Draw posterior samples using Hamiltonian Monte Carlo (HMC)

Markov chain Monte Carlo (MCMC) methods are considered the gold standard of Bayesian inference; under suitable conditions and in the limit of infinitely many draws they generate samples from the true posterior distribution. HMC (Neal, 2011) uses gradients of the model's log-density function to propose samples, allowing it to exploit posterior geometry. However, it is computationally more expensive than variational inference and relatively sensitive to tuning.

sts_fit_with_hmc(
  observed_time_series,
  model,
  num_results = 100,
  num_warmup_steps = 50,
  num_leapfrog_steps = 15,
  initial_state = NULL,
  initial_step_size = NULL,
  chain_batch_shape = list(),
  num_variational_steps = 150,
  variational_optimizer = NULL,
  variational_sample_size = 5,
  seed = NULL,
  name = NULL
)

Arguments

observed_time_series	float tensor of shape concat([sample_shape, model.batch_shape, [num_timesteps, 1]]) where sample_shape corresponds to i.i.d. observations, and the trailing [1] dimension may (optionally) be omitted if num_timesteps > 1. May optionally be an instance of sts_masked_time_series, which includes a mask tensor to specify timesteps with missing observations.
model	An instance of StructuralTimeSeries representing a time-series model. This represents a joint distribution over time-series and their parameters with batch shape [b1, ..., bN].
num_results	Integer number of Markov chain draws. Default value: 100.
num_warmup_steps	Integer number of steps to take before starting to collect results. The warmup steps are also used to adapt the step size towards a target acceptance rate of 0.75. Default value: 50.
num_leapfrog_steps	Integer number of steps to run the leapfrog integrator for. Total progress per HMC step is roughly proportional to step_size * num_leapfrog_steps. Default value: 15.
initial_state	Optional Python list of Tensors, one for each model parameter, representing the initial state(s) of the Markov chain(s). These should have shape tf$concat(list(chain_batch_shape, param$prior$batch_shape, param$prior$event_shape)). If NULL, the initial state is set automatically using a sample from a variational posterior. Default value: NULL.
initial_step_size	list of tensors, one for each model parameter, representing the step size for the leapfrog integrator. Must broadcast with the shape of initial_state. Larger step sizes lead to faster progress, but too-large step sizes make rejection exponentially more likely. If NULL, the step size is set automatically using the standard deviation of a variational posterior. Default value: NULL.
chain_batch_shape	Batch shape (list or int) of chains to run in parallel. Default value: list() (i.e., a single chain).
num_variational_steps	int number of steps to run the variational optimization to determine the initial state and step sizes. Default value: 150.
variational_optimizer	Optional tf$train$Optimizer instance to use in the variational optimization. If NULL, defaults to tf$train$AdamOptimizer(0.1). Default value: NULL.
variational_sample_size	integer number of Monte Carlo samples to use in estimating the variational divergence. Larger values may stabilize the optimization, but at higher cost per step in time and memory. Default value: 1.
seed	integer to seed the random number generator.
name	name prefixed to ops created by this function. Default value: NULL (i.e., 'fit_with_hmc').

Value

list of:

• samples: list of Tensors representing posterior samples of model parameters, with shapes [concat([[num_results], chain_batch_shape, param.prior.batch_shape, param.prior.event_shape]) for param in model.parameters].

• kernel_results: A (possibly nested) list of Tensors representing internal calculations made within the HMC sampler.

Details

This method attempts to provide a sensible default approach for fitting StructuralTimeSeries models using HMC. It first runs variational inference as a fast posterior approximation, and initializes the HMC sampler from the variational posterior, using the posterior standard deviations to set per-variable step sizes (equivalently, a diagonal mass matrix). During the warmup phase, it adapts the step size to target an acceptance rate of 0.75, which is thought to be in the desirable range for optimal mixing (Betancourt et al., 2014).

References

• Radford Neal. MCMC Using Hamiltonian Dynamics. Handbook of Markov Chain Monte Carlo, 2011.

• M.J. Betancourt, Simon Byrne, and Mark Girolami. Optimizing The Integrator Step Size for Hamiltonian Monte Carlo.

See also

Other sts-functions: sts_build_factored_surrogate_posterior(), sts_build_factored_variational_loss(), sts_decompose_by_component(), sts_decompose_forecast_by_component(), sts_forecast(), sts_one_step_predictive(), sts_sample_uniform_initial_state()

Examples

# \donttest{
observed_time_series <-
  rep(c(3.5, 4.1, 4.5, 3.9, 2.4, 2.1, 1.2), 5) +
  rep(c(1.1, 1.5, 2.4, 3.1, 4.0), each = 7) %>%
  tensorflow::tf$convert_to_tensor(dtype = tensorflow::tf$float64)
day_of_week <- observed_time_series %>% sts_seasonal(num_seasons = 7)
local_linear_trend <- observed_time_series %>% sts_local_linear_trend()
model <- observed_time_series %>%
  sts_sum(components = list(day_of_week, local_linear_trend))
states_and_results <- observed_time_series %>%
  sts_fit_with_hmc(
    model,
    num_results = 10,
    num_warmup_steps = 5,
    num_variational_steps = 15)
# }