Bayesian Estimation or Edge Selection for Markov Random Fields

The bgm function estimates the pseudoposterior distribution of category thresholds (main effects) and pairwise interaction parameters of a Markov Random Field (MRF) model for binary and/or ordinal variables. Optionally, it performs Bayesian edge selection using spike-and-slab priors to infer the network structure.

Usage

bgm(
  x,
  variable_type = "ordinal",
  baseline_category,
  iter = 1000,
  warmup = 1000,
  pairwise_scale = 2.5,
  main_alpha = 0.5,
  main_beta = 0.5,
  edge_selection = TRUE,
  edge_prior = c("Bernoulli", "Beta-Bernoulli", "Stochastic-Block"),
  inclusion_probability = 0.5,
  beta_bernoulli_alpha = 1,
  beta_bernoulli_beta = 1,
  dirichlet_alpha = 1,
  lambda = 1,
  na_action = c("listwise", "impute"),
  update_method = c("nuts", "adaptive-metropolis", "hamiltonian-mc"),
  target_accept,
  hmc_num_leapfrogs = 100,
  nuts_max_depth = 10,
  learn_mass_matrix = FALSE,
  chains = 4,
  cores = parallel::detectCores(),
  display_progress = c("per-chain", "total", "none"),
  seed = NULL,
  interaction_scale,
  burnin,
  save,
  threshold_alpha,
  threshold_beta
)

Arguments

x

A data frame or matrix with n rows and p columns containing binary and ordinal responses. Variables are automatically recoded to non-negative integers (0, 1, ..., m). For regular ordinal variables, unobserved categories are collapsed; for Blume–Capel variables, all categories are retained.

variable_type

Character or character vector. Specifies the type of each variable in x. Allowed values: "ordinal" or "blume-capel". Binary variables are automatically treated as "ordinal". Default: "ordinal".

baseline_category

Integer or vector. Baseline category used in Blume–Capel variables. Can be a single integer (applied to all) or a vector of length p. Required if at least one variable is of type "blume-capel".

iter

Integer. Number of post–burn-in iterations (per chain). Default: 1e3.

warmup

Integer. Number of warmup iterations before collecting samples. A minimum of 1000 iterations is enforced, with a warning if a smaller value is requested. Default: 1e3.

pairwise_scale

Double. Scale of the Cauchy prior for pairwise interaction parameters. Default: 2.5.

main_alpha, main_beta

Double. Shape parameters of the beta-prime prior for threshold parameters. Must be positive. If equal, the prior is symmetric. Defaults: main_alpha = 0.5 and main_beta = 0.5.

edge_selection

Logical. Whether to perform Bayesian edge selection. If FALSE, the model estimates all edges. Default: TRUE.

edge_prior

Character. Specifies the prior for edge inclusion. Options: "Bernoulli", "Beta-Bernoulli", or "Stochastic-Block". Default: "Bernoulli".

inclusion_probability

Numeric scalar. Prior inclusion probability of each edge (used with the Bernoulli prior). Default: 0.5.

beta_bernoulli_alpha, beta_bernoulli_beta

Double. Shape parameters for the beta distribution in the Beta–Bernoulli and the Stochastic-Block priors. Must be positive. Defaults: beta_bernoulli_alpha = 1 and beta_bernoulli_beta = 1.

dirichlet_alpha

Double. Concentration parameter of the Dirichlet prior on block assignments (used with the Stochastic Block model). Default: 1.

lambda

Double. Rate of the zero-truncated Poisson prior on the number of clusters in the Stochastic Block Model. Default: 1.

na_action

Character. Specifies missing data handling. Either "listwise" (drop rows with missing values) or "impute" (perform single imputation during sampling). Default: "listwise".

update_method

Character. Specifies how the MCMC sampler updates the model parameters:

"adaptive-metropolis": Componentwise adaptive Metropolis–Hastings with Robbins–Monro proposal adaptation.
"hamiltonian-mc": Hamiltonian Monte Carlo with fixed path length (number of leapfrog steps set by hmc_num_leapfrogs).
"nuts": The No-U-Turn Sampler, an adaptive form of HMC with dynamically chosen trajectory lengths.

Default: "nuts".

target_accept

Numeric between 0 and 1. Target acceptance rate for the sampler. Defaults are set automatically if not supplied: 0.44 for adaptive Metropolis, 0.65 for HMC, and 0.60 for NUTS.

hmc_num_leapfrogs

Integer. Number of leapfrog steps for Hamiltonian Monte Carlo. Must be positive. Default: 100.

nuts_max_depth

Integer. Maximum tree depth in NUTS. Must be positive. Default: 10.

learn_mass_matrix

Logical. If TRUE, adapt a diagonal mass matrix during warmup (HMC/NUTS only). If FALSE, use the identity matrix. Default: FALSE.

chains

Integer. Number of parallel chains to run. Default: 4.

cores

Integer. Number of CPU cores for parallel execution. Default: parallel::detectCores().

display_progress

Logical. Whether to show a progress bar during sampling. Default: TRUE.

seed

Optional integer. Random seed for reproducibility. Must be a single non-negative integer.

interaction_scale, burnin, save, threshold_alpha, threshold_beta

Deprecated arguments. Use pairwise_scale, warmup, main_alpha, and main_beta instead.

