Inference modes: the three tiers and how to choose one

library(tulpa)

Why the inference choice matters

A fitted model is two things at once: a set of parameter estimates and a claim about how certain those estimates are. The estimates often agree across methods. The uncertainty does not. A Laplace approximation, a gradient sampler, and a variational fit can return the same posterior mean for a slope and still disagree about the width of its credible interval, the shape of its tail, and whether two parameters are correlated. The number you report as a 95% interval depends on which method produced it, and the methods do not all mean the same thing by “95%”.

Most fitting interfaces hide this. You call a fit function, a default runs, and the output looks identical no matter which numerical engine did the work. The default is usually a reasonable one. The problem is that it is silent: when the default is wrong for your model, nothing in the output tells you, and the failure shows up downstream as an interval that is too narrow or a correlation that was never there.

tulpa makes the choice explicit. Every fit carries the mode it ran, the tier that mode belongs to, and a recorded reason for the selection. The tier carries the weight, because it pins down what the credible intervals mean. Two backends sharing a tier offer the same epistemic warranty; a backend one rung higher offers a weaker one. That warranty reads straight off the fitted object, and mode = "auto" picks among tiers by a rule you can inspect rather than by a buried heuristic.

This vignette walks through the tier system, the automatic selection rule, and the backends you can drive from R. It fits one model three ways and compares the estimates and the timing, so the cost-versus-guarantee trade is concrete rather than abstract. It closes with practical guidance: which mode to reach for, and at what point to move up a tier.

The three tiers

tulpa sorts every inference backend into one of three tiers. The tier is not a speed label. The tier states how correct the uncertainty a backend reports actually is. inference_mode_info() prints the full map, tier by tier, with the guarantee attached to each.

inference_mode_info()
#> tulpa Inference Modes
#> ========================
#> 
#> Modes encode EPISTEMIC GUARANTEES, not just runtime characteristics.
#> 
#> TIER 1: EXACT
#>   Asymptotically correct posterior inference
#>   Guarantee: Credible intervals interpretable as posterior uncertainty
#>   Backends: hmc, ess, pg, gibbs, re_cov_gibbs, smc, imh_laplace, mala
#>   Note: Reference standard
#> 
#> TIER 2: STRUCTURED
#>   Accurate inference conditional on structural assumptions
#>   Guarantee: Correct if model meets structural assumptions
#>   Backends: laplace, re_cov_nested, pathfinder, agq, ep, eb, nested_laplace, nested_laplace_joint, spde
#>   Note: Controlled approximation, not heuristics
#> 
#> TIER 3: OPTIMIZED
#>   Optimization-based approximate inference
#>   Guarantee: No general correctness guarantee
#>   Backends: sghmc, sgld, mclmc, vi
#>   Note: Uncertainty often underestimated; requires explicit opt-in
#> 
#> AUTO MODE
#>   Selects between Tier 1 and Tier 2 only.
#>   NEVER silently chooses Tier 3 (Optimized).
#>   Contract: 'Use the most reliable method expected to finish.'
#> 
#> Usage (by tier):
#>   tulpa(..., mode = 'auto')       # Tier 1 or 2 (default)
#>   tulpa(..., mode = 'exact')      # Tier 1
#>   tulpa(..., mode = 'structured') # Tier 2
#>   tulpa(..., mode = 'optimized')  # Tier 3 (explicit opt-in)
#> 
#> Backends ([R] = callable from R, [C-ABI] = model-package kernel only):
#>   [R]     mode = 'hmc'        # Tier 1 (Exact)
#>   [R]     mode = 'ess'        # Tier 1 (Exact)
#>   [C-ABI] mode = 'pg'         # Tier 1 (Exact)
#>   [R]     mode = 'gibbs'      # Tier 1 (Exact)
#>   [R]     mode = 're_cov_gibbs' # Tier 1 (Exact)
#>   [R]     mode = 'smc'        # Tier 1 (Exact)
#>   [R]     mode = 'imh_laplace' # Tier 1 (Exact)
#>   [R]     mode = 'mala'       # Tier 1 (Exact)
#>   [R]     mode = 'laplace'    # Tier 2 (Structured)
#>   [R]     mode = 're_cov_nested' # Tier 2 (Structured)
#>   [R]     mode = 'pathfinder' # Tier 2 (Structured)
#>   [R]     mode = 'agq'        # Tier 2 (Structured)
#>   [R]     mode = 'ep'         # Tier 2 (Structured)
#>   [R]     mode = 'eb'         # Tier 2 (Structured)
#>   [R]     mode = 'nested_laplace' # Tier 2 (Structured)
#>   [R]     mode = 'nested_laplace_joint' # Tier 2 (Structured)
#>   [R]     mode = 'spde'       # Tier 2 (Structured)
#>   [R]     mode = 'sghmc'      # Tier 3 (Optimized)
#>   [R]     mode = 'sgld'       # Tier 3 (Optimized)
#>   [R]     mode = 'mclmc'      # Tier 3 (Optimized)
#>   [R]     mode = 'vi'         # Tier 3 (Optimized)

The output names three tiers. Read them as a ladder of promises.

