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  "Title": "Templated Unified Library for Posterior Approximation in\nBayesian Hierarchical Models",
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  "Description": "A general-purpose engine for fitting Bayesian hierarchical\nmodels with spatial fields, temporal effects, spatially varying\ncoefficients, and multiple inference backends. Scalable spatial\nstructure includes Hilbert space approximate Gaussian processes\n(HSGP; Riutort-Mayol et al. 2023\n<doi:10.1007/s11222-022-10167-2>), nearest-neighbor Gaussian\nprocesses (NNGP; Datta et al. 2016\n<doi:10.1080/01621459.2015.1044091>), intrinsic conditional\nautoregressive models (ICAR; Besag, York, and Mollie 1991\n<doi:10.1007/BF00116466>), the reparameterized\nBesag-York-Mollie model (BYM2; Riebler et al. 2016\n<doi:10.1177/0962280216660421>), and stochastic partial\ndifferential equation fields (SPDE; Lindgren, Rue, and\nLindstrom 2011 <doi:10.1111/j.1467-9868.2011.00777.x>).\nTemporal structure covers random walks, autoregressive\nprocesses, and Gaussian processes. Inference is tiered by\ncorrectness guarantee: exact Hamiltonian Monte Carlo with the\nNo-U-Turn sampler, Laplace and nested Laplace approximations\nwith hyperparameter integration (Rue, Martino, and Chopin 2009\n<doi:10.1111/j.1467-9868.2008.00700.x>), and variational\ninference. Model-specific packages plug observation likelihoods\ninto the engine through a templated C++ callback interface.",
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  "Date/Publication": "2026-10-12 11:04:46 UTC",
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    "message": "fix(summary): no Reliability header without a line under it (#961)\n\n.print_reliability() collects its lines first and prints the section only\nwhen at least one applies. A Laplace fit (kind iid, verdict NA, no band,\nmoment source, acceptance or ELBO) previously ended summary() on a bare\n\"Reliability (iid draws):\" header.\n\nCo-Authored-By: Claude Opus 5.5 <noreply@anthropic.com>\nClaude-Session: https://claude.ai/code/session_016u9terMm7VD2u8RMvvQU1W\n",
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  "_topics": [
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    "hierarchical-models",
    "laplace-approximation",
    "mcmc",
    "spatial",
    "cpp",
    "openmp"
  ],
  "_stars": 0,
  "_contributors": [
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      "count": 1238,
      "uuid": 25009600
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    {
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  ],
  "_userbio": {
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    "type": "user",
    "name": "Gilles Colling",
    "followers": 10,
    "description": "Building tools for ecology in R, C++, and Rust.\nWhy did Microsoft have to buy it. Sadge :("
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  "_downloads": {
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    "source": "https://cranlogs.r-pkg.org/downloads/total/last-month/tulpa"
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  "_devurl": "https://github.com/gcol33/tulpa",
  "_pkgdown": "https://gillescolling.com/tulpa/",
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  "_rbuild": "4.6.1",
  "_assets": [
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    "extra/NEWS.html",
    "extra/NEWS.txt",
    "extra/readme.html",
    "extra/readme.md",
    "extra/tulpa.html",
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  "_homeurl": "https://github.com/gcol33/tulpa",
  "_realowner": "gcol33",
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      "version": "0.2.0",
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    {
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      "date": "2026-09-21"
    },
    {
      "version": "0.6.0",
      "date": "2026-09-24"
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  ],
  "_exports": [
    "adjacency",
    "agq_fit",
    "as_draws",
    "as_draws_array",
    "as_draws_df",
    "as_draws_matrix",
    "as_draws_rvars",
    "auto_grid",
    "auto_grid_place",
    "bayes_R2",
    "bridge_sampling",
    "ccd_grid",
    "ccd_to_theta",
    "ccd_weights",
    "check_adjacency",
    "check_diagnostics",
    "check_model",
    "compare_models",
    "compute_nngp_neighbors",
    "cpo",
    "diagnostic_summary",
    "diagnostics",
    "dic",
    "durbin_watson",
    "energy_bfmi",
    "family_names",
