missingness(model)
  
  replications <- check_intmax(replications, "replications", max = 1000)
  seed <- check_intmax(seed, "seed", FALSE, max = .Machine$integer.max)
  
  if (!test_integerish(candidates, lower = 1, any.missing = FALSE, 
    min.len = 1)) {
    stop("Argument 'candidates' should be vector of positive integers. ")
  } 
  if (max(candidates) > 1e5) 
    stop(paste("I don't believe you want to use over 1e5 particles",
      "If you really do, please file an issue at Github.", sep = " "))
  
  if (missing(theta) | (!is.vector(theta) & !inherits(theta, "mcmc_output"))) {
    stop("'theta' should be either a vector or object of class 'mcmc_output'.")
  }
  if (inherits(theta, "mcmc_output")) {
    theta <- get_map(theta)
  } else {
    if (length(theta) != length(model$theta)) {
      stop("Length of 'theta' does not match length of 'model$theta'.")
    }
  }
  
  if (inherits(model, "nongaussian")) {
    model$distribution <- pmatch(model$distribution,
      c("svm", "poisson", "binomial", "negative binomial", "gamma", 
        "gaussian"), duplicates.ok = TRUE) - 1
    
    out <- suggest_n_nongaussian(model, theta, candidates, replications, seed, 
      model_type(model))
  } else {
    if (inherits(model, "ssm_nlg")) {
      out <- suggest_n_nonlinear(t(model$y), model$Z, model$H, model$T, 
        model$R, model$Z_gn, model$T_gn, model$a1, model$P1, 
        model$theta, model$log_prior_pdf, model$known_params, 
        model$known_tv_params, model$n_states, model$n_etas, 
        as.integer(model$time_varying), 
        theta, candidates, replications, seed)
    } else 
      stop(paste("Function 'suggest_N' is only available for models of",
        "class 'nongaussian' and 'nlg_ssm'.", sep = " "))
  }
  list(N = candidates[which(out < 1)[1]], results = data.frame(N = candidates, 
    sd = out))
}
#' Run Post-correction for Approximate MCMC using \eqn{\psi}-APF
#'
#' Function \code{post_correct} updates previously obtained approximate MCMC 
#' output with post-correction weights leading to asymptotically exact 
#' weighted posterior, and returns updated MCMC output where components
#'  \code{weights}, \code{posterior}, \code{alpha}, \code{alphahat}, and
#'  \code{Vt} are updated (depending on the original output type).
#' 
#' @param model Model of class \code{nongaussian} or \code{ssm_nlg}.
#' @param mcmc_output An output from \code{run_mcmc} used to compute the MAP 
#' estimate of theta. 
#' While the intended use assumes this is from approximate MCMC, it is not 
#' actually checked, i.e., it is also possible to input previous 
#' (asymptotically) exact output.
#' @param particles Number of particles for \eqn{\psi}-APF (positive integer). 
#' Suitable values depend on the model and the data, but often relatively 
#' small value less than say 50 is enough. See also \code{suggest_N}
#' @param threads Number of parallel threads (positive integer, default is 1).
#' @param is_type Type of IS-correction. Possible choices are 
#'\code{"is3"} for simple importance sampling (weight is computed for each 
#'MCMC iteration independently),
#' \code{"is2"} for jump chain importance sampling type weighting (default), or
#' \code{"is1"} for importance sampling type weighting where the number of 
#' particles used forweight computations is proportional to the length of the 
#' jump chain block.
#' @param seed Seed for the C++ RNG (positive integer).
#' @return The original object of class \code{mcmc_output} with updated 
#' weights, log-posterior values and state samples or summaries (depending on 
#' the \code{mcmc_output$mcmc_type}).
#' 
#' @references 
#' Doucet A, Pitt M K, Deligiannidis G, Kohn R (2018). 
#' Efficient implementation of Markov chain Monte Carlo when using an unbiased 
#' likelihood estimator. Biometrika, 102, 2, 295-313, 
#' https://doi.org/10.1093/biomet/asu075
#' 
#' Vihola M, Helske J, Franks J (2020). Importance sampling type estimators 
#' based on approximate marginal Markov chain Monte Carlo. 
#' Scand J Statist. 1-38. https://doi.org/10.1111/sjos.12492
#' @export
#' @examples 
#' \donttest{
#' set.seed(1)
#' n <- 300
#' x1 <- sin((2 * pi / 12) * 1:n)
#' x2 <- cos((2 * pi / 12) * 1:n)
#' alpha <- numeric(n)
#' alpha[1] <- 0
#' rho <- 0.7
#' sigma <- 2
#' mu <- 1
#' for(i in 2:n) {
#'   alpha[i] <- rnorm(1, mu * (1 - rho) + rho * alpha[i-1], sigma)
#' }
#' u <- rpois(n, 50)
#' y <- rbinom(n, size = u, plogis(0.5 * x1 + x2 + alpha))
#' 
#' ts.plot(y / u)
#' 
#' model <- ar1_ng(y, distribution = "binomial", 
#'   rho = uniform(0.5, -1, 1), sigma = gamma_prior(1, 2, 0.001),
#'   mu = normal(0, 0, 10),
#'   xreg = cbind(x1,x2), beta = normal(c(0, 0), 0, 5),
#'   u = u)
#' 
#' out_approx <- run_mcmc(model, mcmc_type = "approx", 
#'   local_approx = FALSE, iter = 50000)
#' 
#' out_is2 <- post_correct(model, out_approx, particles = 30,
#'   threads = 2)
#' out_is2$time
#' 
#' summary(out_approx, return_se = TRUE)
#' summary(out_is2, return_se = TRUE)
#' 
#' # latent state
#' library("dplyr")
#' library("ggplot2")
#' state_approx <- as.data.frame(out_approx, variable = "states") %>% 
#'   group_by(time) %>%
#'   summarise(mean = mean(value))
#'   
#' state_exact <- as.data.frame(out_is2, variable = "states") %>% 
#'   group_by(time) %>%
#'   summarise(mean = weighted.mean(value, weight))
#' 
#' dplyr::bind_rows(approx = state_approx, 
#'   exact = state_exact, .id = "method") %>%
#'   filter(time > 200) %>%
#' ggplot(aes(time, mean, colour = method)) + 
#'   geom_line() + 
#'   theme_bw()
#'
#' # posterior means
#' p_approx <- predict(out_approx, model, type = "mean", 
#'   nsim = 1000, future = FALSE) %>% 
#'   group_by(time) %>%
#'   summarise(mean = mean(value))
#' p_exact <- predict(out_is2, model, type = "mean", 
#'   nsim = 1000, future = FALSE) %>% 
#'   group_by(time) %>%
#'   summarise(mean = mean(value))
#' 
#' dplyr::bind_rows(approx = p_approx, 
#'   exact = p_exact, .id = "method") %>%
#'   filter(time > 200) %>%
#' ggplot(aes(time, mean, colour = method)) + 
#'   geom_line() + 
#'   theme_bw() 
#' }
post_correct <- function(model, mcmc_output, particles, threads = 1L, 
  is_type = "is2", seed = sample(.Machine$integer.max, size = 1)) {
  