Value

A list of class "bgms" with posterior summaries, posterior mean matrices, and access to raw MCMC draws. The object can be passed to print(), summary(), coef(), and as_draws() methods for inspection and analysis.

Main components include:

posterior_summary_main: Data frame with posterior summaries (mean, sd, MCSE, ESS, Rhat) for category threshold parameters.
posterior_summary_pairwise: Data frame with posterior summaries for pairwise interaction parameters.
posterior_summary_indicator: Data frame with posterior summaries for edge inclusion indicators (if edge_selection = TRUE).
posterior_mean_main: Matrix of posterior mean thresholds (rows = variables, cols = categories or parameters).
posterior_mean_pairwise: Symmetric matrix of posterior mean pairwise interaction strengths.
posterior_mean_indicator: Symmetric matrix of posterior mean inclusion probabilities (if edge selection was enabled).
Additional summaries returned when edge_prior = "Stochastic-Block". For more details about this prior see Sekulovski et al. (2025) .
- posterior_summary_pairwise_allocations: Data frame with posterior summaries (mean, sd, MCSE, ESS, Rhat) for the pairwise cluster co-occurrence of the nodes. This serves to indicate whether the estimated posterior allocations,co-clustering matrix and posterior cluster probabilities (see blow) have converged.
- posterior_coclustering_matrix: a symmetric matrix of pairwise proportions of occurrence of every variable. This matrix can be plotted to visually inspect the estimated number of clusters and visually inspect nodes that tend to switch clusters.
- posterior_mean_allocations: A vector with the posterior mean of the cluster allocations of the nodes. This is calculated using the method proposed in Dahl (2009) .
- posterior_mode_allocations: A vector with the posterior mode of the cluster allocations of the nodes.
- posterior_num_blocks: A data frame with the estimated posterior inclusion probabilities for all the possible number of clusters.
raw_samples: A list of raw MCMC draws per chain:
main

List of main effect samples.

pairwise

List of pairwise effect samples.

indicator

List of indicator samples (if edge selection enabled).

allocations

List of cluster allocations (if SBM prior used).

nchains

Number of chains.

niter

Number of post–warmup iterations per chain.

parameter_names

Named lists of parameter labels.
arguments: A list of function call arguments and metadata (e.g., number of variables, warmup, sampler settings, package version).

The summary() method prints formatted posterior summaries, coef() extracts posterior mean matrices, and as_draws() converts the raw samples into a posterior::draws_df object for use with the posterior package.

NUTS diagnostics (tree depth, divergences, energy, E-BFMI) are included in fit$nuts_diag if update_method = "nuts".

Details

This function models the joint distribution of binary and ordinal variables using a Markov Random Field, with support for edge selection through Bayesian variable selection. The statistical foundation of the model is described in Marsman et al. (2025) , where the ordinal MRF model and its Bayesian estimation procedure were first introduced. While the implementation in bgms has since been extended and updated (e.g., alternative priors, parallel chains, HMC/NUTS warmup), it builds on that original framework.

Key components of the model are described in the sections below.

Ordinal Variables

The function supports two types of ordinal variables:

Regular ordinal variables: Assigns a category threshold parameter to each response category except the lowest. The model imposes no additional constraints on the distribution of category responses.

Blume-Capel ordinal variables: Assume a baseline category (e.g., a “neutral” response) and score responses by distance from this baseline. Category thresholds are modeled as:

$$\mu_{c} = \alpha \cdot c + \beta \cdot (c - b)^2$$

where:

$\mu_{c}$: category threshold for category $c$
$\alpha$: linear trend across categories
$\beta$: preference toward or away from the baseline
- If $\beta < 0$, the model favors responses near the baseline category;
- if $\beta > 0$, it favors responses farther away (i.e., extremes).
$b$: baseline category

Edge Selection

When edge_selection = TRUE, the function performs Bayesian variable selection on the pairwise interactions (edges) in the MRF using spike-and-slab priors.

Supported priors for edge inclusion:

Bernoulli: Fixed inclusion probability across edges.
Beta-Bernoulli: Inclusion probability is assigned a Beta prior distribution.
Stochastic-Block: Cluster-based edge priors with Beta, Dirichlet, and Poisson hyperpriors.

All priors operate via binary indicator variables controlling the inclusion or exclusion of each edge in the MRF.

Prior Distributions

Pairwise effects: Modeled with a Cauchy (slab) prior.
Main effects: Modeled using a beta-prime distribution.
Edge indicators: Use either a Bernoulli, Beta-Bernoulli, or Stochastic-Block prior (as above).

Sampling Algorithms and Warmup

Parameters are updated within a Gibbs framework, but the conditional updates can be carried out using different algorithms:

Adaptive Metropolis–Hastings: Componentwise random–walk updates for main effects and pairwise effects. Proposal standard deviations are adapted during burn–in via Robbins–Monro updates toward a target acceptance rate.
Hamiltonian Monte Carlo (HMC): Joint updates of all parameters using fixed–length leapfrog trajectories. Step size is tuned during warmup via dual–averaging; the diagonal mass matrix can also be adapted if learn_mass_matrix = TRUE.
No–U–Turn Sampler (NUTS): An adaptive extension of HMC that dynamically chooses trajectory lengths. Warmup uses a staged adaptation schedule (fast–slow–fast) to stabilize step size and, if enabled, the mass matrix.