    "field_effects",
    "findbars",
    "fit_spde",
    "fit_st_nested",
    "fixef",
    "geweke_test",
    "glance",
    "hyper_axis_spec",
    "imh_laplace",
    "inference_mode_info",
    "is_auto_grid",
    "latent",
    "latent_factor",
    "latent_factors",
    "mala",
    "mcmc_draws",
    "model_average",
    "moran_i",
    "n_divergent",
    "nobars",
    "node_index",
    "pathfinder",
    "pit_residuals",
    "plot_acf",
    "plot_diagnostics",
    "plot_divergences",
    "plot_energy",
    "plot_ess",
    "plot_map",
    "plot_map_panel",
    "plot_pairs",
    "plot_rhat",
    "pointwise_loglik",
    "post_hoc_lm",
    "posterior_predict",
    "posterior_sample",
    "pp_check",
    "prior_beta",
    "prior_exponential",
    "prior_from_spec",
    "prior_gamma",
    "prior_half_cauchy",
    "prior_half_normal",
    "prior_normal",
    "prior_pc",
    "prior_predict",
    "priors_default",
    "ranef",
    "re_cov_pc_lkj_prior",
    "rubins_pool",
    "sbc",
    "sbc_discrete",
    "sbc_draws",
    "sbc_mixture",
    "sbc_normal",
    "sbc_rank",
    "select_main_params",
    "smooth_effects",
    "sn_cdf",
    "sn_match",
    "sn_quantile",
    "spatial",
    "spatial_bym2",
    "spatial_car",
    "spatial_car_proper",
    "spatial_gp",
    "spatial_multiscale",
    "spatial_range",
    "spatial_rsr",
    "spatial_spde",
    "spatial_spde_custom",
    "spatial_svc",
    "spatiotemporal",
    "spatiotemporal_effects",
    "spatiotemporal_gp",
    "svc",
    "temporal",
    "temporal_ar",
    "temporal_ar1",
    "temporal_ar2",
    "temporal_corr",
    "temporal_gp",
    "temporal_multiscale",
    "temporal_rtr",
    "temporal_rw1",
    "temporal_rw2",
    "temporal_tvc",
    "test_dispersion",
    "test_outliers",
    "test_uniformity",
    "test_zero_inflation",
    "tgmrf",
    "tgmrf_cpp",
    "tidy",
    "tulpa",
    "tulpa_bar_field_replicate",
    "tulpa_bar_field_specs",
    "tulpa_batched_pareto_k",
    "tulpa_build_model_data",
    "tulpa_cache_clear",
    "tulpa_cache_dir",
    "tulpa_check_control",
    "tulpa_criteria",
    "tulpa_draws_array",
    "tulpa_eb",
    "tulpa_em_laplace",
    "tulpa_em_mc",
    "tulpa_ep",
    "tulpa_family",
    "tulpa_fit_summary",
    "tulpa_gaussian",
    "tulpa_gibbs",
    "tulpa_grid_axis",
    "tulpa_grid_log_quad",
    "tulpa_hyper_check_copy_slab",
    "tulpa_hyper_copy_slab_density",
    "tulpa_hyper_draws",
    "tulpa_hyper_grid",
    "tulpa_hyper_grid_supports",
    "tulpa_hyper_slice_home",
    "tulpa_integrator",
    "tulpa_is_spatial_bar",
    "tulpa_iter_progress",
    "tulpa_joint_axis_specs_from_grid",
    "tulpa_joint_grid_batch",
    "tulpa_joint_inner_vcov_blocks",
    "tulpa_joint_inner_vcov_mixture",
    "tulpa_kfold",
    "tulpa_laplace",
    "tulpa_laplace_beta",
    "tulpa_loglik",
    "tulpa_multinomial",
    "tulpa_nested_laplace",
    "tulpa_nested_laplace_joint",
    "tulpa_normalise_weights_safe",
    "tulpa_nuts_beta",
    "tulpa_nuts_spde",
    "tulpa_ordinal",
    "tulpa_outer_grid",
    "tulpa_parse_formula",
    "tulpa_pit",
    "tulpa_posterior_draws",
    "tulpa_powerscale_sensitivity",
    "tulpa_priors",
    "tulpa_profile",
    "tulpa_psis",
    "tulpa_re_aghq",
    "tulpa_re_cov_gibbs",
    "tulpa_re_cov_nested",
    "tulpa_reloo",
    "tulpa_rpg",
    "tulpa_simulate",
    "tulpa_spde_log_hyperprior",
    "tulpa_spde_precision_Q",
    "tulpa_tgmrf",
    "tulpa_theta_matrix",
    "tulpa_variogram",
    "tvc",
    "validate_gp",
    "validate_mode",
    "validate_temporal_multiscale",
    "VarCorr",
    "wbic",
    "with_tulpa_integrator"
  ],
  "_help": [
    {
      "page": "adjacency",
      "title": "Construct a spatial adjacency graph for areal models",
      "topics": [
        "adjacency",
        "adjacency.data.frame",
        "adjacency.default",
        "adjacency.sf",
        "adjacency.SpatRaster",
        "adjacency.stars",
        "print.tulpa_adjacency"
      ]
    },
    {
      "page": "agq_fit",
      "title": "Adaptive Gauss-Hermite quadrature for one-RE GLMMs",
      "topics": [
        "agq_fit"
      ]
    },
    {
      "page": "AIC.tulpa_fit",
      "title": "Information criteria on a tulpa fit",
      "topics": [
        "AIC.tulpa_fit",
        "BIC.tulpa_fit"
      ]
    },
    {
      "page": "as_draws",
      "title": "Posterior draws in the posterior package's format",
      "topics": [
        "as_draws",
        "as_draws.tulpa_fit",
        "as_draws_array",