  check_missingness(model)
  
  particles <- check_intmax(particles, "particles")
  threads <- check_intmax(threads, "threads")
  
  seed <- check_intmax(seed, "seed", FALSE, max = .Machine$integer.max)
  
  if (!inherits(mcmc_output, "mcmc_output")) 
    stop("Object 'mcmc_output' is not valid output from 'run_mcmc'.")
  is_type <- pmatch(match.arg(tolower(is_type), paste0("is", 1:3)), 
    paste0("is", 1:3))
  
  a <- proc.time()
  if (inherits(model, "nongaussian")) {
    model$distribution <- pmatch(model$distribution,
      c("svm", "poisson", "binomial", "negative binomial", "gamma", 
        "gaussian"), duplicates.ok = TRUE) - 1
    
    out <- postcorrection_nongaussian(model, model_type(model), 
      mcmc_output$output_type, 
      particles,
      seed, threads, is_type, mcmc_output$counts, t(mcmc_output$theta), 
      mcmc_output$modes)
  } else {
    if (inherits(model, "ssm_nlg")) {
      out <- postcorrection_nonlinear(t(model$y), model$Z, model$H, model$T, 
        model$R, model$Z_gn, model$T_gn, model$a1, model$P1, 
        model$theta, model$log_prior_pdf, model$known_params, 
        model$known_tv_params, model$n_states, model$n_etas, 
        as.integer(model$time_varying),
        mcmc_output$output_type, 
        particles,
        seed, threads, is_type,
        mcmc_output$counts, t(mcmc_output$theta), mcmc_output$modes)
    } else 
      stop(paste("Function 'post_correct' is only available for models of", 
        "class 'nongaussian' and 'ssm_nlg'.", sep = " "))
  }
  mcmc_output$weights <- out$weights
  mcmc_output$posterior <- mcmc_output$posterior + out$posterior
  if (mcmc_output$output_type == 1) {
    mcmc_output$alpha <- out$alpha
    colnames(mcmc_output$alpha) <- names(model$a1)
  } else {
    if (mcmc_output$output_type == 2) {
      mcmc_output$alphahat <- out$alphahat
      mcmc_output$Vt <- out$Vt
      colnames(mcmc_output$alphahat) <- colnames(mcmc_output$Vt) <- 
        rownames(mcmc_output$Vt) <- names(model$a1)
      mcmc_output$alphahat <- ts(mcmc_output$alphahat, start = start(model$y),
        frequency = frequency(model$y))
    }
  }
  mcmc_output$time <- 
    rbind("approx" = mcmc_output$time, 
      "postcorrection" = proc.time() - a)[, 1:3]
  mcmc_output$mcmc_type <- paste0("is", is_type)
  mcmc_output$seed <- c(mcmc_output$seed, seed)
  mcmc_output$call <- c(mcmc_output$call, match.call()) 
  mcmc_output
}