When edge_selection = TRUE, updates of edge–inclusion indicators are carried out with Metropolis–Hastings steps. These are switched on after the core warmup phase, ensuring that graph updates occur only once the samplers’ tuning parameters (step size, mass matrix, proposal SDs) have stabilized.

After warmup, adaptation is disabled. Step size and mass matrix are fixed at their learned values, and proposal SDs remain constant.

Warmup and Adaptation

The warmup procedure in bgm is based on the multi–stage adaptation schedule used in Stan (Stan Development Team 2023) . Warmup iterations are split into several phases:

Stage 1 (fast adaptation): A short initial interval where only step size (for HMC/NUTS) is adapted, allowing the chain to move quickly toward the typical set.
Stage 2 (slow windows): A sequence of expanding, memoryless windows where both step size and, if learn_mass_matrix = TRUE, the diagonal mass matrix are adapted. Each window ends with a reset of the dual–averaging scheme for improved stability.
Stage 3a (final fast interval): A short interval at the end of the core warmup where the step size is adapted one final time.
Stage 3b (proposal–SD tuning): Only active when edge_selection = TRUE under HMC/NUTS. In this phase, Robbins–Monro adaptation of proposal standard deviations is performed for the Metropolis steps used in edge–selection moves.
Stage 3c (graph selection warmup): Also only relevant when edge_selection = TRUE. At the start of this phase, a random graph structure is initialized, and Metropolis–Hastings updates for edge inclusion indicators are switched on.

When edge_selection = FALSE, the total number of warmup iterations equals the user–specified burnin. When edge_selection = TRUE and update_method is "nuts" or "hamiltonian-mc", the schedule automatically appends additional Stage-3b and Stage-3c intervals, so the total warmup is strictly greater than the requested burnin.

After all warmup phases, the sampler transitions to the sampling phase with adaptation disabled. Step size and mass matrix (for HMC/NUTS) are fixed at their learned values, and proposal SDs remain constant.

This staged design improves stability of proposals and ensures that both local parameters (step size) and global parameters (mass matrix, proposal SDs) are tuned before collecting posterior samples.

For adaptive Metropolis–Hastings runs, step size and mass matrix adaptation are not relevant. Proposal SDs are tuned continuously during burn–in using Robbins–Monro updates, without staged fast/slow intervals.

Missing Data

If na_action = "listwise", observations with missing values are removed. If na_action = "impute", missing values are imputed during Gibbs sampling.

References

Dahl DB (2009). “Modal clustering in a class of product partition models.” Bayesian Analysis, 4(2), 243–264. doi:10.1214/09-BA409 .

Marsman M, van den Bergh D, Haslbeck JMB (2025). “Bayesian analysis of the ordinal Markov random field.” Psychometrika, 90, 146–-182.

Sekulovski N, Arena G, Haslbeck JMB, Huth KBS, Friel N, Marsman M (2025). “A Stochastic Block Prior for Clustering in Graphical Models.” Retrieved from https://osf.io/preprints/psyarxiv/29p3m_v1. OSF preprint.

Stan Development Team (2023). Stan Modeling Language Users Guide and Reference Manual. Version 2.33, https://mc-stan.org/docs/.