        "as_draws_array.tulpa_fit",
        "as_draws_df",
        "as_draws_df.tulpa_fit",
        "as_draws_matrix",
        "as_draws_matrix.tulpa_fit",
        "as_draws_rvars",
        "as_draws_rvars.tulpa_fit"
      ]
    },
    {
      "page": "auto_grid",
      "title": "Mark an outer-grid setting as a default rather than a pin",
      "topics": [
        "auto_grid"
      ]
    },
    {
      "page": "auto_grid_place",
      "title": "May the placement pass move a marked outer-grid axis?",
      "topics": [
        "auto_grid_place"
      ]
    },
    {
      "page": "bayes_R2",
      "title": "Bayesian R-squared",
      "topics": [
        "bayes_R2",
        "bayes_R2.tulpa_fit"
      ]
    },
    {
      "page": "bridge_sampling",
      "title": "Bridge sampling for marginal likelihood",
      "topics": [
        "bridge_sampling"
      ]
    },
    {
      "page": "categorical_accessors",
      "title": "Observation-level accessors for categorical fits",
      "topics": [
        "categorical_accessors",
        "fitted.tulpa_categorical",
        "posterior_predict.tulpa_categorical",
        "pp_check.tulpa_categorical",
        "predict.tulpa_categorical",
        "residuals.tulpa_categorical"
      ]
    },
    {
      "page": "check_adjacency",
      "title": "Validate a spatial adjacency matrix",
      "topics": [
        "check_adjacency",
        "print.tulpa_adjacency_check"
      ]
    },
    {
      "page": "check_diagnostics",
      "title": "Quick convergence check",
      "topics": [
        "check_diagnostics"
      ]
    },
    {
      "page": "check_model",
      "title": "Diagnostic panel plot",
      "topics": [
        "check_model",
        "check_model.default"
      ]
    },
    {
      "page": "coef.tulpa_fit",
      "title": "Fixed-effect coefficients",
      "topics": [
        "coef.tulpa_fit"
      ]
    },
    {
      "page": "compare_models",
      "title": "Compare models by information criteria",
      "topics": [
        "compare_models"
      ]
    },
    {
      "page": "confint.tulpa_fit",
      "title": "Credible intervals for the fixed effects",
      "topics": [
        "confint.tulpa_fit"
      ]
    },
    {
      "page": "criteria_doors",
      "title": "DIC, CPO, WAIC and PSIS-LOO on a fit",
      "topics": [
        "cpo",
        "cpo.default",
        "cpo.tulpa_fit",
        "criteria_doors",
        "dic",
        "dic.default",
        "dic.tulpa_fit",
        "loo.tulpa_fit",
        "pointwise_loglik",
        "pointwise_loglik.default",
        "pointwise_loglik.tulpa_fit",
        "waic.tulpa_fit"
      ]
    },
    {
      "page": "diagnostic_summary",
      "title": "Comprehensive diagnostic summary",
      "topics": [
        "diagnostic_summary",
        "print.tulpa_diagnostic_summary"
      ]
    },
    {
      "page": "diagnostics",
      "title": "Posterior diagnostics for a fitted model",
      "topics": [
        "diagnostics",
        "diagnostics.default",
        "diagnostics.sbc",
        "print.laplace_diagnostics"
      ]
    },
    {
      "page": "durbin_watson",
      "title": "Durbin-Watson test for temporal autocorrelation",
      "topics": [
        "durbin_watson"
      ]
    },
    {
      "page": "energy_bfmi",
      "title": "Energy Bayesian Fraction of Missing Information (E-BFMI)",
      "topics": [
        "energy_bfmi"
      ]
    },
    {
      "page": "family_names",
      "title": "Supported R-level family names",
      "topics": [
        "family_names"
      ]
    },
    {
      "page": "field_effects",
      "title": "Spatial field effects per unit",
      "topics": [
        "field_effects"
      ]
    },
    {
      "page": "fit_spde",
      "title": "Fit a spatial model using SPDE Laplace approximation",
      "topics": [
        "fit_spde"
      ]
    },
    {
      "page": "fit_st_nested",
      "title": "Fit an additive spatiotemporal GLM by nested Laplace",
      "topics": [
        "fit_st_nested",
        "print.tulpa_st_nested"
      ]
    },
    {
      "page": "fitted.tulpa_fit",
      "title": "Fitted values (population level)",
      "topics": [
        "fitted.tulpa_fit"
      ]
    },
    {
      "page": "fixef",
      "title": "Fixed-effect coefficients (lme4-compatible)",
      "topics": [
        "fixef",
        "fixef.tulpa_fit"
      ]
    },
    {
      "page": "geweke_test",
      "title": "Geweke convergence test",
      "topics": [
        "geweke_test",
        "print.tulpa_geweke"
