bayesSSM

\[\begin{align*} X_0 &\sim N(0,1) \\ X_t&=\sin(X_{t-1})+\sigma_x V_t, \quad V_t \sim N(0,1), \quad t\geq 1 \\ Y_t&=X_t+\sigma_y W_t, \quad W_t \sim N(0, 1), \quad t\geq 1 \end{align*}\] that is \(X_t\) is a latent state and \(Y_t\) is an observed value. The parameters of the model are \(\sigma_x\) and \(\sigma_y\).

Note: While this example uses pmmh, the model is simple enough that standard MCMC methods could also be applied. For a more complicated example where standard MCMC methods cannot be used, see the article Stochastic SIR Model article here.

set.seed(1405)
t_val <- 20
sigma_x <- 1
sigma_y <- 0.5

init_state <- rnorm(1, mean = 0, sd = 1)
x <- numeric(t_val)
y <- numeric(t_val)
x[1] <- sin(init_state) + rnorm(1, mean = 0, sd = sigma_x)
y[1] <- x[1] + rnorm(1, mean = 0, sd = sigma_y)
for (t in 2:t_val) {
  x[t] <- sin(x[t - 1]) + rnorm(1, mean = 0, sd = sigma_x)
  y[t] <- x[t] + rnorm(1, mean = 0, sd = sigma_y)
}
x <- c(init_state, x)

ggplot() +
  geom_line(aes(x = 0:t_val, y = x), color = "blue", linewidth = 1) + # Latent
  geom_point(aes(x = 1:t_val, y = y), color = "red", size = 2) + # Observed
  labs(
    title = "Simulated Data: Latent State and Observations",
    x = "Time",
    y = "Value",
    caption = "Blue line: Latent state (x), Red points: Observed values (y)"
  ) +
  theme_minimal()

To fit the model using pmmh we need to specify the initialization, transition, and log-likelihood functions. The initialization function init_fn must take the argument num_particles and return a vector or matrix of particles. The transition function transition_fn must take the argument particles and return a vector or matrix of particles. The log-likelihood function log_likelihood_fn must take the arguments y, particles, and return a vector of log-likelihood values.

All the functions can take model-specific parameters as arguments. Time-dependency can be implemented by giving a t argument in transition_fn and log_likelihood_fn.

init_fn <- function(num_particles) {
  rnorm(num_particles, mean = 0, sd = 1)
}

transition_fn <- function(particles, sigma_x) {
  sin(particles) + rnorm(length(particles), mean = 0, sd = sigma_x)
}

log_likelihood_fn <- function(y, particles, sigma_y) {
  dnorm(y, mean = particles, sd = sigma_y, log = TRUE)
}

Since we are interested in Bayesian inference, we need to specify the priors for our parameters. We will use exponential priors for \(\sigma_x\) and \(\sigma_y\) in this vignette. pmmh needs the priors to be specified on the \(\log\)-scale and takes the priors as a list of functions.

log_prior_sigma_x <- function(sigma) {
  dexp(sigma, rate = 1, log = TRUE)
}

log_prior_sigma_y <- function(sigma) {
  dexp(sigma, rate = 1, log = TRUE)
}

log_priors <- list(
  sigma_x = log_prior_sigma_x,
  sigma_y = log_prior_sigma_y
)

The pmmh function automatically tunes the number of particles and proposal distribution for the parameters. The tuning can be modified by the the function default_tune_control.

We fit 2 chains with \(m=1000\) iterations for each, with a burn_in of \(500\). We also modify the tuning to only use a pilot run of \(100\) iterations and \(10\) burn-in iterations. In practice you should run more iterations and chains. To improve sampling we specify that proposals for \(\sigma_x\) and \(\sigma_y\) should be on the \(\log\)-scale.

result <- pmmh(
  y = y,
  m = 1000,
  init_fn = init_fn,
  transition_fn = transition_fn,
  log_likelihood_fn = log_likelihood_fn,
  log_priors = log_priors,
  pilot_init_params = list(
    c(sigma_x = 0.4, sigma_y = 0.4),
    c(sigma_x = 0.8, sigma_y = 0.8)
  ),
  burn_in = 500,
  num_chains = 2,
  seed = 1405,
  param_transform = list(
    sigma_x = "log",
    sigma_y = "log"
  ),
  tune_control = default_tune_control(pilot_m = 100, pilot_burn_in = 10),
  verbose = TRUE
)
#> Running chain 1...
#> Running pilot chain for tuning...
#> Pilot chain posterior mean:
#>   sigma_x   sigma_y 
#> 0.3141062 1.3603254
#> Pilot chain posterior covariance (on transformed space):
#>             sigma_x     sigma_y
#> sigma_x  0.04474149 -0.01090838
#> sigma_y -0.01090838  0.02731918
#> Using 50 particles for PMMH:
#> Running Particle MCMC chain with tuned settings...
#> Running chain 2...
#> Running pilot chain for tuning...
#> Pilot chain posterior mean:
#>   sigma_x   sigma_y 
#> 0.8389468 1.1278880
#> Pilot chain posterior covariance (on transformed space):
#>             sigma_x     sigma_y
#> sigma_x  0.20351954 -0.07524757
#> sigma_y -0.07524757  0.04146851
#> Using 50 particles for PMMH:
#> Running Particle MCMC chain with tuned settings...
#> PMMH Results Summary:
#>  Parameter Mean   SD Median 2.5% 97.5% ESS  Rhat
#>    sigma_x 0.74 0.49   0.69 0.05  1.71  17 1.145
#>    sigma_y 1.11 0.39   1.17 0.39  1.77   4 1.265
#> Warning in pmmh(y = y, m = 1000, init_fn = init_fn, transition_fn =
#> transition_fn, : Some ESS values are below 400, indicating poor mixing.
#> Consider running the chains for more iterations.
#> Warning in pmmh(y = y, m = 1000, init_fn = init_fn, transition_fn = transition_fn, : 
#> Some Rhat values are above 1.01, indicating that the chains have not converged. 
#> Consider running the chains for more iterations and/or increase burn_in.

We see that the chains gives convergence issues, indicating that we should run it for more iterations, but we ignore this issue in this Vignette.

It automatically prints data frame summarizing the results, which can be printed from any pmmh_output object by calling print.

print(result)
#> PMMH Results Summary:
#>  Parameter Mean   SD Median 2.5% 97.5% ESS  Rhat
#>    sigma_x 0.74 0.49   0.69 0.05  1.71  17 1.145
#>    sigma_y 1.11 0.39   1.17 0.39  1.77   4 1.265

chains <- result$theta_chain

Let’s collect the chains for \(\sigma_x\) from the chains and visualize the densities

ggplot(chains, aes(x = sigma_x, fill = factor(chain))) +
  geom_density(alpha = 0.5) +
  labs(
    title = "Density plot of sigma_x chains",
    x = "Value",
    y = "Density",
    fill = "Chain"
  ) +
  theme_minimal()

We have now fitted a simple SSM model using bayesSSM. Feel free to explore the package further and try out different models.