Examples

# \donttest{
# Run bgm on subset of the Wenchuan dataset
fit = bgm(x = Wenchuan[, 1:5])
#> Warning: There were 7 rows with missing observations in the input matrix x.
#> Since na_action = listwise these rows were excluded from the analysis.
#> Chain 1 (Warmup): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 50/2200 (2.3%)
#> Chain 2 (Warmup): ⦗━╺━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 60/2200 (2.7%)
#> Chain 3 (Warmup): ⦗━╺━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 65/2200 (3.0%)
#> Chain 4 (Warmup): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 43/2200 (2.0%)
#> Total   (Warmup): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 218/8800 (2.5%)
#> Elapsed: 2s | ETA: 1m 18s
#> Chain 1 (Warmup): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 100/2200 (4.5%)
#> Chain 2 (Warmup): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 104/2200 (4.7%)
#> Chain 3 (Warmup): ⦗━━╺━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 118/2200 (5.4%)
#> Chain 4 (Warmup): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 85/2200 (3.9%)
#> Total   (Warmup): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 407/8800 (4.6%)
#> Elapsed: 4s | ETA: 1m 22s
#> Chain 1 (Warmup): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 150/2200 (6.8%)
#> Chain 2 (Warmup): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 155/2200 (7.0%)
#> Chain 3 (Warmup): ⦗━━━╺━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 170/2200 (7.7%)
#> Chain 4 (Warmup): ⦗━━╺━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 127/2200 (5.8%)
#> Total   (Warmup): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 602/8800 (6.8%)
#> Elapsed: 6s | ETA: 1m 21s
#> Chain 1 (Warmup): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 200/2200 (9.1%)
#> Chain 2 (Warmup): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 196/2200 (8.9%)
#> Chain 3 (Warmup): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 219/2200 (10.0%)
#> Chain 4 (Warmup): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 161/2200 (7.3%)
#> Total   (Warmup): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 776/8800 (8.8%)
#> Elapsed: 8s | ETA: 1m 22s
#> Chain 1 (Warmup): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 250/2200 (11.4%)
#> Chain 2 (Warmup): ⦗━━━━╺━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 242/2200 (11.0%)
#> Chain 3 (Warmup): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 263/2200 (12.0%)
#> Chain 4 (Warmup): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 204/2200 (9.3%)
#> Total   (Warmup): ⦗━━━━╺━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 959/8800 (10.9%)
#> Elapsed: 10s | ETA: 1m 21s
#> Chain 1 (Warmup): ⦗━━━━━╺━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 300/2200 (13.6%)
#> Chain 2 (Warmup): ⦗━━━━━╺━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 281/2200 (12.8%)
#> Chain 3 (Warmup): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 312/2200 (14.2%)
#> Chain 4 (Warmup): ⦗━━━━╺━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 244/2200 (11.1%)
#> Total   (Warmup): ⦗━━━━━╺━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 1137/8800 (12.9%)
#> Elapsed: 11s | ETA: 1m 14s
#> Chain 1 (Warmup): ⦗━━━━━━╺━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 350/2200 (15.9%)
#> Chain 2 (Warmup): ⦗━━━━━━╺━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 332/2200 (15.1%)
#> Chain 3 (Warmup): ⦗━━━━━━╺━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 356/2200 (16.2%)
#> Chain 4 (Warmup): ⦗━━━━━╺━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 292/2200 (13.3%)
#> Total   (Warmup): ⦗━━━━━━╺━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 1330/8800 (15.1%)
#> Elapsed: 13s | ETA: 1m 13s
#> Chain 1 (Warmup): ⦗━━━━━━━╺━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 400/2200 (18.2%)
#> Chain 2 (Warmup): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 382/2200 (17.4%)
#> Chain 3 (Warmup): ⦗━━━━━━━╺━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 399/2200 (18.1%)
#> Chain 4 (Warmup): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 328/2200 (14.9%)
#> Total   (Warmup): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 1509/8800 (17.1%)
#> Elapsed: 15s | ETA: 1m 12s
#> Chain 1 (Warmup): ⦗━━━━━━━━╺━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 450/2200 (20.5%)
#> Chain 2 (Warmup): ⦗━━━━━━━━╺━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 443/2200 (20.1%)
#> Chain 3 (Warmup): ⦗━━━━━━━━╺━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 462/2200 (21.0%)
#> Chain 4 (Warmup): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 383/2200 (17.4%)
#> Total   (Warmup): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 1738/8800 (19.8%)
#> Elapsed: 17s | ETA: 1m 9s
#> Chain 1 (Warmup): ⦗━━━━━━━━━╺━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 500/2200 (22.7%)
#> Chain 2 (Warmup): ⦗━━━━━━━━━╺━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 501/2200 (22.8%)
#> Chain 3 (Warmup): ⦗━━━━━━━━━╺━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 514/2200 (23.4%)
#> Chain 4 (Warmup): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 433/2200 (19.7%)
#> Total   (Warmup): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 1948/8800 (22.1%)
#> Elapsed: 19s | ETA: 1m 6s
#> Chain 1 (Warmup): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 550/2200 (25.0%)
#> Chain 2 (Warmup): ⦗━━━━━━━━━━╺━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 551/2200 (25.0%)
#> Chain 3 (Warmup): ⦗━━━━━━━━━━╺━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 564/2200 (25.6%)
#> Chain 4 (Warmup): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 488/2200 (22.2%)
#> Total   (Warmup): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 2153/8800 (24.5%)
#> Elapsed: 21s | ETA: 1m 4s
#> Chain 1 (Warmup): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 600/2200 (27.3%)
#> Chain 2 (Warmup): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 601/2200 (27.3%)
#> Chain 3 (Warmup): ⦗━━━━━━━━━━━╺━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 630/2200 (28.6%)
#> Chain 4 (Warmup): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 542/2200 (24.6%)
#> Total   (Warmup): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 2373/8800 (27.0%)
#> Elapsed: 24s | ETA: 1m 5s
#> Chain 1 (Warmup): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 650/2200 (29.5%)
#> Chain 2 (Warmup): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 659/2200 (30.0%)
#> Chain 3 (Warmup): ⦗━━━━━━━━━━━━╺━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 682/2200 (31.0%)
#> Chain 4 (Warmup): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 598/2200 (27.2%)
#> Total   (Warmup): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 2589/8800 (29.4%)
#> Elapsed: 26s | ETA: 1m 2s
#> Chain 1 (Warmup): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 700/2200 (31.8%)
#> Chain 2 (Warmup): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 709/2200 (32.2%)
#> Chain 3 (Warmup): ⦗━━━━━━━━━━━━━╺━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 737/2200 (33.5%)
#> Chain 4 (Warmup): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 659/2200 (30.0%)
#> Total   (Warmup): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 2805/8800 (31.9%)
#> Elapsed: 28s | ETA: 60s
#> Chain 1 (Warmup): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 750/2200 (34.1%)
#> Chain 2 (Warmup): ⦗━━━━━━━━━━━━━━╺━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 773/2200 (35.1%)
#> Chain 3 (Warmup): ⦗━━━━━━━━━━━━━━╺━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 790/2200 (35.9%)
#> Chain 4 (Warmup): ⦗━━━━━━━━━━━━━╺━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 718/2200 (32.6%)