      ]
    },
    {
      "page": "glance.tulpa_fit",
      "title": "Model-level summary statistics (broom-compatible)",
      "topics": [
        "glance.tulpa_fit"
      ]
    },
    {
      "page": "imh_laplace",
      "title": "Independence Metropolis-Hastings with Laplace proposal",
      "topics": [
        "imh_laplace"
      ]
    },
    {
      "page": "inference_mode_info",
      "title": "Print inference mode information",
      "topics": [
        "inference_mode_info"
      ]
    },
    {
      "page": "is_auto_grid",
      "title": "Is an outer-grid setting marked as a default?",
      "topics": [
        "is_auto_grid"
      ]
    },
    {
      "page": "latent",
      "title": "Mark an expression as a latent block in a tulpa formula",
      "topics": [
        "latent"
      ]
    },
    {
      "page": "latent_factor",
      "title": "Create a latent factor specification",
      "topics": [
        "latent_factor",
        "print.tulpa_latent"
      ]
    },
    {
      "page": "latent_factors",
      "title": "Extract latent factor posteriors from fit",
      "topics": [
        "latent_factors",
        "latent_factors.default"
      ]
    },
    {
      "page": "logLik.tulpa_fit",
      "title": "The fit's log-scale goodness quantity",
      "topics": [
        "logLik.tulpa_fit"
      ]
    },
    {
      "page": "mala",
      "title": "Metropolis-adjusted Langevin algorithm (MALA)",
      "topics": [
        "mala"
      ]
    },
    {
      "page": "mcmc_draws",
      "title": "MCMC chain draws from a fit",
      "topics": [
        "mcmc_draws"
      ]
    },
    {
      "page": "model_average",
      "title": "Model-averaged predictions",
      "topics": [
        "model_average"
      ]
    },
    {
      "page": "moran_i",
      "title": "Moran's I test for spatial autocorrelation in residuals",
      "topics": [
        "moran_i"
      ]
    },
    {
      "page": "n_divergent",
      "title": "Number of divergent transitions",
      "topics": [
        "n_divergent"
      ]
    },
    {
      "page": "nobs.tulpa_fit",
      "title": "Number of observations in a tulpa fit",
      "topics": [
        "nobs.tulpa_fit"
      ]
    },
    {
      "page": "node_index",
      "title": "Map cell identifiers to graph node indices",
      "topics": [
        "node_index"
      ]
    },
    {
      "page": "pathfinder",
      "title": "Pathfinder: variational warm-start via L-BFGS + ELBO scoring",
      "topics": [
        "pathfinder"
      ]
    },
    {
      "page": "pit_residuals",
      "title": "PIT (probability integral transform) residuals",
      "topics": [
        "pit_residuals",
        "pit_residuals.default"
      ]
    },
    {
      "page": "plot_acf",
      "title": "Plot autocorrelation functions",
      "topics": [
        "plot_acf"
      ]
    },
    {
      "page": "plot_divergences",
      "title": "Plot divergent transitions",
      "topics": [
        "plot_divergences"
      ]
    },
    {
      "page": "plot_energy",
      "title": "Plot energy diagnostic (E-BFMI)",
      "topics": [
        "plot_energy"
      ]
    },
    {
      "page": "plot_ess",
      "title": "Plot effective sample size diagnostic",
      "topics": [
        "plot_ess"
      ]
    },
    {
      "page": "plot_map",
      "title": "Plot spatial predictions as a map",
      "topics": [
        "plot_map"
      ]
    },
    {
      "page": "plot_map_panel",
      "title": "Plot multiple maps in a grid",
      "topics": [
        "plot_map_panel"
      ]
    },
    {
      "page": "plot_pairs",
      "title": "Plot bivariate parameter posteriors (pairs plot)",
      "topics": [
        "plot_pairs"
      ]
    },
    {
      "page": "plot_rhat",
      "title": "Plot Rhat convergence diagnostic",
      "topics": [
        "plot_rhat"
      ]
    },
    {
      "page": "plot.tulpa_fit",
      "title": "Plot fixed-effect posteriors",
      "topics": [
        "plot.tulpa_fit"
      ]
    },
    {
      "page": "plot.tulpa_prior_predict",
      "title": "Plot method for tulpa_prior_predict",
      "topics": [
        "plot.tulpa_prior_predict"
      ]
    },
    {
      "page": "plot.tulpa_st_summary",
      "title": "Plot method for spatiotemporal effects",
      "topics": [
        "plot.tulpa_st_summary"
      ]
    },
    {
      "page": "plot.tulpa_svc_posterior",
      "title": "Plot method for tulpa_svc_posterior",
      "topics": [
        "plot.tulpa_svc_posterior"