Tier 1, Exact. A Tier 1 backend draws from the posterior and its credible intervals are interpretable as posterior uncertainty, up to Monte Carlo error that shrinks as you draw more samples. This is the reference standard: if you want an interval you can quote without an asterisk, a Tier 1 fit gives it to you. The cost is iteration. Every draw is a likelihood evaluation, sometimes a gradient evaluation, and you need many of them. MALA and the independence-MH samplers live here, along with the Polya-Gamma Gibbs sampler for the families that admit it.

Tier 2, Structured. A Tier 2 backend is accurate conditional on an explicit structural assumption. The Laplace approximation assumes the posterior is Gaussian near its mode. Pathfinder fits a Gaussian along an optimisation path. When the assumption holds, and for a latent Gaussian model with enough data it usually does, the fit is fast and its intervals are close to the exact ones. When the assumption fails, the failure is predictable: a skewed posterior gets symmetrised, a heavy tail gets clipped. You can see it coming from the model, and you can check it by comparing against a Tier 1 fit. The promise is conditional, and the condition is stated.

Tier 3, Optimized. A Tier 3 backend gives a point estimate and a covariance from an optimisation objective with no general correctness guarantee on the uncertainty. The mode is usually good. The spread is often too small, the tails unreliable, the correlations approximate, and the failure is typically silent. Generic variational inference sits here. Tier 3 is optimisation, not sampling, and tulpa treats it as a deliberate choice rather than a fallback, reachable only by asking for it by name.

The ladder runs from a promise you can quote, through a promise with a stated condition, to a tool with no promise on its uncertainty. Moving up a tier costs computation and buys a stronger guarantee. The point of the system is that the guarantee is never traded away without a record of the trade.

The design follows four rules, and they are worth stating because they explain why the tier shows up on every fit. First, a mode changes the meaning of the output, not just its runtime: an interval from the optimised tier does not mean what an interval from the exact tier means, so the two must be distinguishable. Second, the mode is always visible in the result. Third, there is no silent upgrading or downgrading between tiers; if an exact fit fails, the call errors rather than quietly substituting a structured approximation and handing back numbers that look the same but promise less. Fourth, adding a backend slots it into an existing tier and never invents a new kind of promise, so the three guarantees above are the whole vocabulary. Tier membership for each backend is derived from a single registry, which is why inference_mode_info() and the field on a fit can never drift apart.

How `mode = "auto"` chooses

The default mode is "auto". Its contract is one sentence: use the most reliable method that is expected to finish for this model. Two parts of that sentence carry weight. “Most reliable” means it prefers Tier 1 over Tier 2 when both are feasible. “Expected to finish” means it steps down to Tier 2 when a Tier 1 sampler would be too slow to be practical. The rule never reaches Tier 3. The optimisation tier is never selected on your behalf, because its silent under-coverage is exactly the failure an automatic default should not introduce.

The decision is deterministic and depends on the model, not on chance. Two properties drive it: structure and size. A latent prior block goes to nested Laplace, the designed Tier 2 hot path for latent Gaussian structure. A spatial field branches by its kind, with a binomial areal field handed to the exact component-wise Polya-Gamma Gibbs sampler (Tier 1) and most other fields sent to nested-Laplace integration of the spatial hyperparameter (Tier 2). An ordinary model carrying neither spatial nor latent structure heads for a full sampler, unless the dataset grows past tens of thousands of rows, where the Laplace path takes over because a sampler would no longer finish quickly enough to serve as a default.