#> Total   (Warmup): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 3031/8800 (34.4%)
#> Elapsed: 30s | ETA: 57s
#> Chain 1 (Warmup): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 800/2200 (36.4%)
#> Chain 2 (Warmup): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 823/2200 (37.4%)
#> Chain 3 (Warmup): ⦗━━━━━━━━━━━━━━━╺━━━━━━━━━━━━━━━━━━━━━━━━⦘ 827/2200 (37.6%)
#> Chain 4 (Warmup): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 767/2200 (34.9%)
#> Total   (Warmup): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 3217/8800 (36.6%)
#> Elapsed: 31s | ETA: 54s
#> Chain 1 (Warmup): ⦗━━━━━━━━━━━━━━━╺━━━━━━━━━━━━━━━━━━━━━━━━⦘ 850/2200 (38.6%)
#> Chain 2 (Warmup): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 859/2200 (39.0%)
#> Chain 3 (Warmup): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 873/2200 (39.7%)
#> Chain 4 (Warmup): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 807/2200 (36.7%)
#> Total   (Warmup): ⦗━━━━━━━━━━━━━━━╺━━━━━━━━━━━━━━━━━━━━━━━━⦘ 3389/8800 (38.5%)
#> Elapsed: 33s | ETA: 53s
#> Chain 1 (Warmup): ⦗━━━━━━━━━━━━━━━━╺━━━━━━━━━━━━━━━━━━━━━━━⦘ 900/2200 (40.9%)
#> Chain 2 (Warmup): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 911/2200 (41.4%)
#> Chain 3 (Warmup): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 912/2200 (41.5%)
#> Chain 4 (Warmup): ⦗━━━━━━━━━━━━━━━╺━━━━━━━━━━━━━━━━━━━━━━━━⦘ 849/2200 (38.6%)
#> Total   (Warmup): ⦗━━━━━━━━━━━━━━━━╺━━━━━━━━━━━━━━━━━━━━━━━⦘ 3572/8800 (40.6%)
#> Elapsed: 34s | ETA: 50s
#> Chain 1 (Warmup): ⦗━━━━━━━━━━━━━━━━━╺━━━━━━━━━━━━━━━━━━━━━━⦘ 950/2200 (43.2%)
#> Chain 2 (Warmup): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 965/2200 (43.9%)
#> Chain 3 (Warmup): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 974/2200 (44.3%)
#> Chain 4 (Warmup): ⦗━━━━━━━━━━━━━━━━╺━━━━━━━━━━━━━━━━━━━━━━━⦘ 895/2200 (40.7%)
#> Total   (Warmup): ⦗━━━━━━━━━━━━━━━━━╺━━━━━━━━━━━━━━━━━━━━━━⦘ 3784/8800 (43.0%)
#> Elapsed: 36s | ETA: 48s
#> Chain 1 (Warmup): ⦗━━━━━━━━━━━━━━━━━━╺━━━━━━━━━━━━━━━━━━━━━⦘ 1000/2200 (45.5%)
#> Chain 2 (Warmup): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 1019/2200 (46.3%)
#> Chain 3 (Warmup): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 1023/2200 (46.5%)
#> Chain 4 (Warmup): ⦗━━━━━━━━━━━━━━━━━╺━━━━━━━━━━━━━━━━━━━━━━⦘ 953/2200 (43.3%)
#> Total   (Warmup): ⦗━━━━━━━━━━━━━━━━━━╺━━━━━━━━━━━━━━━━━━━━━⦘ 3995/8800 (45.4%)
#> Elapsed: 38s | ETA: 46s
#> Chain 1 (Warmup): ⦗━━━━━━━━━━━━━━━━━━━╺━━━━━━━━━━━━━━━━━━━━⦘ 1050/2200 (47.7%)
#> Chain 2 (Warmup): ⦗━━━━━━━━━━━━━━━━━━━╺━━━━━━━━━━━━━━━━━━━━⦘ 1065/2200 (48.4%)
#> Chain 3 (Warmup): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 1073/2200 (48.8%)
#> Chain 4 (Warmup): ⦗━━━━━━━━━━━━━━━━━━╺━━━━━━━━━━━━━━━━━━━━━⦘ 1003/2200 (45.6%)
#> Total   (Warmup): ⦗━━━━━━━━━━━━━━━━━━━╺━━━━━━━━━━━━━━━━━━━━⦘ 4191/8800 (47.6%)
#> Elapsed: 40s | ETA: 44s
#> Chain 1 (Warmup): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 1100/2200 (50.0%)
#> Chain 2 (Warmup): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 1129/2200 (51.3%)
#> Chain 3 (Warmup): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 1132/2200 (51.5%)
#> Chain 4 (Warmup): ⦗━━━━━━━━━━━━━━━━━━━╺━━━━━━━━━━━━━━━━━━━━⦘ 1062/2200 (48.3%)
#> Total   (Warmup): ⦗━━━━━━━━━━━━━━━━━━━━╺━━━━━━━━━━━━━━━━━━━⦘ 4423/8800 (50.3%)
#> Elapsed: 42s | ETA: 42s
#> Chain 1 (Warmup): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 1150/2200 (52.3%)
#> Chain 2 (Warmup): ⦗━━━━━━━━━━━━━━━━━━━━━╺━━━━━━━━━━━━━━━━━━⦘ 1171/2200 (53.2%)
#> Chain 3 (Warmup): ⦗━━━━━━━━━━━━━━━━━━━━━╺━━━━━━━━━━━━━━━━━━⦘ 1159/2200 (52.7%)
#> Chain 4 (Warmup): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 1091/2200 (49.6%)
#> Total   (Warmup): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 4571/8800 (51.9%)
#> Elapsed: 44s | ETA: 41s
#> Chain 1 (Sampling): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 1200/2200 (54.5%)
#> Chain 2 (Sampling): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 1205/2200 (54.8%)
#> Chain 3 (Sampling): ⦗━━━━━━━━━━━━━━━━━━━━━━╺━━━━━━━━━━━━━━━━━⦘ 1211/2200 (55.0%)
#> Chain 4 (Warmup): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 1147/2200 (52.1%)
#> Total   (Warmup): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 4763/8800 (54.1%)
#> Elapsed: 45s | ETA: 38s
#> Chain 1 (Sampling): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 1250/2200 (56.8%)
#> Chain 2 (Sampling): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 1243/2200 (56.5%)
#> Chain 3 (Sampling): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 1251/2200 (56.9%)
#> Chain 4 (Warmup): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 1198/2200 (54.5%)
#> Total   (Warmup): ⦗━━━━━━━━━━━━━━━━━━━━━━╺━━━━━━━━━━━━━━━━━⦘ 4942/8800 (56.2%)
#> Elapsed: 46s | ETA: 36s
#> Chain 1 (Sampling): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 1300/2200 (59.1%)
#> Chain 2 (Sampling): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 1296/2200 (58.9%)
#> Chain 3 (Sampling): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 1307/2200 (59.4%)
#> Chain 4 (Sampling): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 1245/2200 (56.6%)
#> Total   (Sampling): ⦗━━━━━━━━━━━━━━━━━━━━━━━╺━━━━━━━━━━━━━━━━⦘ 5148/8800 (58.5%)
#> Elapsed: 48s | ETA: 34s
#> Chain 1 (Sampling): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 1350/2200 (61.4%)
#> Chain 2 (Sampling): ⦗━━━━━━━━━━━━━━━━━━━━━━━━╺━━━━━━━━━━━━━━━⦘ 1346/2200 (61.2%)
#> Chain 3 (Sampling): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 1354/2200 (61.5%)
#> Chain 4 (Sampling): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 1297/2200 (59.0%)
#> Total   (Sampling): ⦗━━━━━━━━━━━━━━━━━━━━━━━━╺━━━━━━━━━━━━━━━⦘ 5347/8800 (60.8%)
#> Elapsed: 50s | ETA: 32s
#> Chain 1 (Sampling): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━╺━━━━━━━━━━━━━━⦘ 1400/2200 (63.6%)
#> Chain 2 (Sampling): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━╺━━━━━━━━━━━━━━⦘ 1401/2200 (63.7%)
#> Chain 3 (Sampling): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 1407/2200 (64.0%)
#> Chain 4 (Sampling): ⦗━━━━━━━━━━━━━━━━━━━━━━━━╺━━━━━━━━━━━━━━━⦘ 1345/2200 (61.1%)
#> Total   (Sampling): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━╺━━━━━━━━━━━━━━⦘ 5553/8800 (63.1%)
#> Elapsed: 51s | ETA: 30s
#> Chain 1 (Sampling): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━╺━━━━━━━━━━━━━⦘ 1450/2200 (65.9%)
#> Chain 2 (Sampling): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━╺━━━━━━━━━━━━━⦘ 1453/2200 (66.0%)
#> Chain 3 (Sampling): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━╺━━━━━━━━━━━━━⦘ 1451/2200 (66.0%)
#> Chain 4 (Sampling): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━╺━━━━━━━━━━━━━━⦘ 1385/2200 (63.0%)
#> Total   (Sampling): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━╺━━━━━━━━━━━━━⦘ 5739/8800 (65.2%)
#> Elapsed: 53s | ETA: 28s
#> Chain 1 (Sampling): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━╺━━━━━━━━━━━━⦘ 1500/2200 (68.2%)
#> Chain 2 (Sampling): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━╺━━━━━━━━━━━━⦘ 1500/2200 (68.2%)
#> Chain 3 (Sampling): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━╺━━━━━━━━━━━━⦘ 1501/2200 (68.2%)
#> Chain 4 (Sampling): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━╺━━━━━━━━━━━━━⦘ 1431/2200 (65.0%)
#> Total   (Sampling): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 5932/8800 (67.4%)
#> Elapsed: 54s | ETA: 26s
#> Chain 1 (Sampling): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━╺━━━━━━━━━━━⦘ 1550/2200 (70.5%)
#> Chain 2 (Sampling): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━╺━━━━━━━━━━━⦘ 1546/2200 (70.3%)