      ]
    },
    {
      "page": "plot.tulpa_temporal_posterior",
      "title": "Plot method for tulpa_temporal_posterior",
      "topics": [
        "plot.tulpa_temporal_posterior"
      ]
    },
    {
      "page": "plot.tulpa_tvc_posterior",
      "title": "Plot method for tulpa_tvc_posterior",
      "topics": [
        "plot.tulpa_tvc_posterior"
      ]
    },
    {
      "page": "post_hoc_lm",
      "title": "Fit a post-hoc linear model on estimated parameters",
      "topics": [
        "post_hoc_lm",
        "print.post_hoc_lm"
      ]
    },
    {
      "page": "posterior_predict",
      "title": "Posterior predictive replicates",
      "topics": [
        "posterior_predict",
        "posterior_predict.tulpa_fit"
      ]
    },
    {
      "page": "posterior_sample",
      "title": "Posterior parameter sample from a fit",
      "topics": [
        "posterior_sample"
      ]
    },
    {
      "page": "pp_check",
      "title": "Posterior predictive check",
      "topics": [
        "pp_check",
        "pp_check.tulpa_fit"
      ]
    },
    {
      "page": "predict.tulpa_fit",
      "title": "Predict at new covariate values (population level)",
      "topics": [
        "predict.tulpa_fit"
      ]
    },
    {
      "page": "summary.tulpa_fit",
      "title": "Summarize and print a tulpa fit",
      "topics": [
        "coef.summary.tulpa_fit",
        "print.summary.tulpa_fit",
        "print.tulpa_fit",
        "summary.tulpa_fit"
      ]
    },
    {
      "page": "tulpa_nested_laplace",
      "title": "Nested Laplace approximation for latent Gaussian models",
      "topics": [
        "print.tulpa_nested_laplace",
        "tulpa_nested_laplace"
      ]
    },
    {
      "page": "tulpa_nested_laplace_joint",
      "title": "Joint multi-likelihood nested Laplace approximation",
      "topics": [
        "print.tulpa_nested_laplace_joint",
        "print.tulpa_nested_laplace_joint_multi",
        "tulpa_nested_laplace_joint"
      ]
    },
    {
      "page": "temporal",
      "title": "Extract temporal effects from a fitted model",
      "topics": [
        "print.tulpa_temporal_field",
        "print.tulpa_temporal_field_fit",
        "print.tulpa_temporal_posterior",
        "print.tulpa_temporal_summary",
        "summary.tulpa_temporal_posterior",
        "temporal",
        "temporal.tulpa_fit"
      ]
    },
    {
      "page": "prior_beta",
      "title": "Beta prior",
      "topics": [
        "prior_beta"
      ]
    },
    {
      "page": "prior_exponential",
      "title": "Exponential prior",
      "topics": [
        "prior_exponential"
      ]
    },
    {
      "page": "prior_from_spec",
      "title": "Build a 'prior' list for 'tulpa_nested_laplace()' from a tulpa spec object",
      "topics": [
        "prior_from_spec"
      ]
    },
    {
      "page": "prior_gamma",
      "title": "Gamma prior",
      "topics": [
        "prior_gamma"
      ]
    },
    {
      "page": "prior_half_cauchy",
      "title": "Half-Cauchy prior",
      "topics": [
        "prior_half_cauchy"
      ]
    },
    {
      "page": "prior_half_normal",
      "title": "Half-normal prior",
      "topics": [
        "prior_half_normal"
      ]
    },
    {
      "page": "prior_normal",
      "title": "Normal prior",
      "topics": [
        "prior_normal"
      ]
    },
    {
      "page": "prior_pc",
      "title": "Penalized complexity (PC) prior",
      "topics": [
        "prior_pc"
      ]
    },
    {
      "page": "prior_predict",
      "title": "Prior predictive simulation",
      "topics": [
        "print.tulpa_prior_predict",
        "prior_predict"
      ]
    },
    {
      "page": "priors_default",
      "title": "Show default priors for a tulpa family",
      "topics": [
        "priors_default"
      ]
    },
    {
      "page": "ranef",
      "title": "Random-effect summaries",
      "topics": [
        "ranef",
        "ranef.tulpa_fit"
      ]
    },
    {
      "page": "re_cov_pc_lkj_prior",
      "title": "PC + LKJ hyperprior for a random-effect covariance",
      "topics": [
        "re_cov_pc_lkj_prior"
      ]
    },
    {
      "page": "residuals.tulpa_fit",
      "title": "Residuals from a tulpa fit",
      "topics": [
        "residuals.tulpa_fit"
      ]
    },
    {
      "page": "rubins_pool",
      "title": "Pool multiple imputation draws via Rubin's rules",
      "topics": [
        "rubins_pool"
      ]
    },
    {
      "page": "sbc",