Every fit records the reason. The selection_reason field is a short string explaining the branch that was taken, and backend and inference_tier report the result. Take a binomial areal model: twenty regions on a ring, each adjacent to its two neighbours, with a latent field that varies smoothly around the loop.

n <- 400L; K <- 20L
W <- matrix(0, K, K)
for (i in 1:K) { j <- if (i < K) i + 1L else 1L; W[i, j] <- W[j, i] <- 1 }
x      <- rnorm(n)
region <- factor(sample(1:K, n, replace = TRUE))
field  <- as.numeric(scale(sin(2 * pi * (1:K) / K)))[region]
ds <- data.frame(y = rbinom(n, 1, plogis(-0.2 + 0.7 * x + field)),
                 x = x, region = region)

Fitting with mode = "auto" lets the rule pick. For a binomial ICAR field it selects the exact Polya-Gamma Gibbs sampler, a Tier 1 backend, and says so.

fit_auto <- tulpa(y ~ x + spatial(region), data = ds, family = "binomial",
                  spatial = list(type = "icar", adjacency = W), mode = "auto")
c(backend = fit_auto$backend, tier = fit_auto$inference_tier)
#> backend    tier 
#> "gibbs"     "1"
fit_auto$selection_reason
#> [1] "icar spatial model (binomial Polya-Gamma Gibbs)"

Auto reached for the most reliable method, Tier 1, because for this model class an exact sampler is both available and fast. A binomial likelihood with an areal field admits the Polya-Gamma data augmentation, which turns each conditional update into a Gaussian draw, and the field components update one at a time. Those component-wise updates sidestep the dimensionality that would slow a general gradient sampler on a field of this size, so the exact route is also the practical one. The rule saw a case where the strongest guarantee was affordable and took it. Now change the situation to a large, plain Gaussian model where a full sampler would be slow.

nL <- 60000L
xL <- rnorm(nL)
dL <- data.frame(y = 0.5 + 1.2 * xL + rnorm(nL, sd = 0.8), x = xL)
fit_big <- tulpa(y ~ x, data = dL, family = "gaussian", mode = "auto",
                 phi = 0.8^2)
c(backend = fit_big$backend, tier = fit_big$inference_tier)
#>   backend      tier 
#> "laplace"       "2"
fit_big$selection_reason
#> [1] "large dataset (n=60,000)"

Here auto stepped down to the Laplace path. The reason names the dataset size: at sixty thousand rows the rule judges a full sampler too slow to be the sensible default and takes the Tier 2 route instead, which for a Gaussian likelihood is exact anyway. The estimates land on the truth.

coef(fit_big)
#> (Intercept)           x 
#>    0.506521    1.203624

Two models, two different branches, each recorded. The same rule that picked an exact sampler for the small spatial model picked the structured approximation for the large plain one, and in both cases the choice traces to a property of the model you can name in advance. That predictability is the whole point. You are never left guessing which engine ran, and you can reason about what auto will do before you call it. When you disagree with the choice, override it by passing a tier ("exact" or "structured") or a backend name directly, and the fit records that you overrode it rather than papering over the change.

The R-callable backends

A backend is the concrete algorithm a mode resolves to. tulpa registers more backends than it exposes from R: several ship a C++ kernel reachable by model packages that link against tulpa, with no R entry point yet. The inference_mode_info() map tags each one, [R] for callable from R and [C-ABI] for the kernel-only ones. Asking for a C-ABI backend by name from R errors with a message that names the missing entry point rather than pretending to dispatch. This section covers the backends you can drive from R today.

laplace (Tier 2)

The workhorse. mode = "laplace" finds the posterior mode and the curvature there, then reports a Gaussian centred at the mode with that curvature as its precision. For a Gaussian likelihood this is exact and returns immediately. For other families it is the Gaussian approximation to the posterior, accurate when the posterior is close to Gaussian.

set.seed(101)
g  <- factor(sample(1:12, n, replace = TRUE))
u  <- rnorm(12, sd = 0.6)
db <- data.frame(y = rbinom(n, 1, plogis(-0.3 + 1.0 * x + u[g])),
                 x = x, g = g)
fit_lap <- tulpa(y ~ x + (1 | g), data = db, family = "binomial",
                 mode = "laplace", sigma_re = 0.6)
coef(fit_lap)
#> (Intercept)           x 
#>  -0.2581721   0.8021130

The Laplace fit is deterministic, carries no Monte Carlo error, and its logLik() is the approximate log marginal likelihood, the model evidence that compare_models() uses. This is the first fit to run, every time, as the fast sanity check and the comparison baseline.

pathfinder (Tier 2)

Pathfinder runs a quasi-Newton optimiser toward the mode, fits a Gaussian at the optimum from the inverse-Hessian estimate it accumulates along the way, and returns draws from that Gaussian plus an ELBO score. It sits in the same tier as Laplace: the output is a Gaussian approximation, not exact samples. What it adds over a plain Laplace fit is a set of draws and a diagnostic. The draws make it a drop-in where downstream code expects samples, and the ELBO gauges how well the Gaussian fits.