#> Chain 3 (Sampling): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 1537/2200 (69.9%)
#> Chain 4 (Sampling): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 1478/2200 (67.2%)
#> Total   (Sampling): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 6111/8800 (69.4%)
#> Elapsed: 56s | ETA: 25s
#> Chain 1 (Sampling): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━╺━━━━━━━━━━⦘ 1600/2200 (72.7%)
#> Chain 2 (Sampling): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━╺━━━━━━━━━━⦘ 1597/2200 (72.6%)
#> Chain 3 (Sampling): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 1592/2200 (72.4%)
#> Chain 4 (Sampling): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 1533/2200 (69.7%)
#> Total   (Sampling): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 6322/8800 (71.8%)
#> Elapsed: 57s | ETA: 22s
#> Chain 1 (Sampling): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 1650/2200 (75.0%)
#> Chain 2 (Sampling): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 1650/2200 (75.0%)
#> Chain 3 (Sampling): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 1636/2200 (74.4%)
#> Chain 4 (Sampling): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 1575/2200 (71.6%)
#> Total   (Sampling): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 6511/8800 (74.0%)
#> Elapsed: 59s | ETA: 21s
#> Chain 1 (Sampling): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 1700/2200 (77.3%)
#> Chain 2 (Sampling): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 1699/2200 (77.2%)
#> Chain 3 (Sampling): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━╺━━━━━━━━━⦘ 1672/2200 (76.0%)
#> Chain 4 (Sampling): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━╺━━━━━━━━━━⦘ 1620/2200 (73.6%)
#> Total   (Sampling): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━╺━━━━━━━━━⦘ 6691/8800 (76.0%)
#> Elapsed: 1m | ETA: 19s
#> Chain 1 (Sampling): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 1750/2200 (79.5%)
#> Chain 2 (Sampling): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━╺━━━━━━━⦘ 1768/2200 (80.4%)
#> Chain 3 (Sampling): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━╺━━━━━━━━⦘ 1730/2200 (78.6%)
#> Chain 4 (Sampling): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━╺━━━━━━━━━⦘ 1674/2200 (76.1%)
#> Total   (Sampling): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━╺━━━━━━━━⦘ 6922/8800 (78.7%)
#> Elapsed: 1m 2s | ETA: 17s
#> Chain 1 (Sampling): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 1800/2200 (81.8%)
#> Chain 2 (Sampling): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━╺━━━━━━⦘ 1817/2200 (82.6%)
#> Chain 3 (Sampling): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━╺━━━━━━━⦘ 1776/2200 (80.7%)
#> Chain 4 (Sampling): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━╺━━━━━━━━⦘ 1725/2200 (78.4%)
#> Total   (Sampling): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━╺━━━━━━━⦘ 7118/8800 (80.9%)
#> Elapsed: 1m 4s | ETA: 15s
#> Chain 1 (Sampling): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 1850/2200 (84.1%)
#> Chain 2 (Sampling): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 1865/2200 (84.8%)
#> Chain 3 (Sampling): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━╺━━━━━━⦘ 1835/2200 (83.4%)
#> Chain 4 (Sampling): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━╺━━━━━━━⦘ 1770/2200 (80.5%)
#> Total   (Sampling): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━╺━━━━━━⦘ 7320/8800 (83.2%)
#> Elapsed: 1m 5s | ETA: 13s
#> Chain 1 (Sampling): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 1900/2200 (86.4%)
#> Chain 2 (Sampling): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 1908/2200 (86.7%)
#> Chain 3 (Sampling): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━╺━━━━━⦘ 1871/2200 (85.0%)
#> Chain 4 (Sampling): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 1805/2200 (82.0%)
#> Total   (Sampling): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━╺━━━━━⦘ 7484/8800 (85.0%)
#> Elapsed: 1m 7s | ETA: 12s
#> Chain 1 (Sampling): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━╺━━━━⦘ 1950/2200 (88.6%)
#> Chain 2 (Sampling): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 1957/2200 (89.0%)
#> Chain 3 (Sampling): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 1920/2200 (87.3%)
#> Chain 4 (Sampling): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 1855/2200 (84.3%)
#> Total   (Sampling): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 7682/8800 (87.3%)
#> Elapsed: 1m 8s | ETA: 10s
#> Chain 1 (Sampling): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━╺━━━⦘ 2000/2200 (90.9%)
#> Chain 2 (Sampling): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 2008/2200 (91.3%)
#> Chain 3 (Sampling): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 1969/2200 (89.5%)
#> Chain 4 (Sampling): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 1908/2200 (86.7%)
#> Total   (Sampling): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 7885/8800 (89.6%)
#> Elapsed: 1m 10s | ETA: 8s
#> Chain 1 (Sampling): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━╺━━⦘ 2050/2200 (93.2%)
#> Chain 2 (Sampling): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 2067/2200 (94.0%)
#> Chain 3 (Sampling): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 2021/2200 (91.9%)
#> Chain 4 (Sampling): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━╺━━━━⦘ 1951/2200 (88.7%)
#> Total   (Sampling): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 8089/8800 (91.9%)
#> Elapsed: 1m 11s | ETA: 6s
#> Chain 1 (Sampling): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━╺━⦘ 2100/2200 (95.5%)
#> Chain 2 (Sampling): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━╺━⦘ 2108/2200 (95.8%)
#> Chain 3 (Sampling): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 2067/2200 (94.0%)
#> Chain 4 (Sampling): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━╺━━━⦘ 1989/2200 (90.4%)
#> Total   (Sampling): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 8264/8800 (93.9%)
#> Elapsed: 1m 13s | ETA: 5s
#> Chain 1 (Sampling): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━╺⦘ 2150/2200 (97.7%)
#> Chain 2 (Sampling): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━╺⦘ 2154/2200 (97.9%)
#> Chain 3 (Sampling): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━╺━⦘ 2114/2200 (96.1%)
#> Chain 4 (Sampling): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━╺━━⦘ 2039/2200 (92.7%)
#> Total   (Sampling): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━╺━⦘ 8457/8800 (96.1%)
#> Elapsed: 1m 14s | ETA: 3s
#> Chain 1 (Sampling): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 2200/2200 (100.0%)
#> Chain 2 (Sampling): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 2200/2200 (100.0%)
#> Chain 3 (Sampling): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 2173/2200 (98.8%)
#> Chain 4 (Sampling): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━╺━⦘ 2091/2200 (95.0%)
#> Total   (Sampling): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━╺⦘ 8664/8800 (98.5%)
#> Elapsed: 1m 16s | ETA: 1s
#> Chain 1 (Sampling): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 2200/2200 (100.0%)
#> Chain 2 (Sampling): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 2200/2200 (100.0%)
#> Chain 3 (Sampling): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 2200/2200 (100.0%)
#> Chain 4 (Sampling): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 2200/2200 (100.0%)
#> Total   (Sampling): ⦗━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━━⦘ 8800/8800 (100.0%)
#> Elapsed: 1m 18s | ETA: 0s
#> NUTS Diagnostics Summary:
#>   Total divergences:         0 
#>   Max tree depth hits:       0 
#>   Min E-BFMI across chains:  1.392 