      "title": "Simulation-based calibration",
      "topics": [
        "plot.sbc",
        "print.sbc",
        "print.sbc_summary",
        "sbc",
        "sbc.character",
        "sbc.default",
        "summary.sbc"
      ]
    },
    {
      "page": "sbc_predictive",
      "title": "Predictive shapes an SBC fitter reports",
      "topics": [
        "sbc_discrete",
        "sbc_draws",
        "sbc_mixture",
        "sbc_normal",
        "sbc_predictive",
        "sbc_rank"
      ]
    },
    {
      "page": "select_main_params",
      "title": "Select the \"main\" model parameters for diagnostic display",
      "topics": [
        "select_main_params"
      ]
    },
    {
      "page": "simulate.tulpa_fit",
      "title": "Simulate responses from a fitted tulpa model",
      "topics": [
        "simulate.tulpa_fit"
      ]
    },
    {
      "page": "smooth_effects",
      "title": "Extract fitted covariate smooths",
      "topics": [
        "smooth_effects"
      ]
    },
    {
      "page": "spatial",
      "title": "Areal spatially varying coefficient field",
      "topics": [
        "print.tulpa_spatial_field",
        "print.tulpa_spatial_field_fit",
        "spatial"
      ]
    },
    {
      "page": "spatial_bym2",
      "title": "BYM2 spatial structure",
      "topics": [
        "spatial_bym2"
      ]
    },
    {
      "page": "spatial_car",
      "title": "CAR / ICAR spatial structure",
      "topics": [
        "spatial_car"
      ]
    },
    {
      "page": "spatial_car_proper",
      "title": "Proper CAR spatial structure",
      "topics": [
        "spatial_car_proper"
      ]
    },
    {
      "page": "spatial_gp",
      "title": "Gaussian process spatial structure (NNGP)",
      "topics": [
        "print.tulpa_gp",
        "print.tulpa_hsgp",
        "spatial_gp"
      ]
    },
    {
      "page": "spatial_multiscale",
      "title": "Multi-scale Gaussian process spatial structure",
      "topics": [
        "print.tulpa_multiscale",
        "spatial_multiscale"
      ]
    },
    {
      "page": "spatial_range",
      "title": "Extract spatial range and variance from a fitted spatial model",
      "topics": [
        "spatial_range"
      ]
    },
    {
      "page": "spatial_rsr",
      "title": "Restricted spatial regression (RSR)",
      "topics": [
        "print.tulpa_rsr",
        "spatial_rsr"
      ]
    },
    {
      "page": "spatial_spde",
      "title": "SPDE spatial field (Matern via triangular mesh)",
      "topics": [
        "spatial_spde"
      ]
    },
    {
      "page": "spatial_spde_custom",
      "title": "SPDE spatial field from custom matrices",
      "topics": [
        "spatial_spde_custom"
      ]
    },
    {
      "page": "spatial_svc",
      "title": "Spatially varying coefficient structure",
      "topics": [
        "print.tulpa_svc",
        "spatial_svc"
      ]
    },
    {
      "page": "spatiotemporal",
      "title": "Spatiotemporal interaction specifications for tulpa",
      "topics": [
        "spatiotemporal"
      ]
    },
    {
      "page": "spatiotemporal_effects",
      "title": "Extract spatiotemporal effects from fitted model",
      "topics": [
        "print.tulpa_st_summary",
        "spatiotemporal_effects",
        "spatiotemporal_effects.default"
      ]
    },
    {
      "page": "spatiotemporal_gp",
      "title": "Non-separable spatiotemporal GP",
      "topics": [
        "spatiotemporal_gp"
      ]
    },
    {
      "page": "svc",
      "title": "Extract spatially-varying coefficients from a fitted model",
      "topics": [
        "print.tulpa_svc_posterior",
        "print.tulpa_svc_summary",
        "summary.tulpa_svc_posterior",
        "svc",
        "svc.tulpa_fit"
      ]
    },
    {
      "page": "temporal_ar",
      "title": "AR(p) temporal latent field (general-order autoregressive)",
      "topics": [
        "print.tulpa_temporal_latent_block",
        "temporal_ar"
      ]
    },
    {
      "page": "temporal_ar1",
      "title": "AR1 temporal structure (first-order autoregressive)",
      "topics": [
        "temporal_ar1"
      ]
    },
    {
      "page": "temporal_ar2",
      "title": "AR(2) temporal latent field (second-order autoregressive)",
      "topics": [
        "temporal_ar2"
      ]
    },
    {
      "page": "temporal_corr",
      "title": "Extract temporal correlation parameters from a fitted model",
      "topics": [
        "temporal_corr"
      ]
    },
    {
      "page": "temporal_gp",