fit_pf <- tulpa(y ~ x + (1 | g), data = db, family = "binomial",
                mode = "pathfinder", sigma_re = 0.6,
                control = list(n_draws = 450))
coef(fit_pf)
#> (Intercept)           x 
#>  -0.2313448   0.7959445
fit_pf$elbo
#> [1] -252.4946

Pathfinder fits three situations well: you want draws but not the cost of full MCMC, you need a warm start for a sampler, or you want a quick check on whether the Laplace Gaussian is reasonable.

mala (Tier 1)

The Metropolis-adjusted Langevin algorithm is a gradient sampler. Each step proposes a move along the gradient of the log posterior and accepts or rejects it with a Metropolis step, which makes the chain asymptotically correct. Each iteration runs cheaper than full Hamiltonian Monte Carlo because no leapfrog trajectory needs integrating, and mixing beats a random walk because the drift term nudges proposals toward higher density. The step size adapts during warmup toward an acceptance rate near 0.574.

fit_mala <- tulpa(y ~ x + (1 | g), data = db, family = "binomial",
                  mode = "mala", sigma_re = 0.6,
                  control = list(n_iter = 450, warmup = 150))
coef(fit_mala)
#> (Intercept)           x 
#>   -0.330130    0.805812
fit_mala$mean_accept
#> [1] 0.5493333

MALA earns its keep when you want exact posterior moments at moderate dimension and the posterior is not so badly scaled that a single step size struggles across all directions. The acceptance rate is the diagnostic: far below the target means the chain needs a smaller step or more warmup.

imh_laplace (Tier 1)

Independence Metropolis-Hastings with a Laplace proposal is the cheapest route to exact-tier draws when Laplace is almost right. It builds the Laplace mode and precision once, then proposes from that Gaussian and corrects with a Metropolis step. The proposal ignores the current state, so the iterations are nearly free and the acceptance rate doubles as a verdict on the Laplace approximation: high acceptance means the posterior is close to the Gaussian, low acceptance means it is far and Laplace was biased.

fit_imh <- tulpa(y ~ x + (1 | g), data = db, family = "binomial",
                 mode = "imh_laplace", sigma_re = 0.6,
                 control = list(n_iter = 450, warmup = 150))
coef(fit_imh)
#> (Intercept)           x 
#>  -0.2746751   0.8342437
fit_imh$mean_accept
#> [1] 0.8833333

The acceptance here is high, which says the Laplace fit was already close. Two jobs suit imh_laplace: debiasing a Laplace fit at low parameter dimension, and repeated-fit workflows such as cross-validation where a full sampler’s startup cost would dominate.

gibbs (Tier 1)

The Gibbs backend is the Polya-Gamma sampler for binomial and negative binomial responses, including the binomial areal spatial models that mode = "auto" selects it for. It samples the random-effect standard deviation rather than conditioning on a fixed sigma_re, which is why the spatial fit earlier reported a Gibbs backend. Its output carries the fixed effects in $beta and the field and variance components alongside.