# Posterior inclusion probabilities
summary(fit)$indicator
#>                      mean         sd        mcse n0->0 n0->1 n1->0
#> intrusion-dreams  1.00000 0.00000000          NA     0     0     0
#> intrusion-flash   1.00000 0.00000000          NA     0     0     0
#> intrusion-upset   0.88675 0.31689815 0.033097148   435    18    18
#> intrusion-physior 0.94625 0.22552370 0.023709256   206     9     9
#> dreams-flash      1.00000 0.00000000          NA     0     0     0
#> dreams-upset      0.99850 0.03870078 0.001367047     4     2     2
#> dreams-physior    0.06700 0.25002200 0.008926280  3649    82    82
#> flash-upset       0.08275 0.27550397 0.010723912  3582    86    86
#> flash-physior     1.00000 0.00000000          NA     0     0     0
#> upset-physior     1.00000 0.00000000          NA     0     0     0
#>                   n1->1     n_eff     Rhat
#> intrusion-dreams   3999        NA       NA
#> intrusion-flash    3999        NA       NA
#> intrusion-upset    3528  91.67655 1.024333
#> intrusion-physior  3775  90.47915 1.145775
#> dreams-flash       3999        NA       NA
#> dreams-upset       3991 801.44296 1.293917
#> dreams-physior      186 784.54068 1.015340
#> flash-upset         245 660.00800 1.016598
#> flash-physior      3999        NA       NA
#> upset-physior      3999        NA       NA