      "title": "Gaussian process temporal structure",
      "topics": [
        "print.tulpa_temporal_gp",
        "temporal_gp"
      ]
    },
    {
      "page": "temporal_multiscale",
      "title": "Multi-scale temporal structure",
      "topics": [
        "print.tulpa_temporal_multiscale",
        "temporal_multiscale"
      ]
    },
    {
      "page": "temporal_rtr",
      "title": "Restricted temporal regression (RTR)",
      "topics": [
        "temporal_rtr"
      ]
    },
    {
      "page": "temporal_rw1",
      "title": "RW1 temporal structure (first-order random walk)",
      "topics": [
        "temporal_rw1"
      ]
    },
    {
      "page": "temporal_rw2",
      "title": "RW2 temporal structure (second-order random walk)",
      "topics": [
        "temporal_rw2"
      ]
    },
    {
      "page": "temporal_tvc",
      "title": "Time-varying coefficient structure",
      "topics": [
        "print.tulpa_tvc",
        "temporal_tvc"
      ]
    },
    {
      "page": "test_dispersion",
      "title": "Test for over- or underdispersion",
      "topics": [
        "test_dispersion",
        "test_dispersion.default"
      ]
    },
    {
      "page": "test_outliers",
      "title": "Test for outliers (simulation envelope)",
      "topics": [
        "test_outliers",
        "test_outliers.default"
      ]
    },
    {
      "page": "test_uniformity",
      "title": "Test uniformity of PIT residuals",
      "topics": [
        "test_uniformity"
      ]
    },
    {
      "page": "test_zero_inflation",
      "title": "Test for zero inflation",
      "topics": [
        "test_zero_inflation",
        "test_zero_inflation.default"
      ]
    },
    {
      "page": "tgmrf",
      "title": "User-defined GMRF latent block",
      "topics": [
        "print.tgmrf",
        "summary.tgmrf",
        "tgmrf"
      ]
    },
    {
      "page": "tgmrf_cpp",
      "title": "User-defined GMRF latent block, compiled C++ backend",
      "topics": [
        "tgmrf_cpp"
      ]
    },
    {
      "page": "tidy.tulpa_fit",
      "title": "Tidy fixed-effect table (broom-compatible)",
      "topics": [
        "tidy.tulpa_fit"
      ]
    },
    {
      "page": "tulpa",
      "title": "Fit a tulpa model",
      "topics": [
        "tulpa"
      ]
    },
    {
      "page": "tulpa_bar_field_replicate",
      "title": "Replicate an areal graph across the levels of a factor (replicated CAR)",
      "topics": [
        "tulpa_bar_field_replicate"
      ]
    },
    {
      "page": "tulpa_bar_field_specs",
      "title": "Expand a varying-coefficient bar into per-column field specs",
      "topics": [
        "tulpa_bar_field_specs"
      ]
    },
    {
      "page": "tulpa_batched_pareto_k",
      "title": "Outer Pareto-k-hat for a batched-target grid-integrated fit",
      "topics": [
        "tulpa_batched_pareto_k"
      ]
    },
    {
      "page": "tulpa_cache_clear",
      "title": "Remove compiled blocks from the tgmrf_cpp() cache",
      "topics": [
        "tulpa_cache_clear"
      ]
    },
    {
      "page": "tulpa_cache_dir",
      "title": "Default cache directory for 'tgmrf_cpp()'-compiled DLLs",
      "topics": [
        "tulpa_cache_dir"
      ]
    },
    {
      "page": "tulpa_check_control",
      "title": "Validate a 'control = list()' surface against its canonical key set",
      "topics": [
        "tulpa_check_control"
      ]
    },
    {
      "page": "tulpa_criteria",
      "title": "Model criteria from a pointwise log-likelihood",
      "topics": [
        "print.tulpa_criteria",
        "tulpa_criteria"
      ]
    },
    {
      "page": "tulpa_diagnostics",
      "title": "Simulation-based diagnostics for tulpa models",
      "topics": [
        "tulpa_diagnostics"
      ]
    },
    {
      "page": "tulpa_draws_array",
      "title": "Posterior draws as a 3D array",
      "topics": [
        "tulpa_draws_array"
      ]
    },
    {
      "page": "tulpa_eb",
      "title": "Empirical-Bayes random-effect covariances",
      "topics": [
        "tulpa_eb"
      ]
    },
    {
      "page": "tulpa_em_laplace",
      "title": "Fit a latent-variable model via EM + Laplace approximation",
      "topics": [
        "tulpa_em_laplace"
      ]
    },
    {
      "page": "tulpa_em_mc",
      "title": "Generic Monte-Carlo EM driver",
      "topics": [
        "tulpa_em_mc"
      ]
    },
    {
      "page": "tulpa_ep",
      "title": "Expectation-Propagation fit for a GLM",
      "topics": [
        "tulpa_ep"
      ]
    },
    {
      "page": "tulpa_family",
      "title": "Construct a minimal tulpa_family for simulation",
      "topics": [
        "print.tulpa_family",
        "tulpa_family"
      ]
    },
    {
      "page": "tulpa_fit_summary",
      "title": "Build the summary of a tulpa fit around a coefficient table",
      "topics": [
        "tulpa_fit_summary"
      ]
    },
    {
      "page": "tulpa_formula",
      "title": "Formula parsing for tulpa models",
      "topics": [
        "tulpa_formula"
      ]
    },
    {
      "page": "tulpa_gaussian",
      "title": "Fit a Gaussian linear model via tulpa's generic engine",
      "topics": [
        "tulpa_gaussian"
      ]
    },
    {
      "page": "tulpa_gibbs",
      "title": "Fit via Polya-Gamma Gibbs sampler",
      "topics": [
        "tulpa_gibbs"
      ]
    },
    {
      "page": "tulpa_grid_axis",
      "title": "Read a default nested-Laplace outer-grid axis",
      "topics": [
        "tulpa_grid_axis"
      ]
    },
    {
      "page": "tulpa_grid_log_quad",
      "title": "Per-cell log quadrature weight of an outer grid",
      "topics": [
        "tulpa_grid_log_quad"
      ]
    },
    {
      "page": "tulpa_hyper_check_copy_slab",
      "title": "Validate/default a copy-scale slab measure choice",
      "topics": [
        "tulpa_hyper_check_copy_slab"
      ]
    },
    {
      "page": "tulpa_hyper_copy_slab_density",
      "title": "The copy-scale axis's PC-prior density",
      "topics": [
        "tulpa_hyper_copy_slab_density"
      ]
    },
    {
      "page": "tulpa_hyper_draws",
      "title": "Hyperparameter draws from a nested-Laplace fit",
      "topics": [
        "tulpa_hyper_draws"
      ]
    },
    {
      "page": "tulpa_hyper_grid",
      "title": "Outer hyperparameter-grid integration with a user-supplied inner fit",
      "topics": [
        "tulpa_hyper_grid"
      ]
    },
    {
      "page": "tulpa_hyper_grid_supports",
      "title": "Per-axis integrated support of an outer grid",
      "topics": [
        "tulpa_hyper_grid_supports"
      ]
    },
    {
      "page": "tulpa_hyper_slice_home",
      "title": "Per-cell refinement tag of an outer grid",
      "topics": [
        "tulpa_hyper_slice_home"
      ]
    },
    {
      "page": "tulpa_integrator",
      "title": "Select the symplectic integrator for HMC and NUTS",
      "topics": [
        "tulpa_integrator"
      ]
    },
    {
      "page": "tulpa_is_spatial_bar",
      "title": "Recognize an inline varying-coefficient bar",
      "topics": [
        "tulpa_is_spatial_bar"
      ]
    },
    {
      "page": "tulpa_joint_axis_specs_from_grid",
      "title": "Rebuild axis specs from an already-assembled outer grid",
      "topics": [
        "tulpa_joint_axis_specs_from_grid"
      ]
    },
    {
      "page": "tulpa_joint_grid_batch",
      "title": "Fit multiple joint nested-Laplace models through one fused grid solve",
      "topics": [
        "tulpa_joint_grid_batch"
      ]
    },
    {
      "page": "tulpa_joint_inner_vcov_blocks",
      "title": "Per-cell constrained inverse-precision sub-block for a joint fit",
      "topics": [
        "tulpa_joint_inner_vcov_blocks"
      ]
    },
    {
      "page": "tulpa_joint_inner_vcov_mixture",
      "title": "Outer-grid posterior covariance of a joint fit's latent coordinates",
      "topics": [
        "tulpa_joint_inner_vcov_mixture"
      ]
    },
    {
      "page": "tulpa_kfold",
      "title": "K-fold cross-validation for a tulpa fit",
      "topics": [
        "tulpa_kfold"
      ]
    },
    {
      "page": "tulpa_laplace",
      "title": "Fit a model via Laplace approximation",
      "topics": [
        "tulpa_laplace"
      ]
    },
    {
      "page": "tulpa_laplace_beta",
      "title": "Fit a beta-regression model via Laplace, estimating the precision",
      "topics": [
        "tulpa_laplace_beta"
      ]
    },
    {
      "page": "tulpa_latent",
      "title": "Latent factor specification for unmeasured confounders",
      "topics": [
        "tulpa_latent"
      ]
    },
    {
      "page": "tulpa_loglik",
      "title": "Streaming pointwise log-likelihood",
      "topics": [
        "tulpa_loglik"
      ]
    },
    {
      "page": "tulpa_multinomial",
      "title": "Multinomial (nominal K-class) logistic regression via Laplace",
      "topics": [
        "print.tulpa_multinomial",
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