# Posterior pairwise effects
summary(fit)$pairwise
#>                          mean          sd         mcse     n_eff
#> intrusion-dreams  0.630011559 0.001608864 0.0661239670 1689.1933
#> intrusion-flash   0.338858635 0.001426958 0.0611948843 1839.1102
#> intrusion-upset   0.182650623 0.082863156 0.0076871038  116.1977
#> intrusion-physior 0.197786720 0.072082734 0.0054443329  175.2965
#> dreams-flash      0.499269400 0.001282161 0.0597517775 2171.7848
#> dreams-upset      0.231409344 0.055056569 0.0015828592 1209.8560
#> dreams-physior    0.006397386 0.024184302 0.0008659395  779.9954
#> flash-upset       0.008209138 0.027710743 0.0010812828  656.7767
#> flash-physior     0.306644868 0.001209639 0.0530945377 1926.5846
#> upset-physior     0.710281824 0.001426547 0.0593256736 1729.4699
#>                        Rhat
#> intrusion-dreams  1.0031150
#> intrusion-flash   1.0009726
#> intrusion-upset   1.0076115
#> intrusion-physior 1.0122796
#> dreams-flash      1.0009953
#> dreams-upset      1.0028343
#> dreams-physior    1.0053788
#> flash-upset       1.0048742
#> flash-physior     1.0064246
#> upset-physior     0.9999516
# }