round(fit_auto$beta, 3)
#>           [,1]  [,2]
#>    [1,] -0.139 0.612
#>    [2,] -0.060 0.653
#>    [3,] -0.013 0.637
#>    [4,] -0.201 0.680
#>    [5,] -0.177 0.611
#>    [6,] -0.239 0.737
#>    [7,] -0.007 0.532
#>    [8,] -0.194 0.803
#>    [9,] -0.261 0.591
#>   [10,] -0.112 0.592
#>   [11,] -0.093 0.529
#>   [12,] -0.200 0.524
#>   [13,] -0.136 0.509
#>   [14,] -0.307 0.628
#>   [15,] -0.285 0.666
#>   [16,] -0.200 0.668
#>   [17,] -0.076 0.771
#>   [18,] -0.374 0.575
#>   [19,] -0.229 0.529
#>   [20,] -0.246 0.944
#>   [21,] -0.168 0.733
#>   [22,] -0.295 0.666
#>   [23,] -0.083 0.617
#>   [24,] -0.068 0.776
#>   [25,] -0.343 0.498
#>   [26,] -0.096 0.529
#>   [27,] -0.356 0.623
#>   [28,] -0.245 0.586
#>   [29,] -0.155 0.724
#>   [30,] -0.068 0.635
#>   [31,] -0.208 0.603
#>   [32,] -0.168 0.634
#>   [33,] -0.216 0.689
#>   [34,] -0.112 0.652
#>   [35,] -0.251 0.667
#>   [36,] -0.179 0.467
#>   [37,] -0.281 0.595
#>   [38,] -0.286 0.621
#>   [39,] -0.341 0.269
#>   [40,] -0.118 0.635
#>   [41,] -0.086 0.719
#>   [42,] -0.126 0.686
#>   [43,]  0.079 0.560
#>   [44,] -0.076 0.594
#>   [45,] -0.043 0.633
#>   [46,] -0.186 0.643
#>   [47,] -0.295 0.506
#>   [48,] -0.252 0.540
#>   [49,] -0.182 0.534
#>   [50,] -0.089 0.776
#>   [51,] -0.322 0.684
#>   [52,] -0.290 0.730
#>   [53,] -0.325 0.704
#>   [54,] -0.221 0.603
#>   [55,] -0.246 0.811
#>   [56,] -0.424 0.518
#>   [57,] -0.211 0.805
#>   [58,] -0.076 0.802
#>   [59,] -0.094 0.743
#>   [60,] -0.035 0.513
#>   [61,] -0.091 0.513
#>   [62,] -0.064 0.573
#>   [63,] -0.124 0.638
#>   [64,] -0.209 0.548
#>   [65,] -0.194 0.436
#>   [66,] -0.091 0.633
#>   [67,] -0.196 0.476
#>   [68,] -0.192 0.542
#>   [69,] -0.162 0.627
#>   [70,] -0.340 0.625
#>   [71,] -0.248 0.697
#>   [72,] -0.198 0.612
#>   [73,] -0.237 0.436
#>   [74,] -0.315 0.561
#>   [75,]  0.008 0.643
#>   [76,] -0.247 0.717
#>   [77,] -0.319 0.569
#>   [78,] -0.123 0.485
#>   [79,] -0.318 0.410
#>   [80,] -0.296 0.698
#>   [81,] -0.050 0.425
#>   [82,] -0.267 0.494
#>   [83,] -0.144 0.688
#>   [84,] -0.259 0.541
#>   [85,] -0.017 0.558
#>   [86,] -0.001 0.654
#>   [87,] -0.380 0.548
#>   [88,] -0.068 0.556
#>   [89,] -0.453 0.631
#>   [90,] -0.380 0.643
#>   [91,] -0.295 0.649
#>   [92,] -0.303 0.516
#>   [93,] -0.088 0.565
#>   [94,] -0.166 0.446
#>   [95,] -0.294 0.658
#>   [96,] -0.111 0.736
#>   [97,] -0.183 0.586
#>   [98,] -0.089 0.879
#>   [99,]  0.035 0.616
#>  [100,] -0.206 0.766
#>  [101,]  0.033 0.685
#>  [102,] -0.088 0.853
#>  [103,] -0.060 0.582
#>  [104,] -0.170 0.622
#>  [105,]  0.087 0.600
#>  [106,] -0.219 0.670
#>  [107,]  0.007 0.559
#>  [108,] -0.290 0.688
#>  [109,] -0.096 0.509
#>  [110,] -0.221 0.318
#>  [111,] -0.244 0.476
#>  [112,]  0.051 0.634
#>  [113,] -0.046 0.700
#>  [114,]  0.086 0.536
#>  [115,] -0.065 0.431
#>  [116,] -0.249 0.575
#>  [117,] -0.254 0.586
#>  [118,] -0.147 0.633
#>  [119,] -0.371 0.422
#>  [120,] -0.103 0.587
#>  [121,] -0.182 0.556
#>  [122,]  0.014 0.545
#>  [123,] -0.024 0.507
#>  [124,] -0.254 0.428
#>  [125,] -0.171 0.545
#>  [126,] -0.109 0.407
#>  [127,] -0.264 0.475
#>  [128,] -0.091 0.530
#>  [129,]  0.018 0.783
#>  [130,] -0.173 0.542
#>  [131,]  0.023 0.677
#>  [132,] -0.211 0.698
#>  [133,] -0.267 0.503
#>  [134,] -0.272 0.676
#>  [135,] -0.243 0.521
#>  [136,] -0.176 0.829
#>  [137,] -0.194 0.751
#>  [138,] -0.372 0.724
#>  [139,] -0.389 0.647
#>  [140,] -0.344 0.727
#>  [141,] -0.287 0.557
#>  [142,] -0.201 0.843
#>  [143,] -0.248 0.631
#>  [144,] -0.175 0.379
#>  [145,] -0.116 0.469
#>  [146,] -0.172 0.476
#>  [147,] -0.075 0.611
#>  [148,] -0.273 0.619
#>  [149,] -0.061 0.764
#>  [150,] -0.063 0.791
#>  [151,] -0.262 0.681
#>  [152,] -0.145 0.834
#>  [153,] -0.208 0.479
#>  [154,] -0.168 0.631
#>  [155,] -0.005 0.592
#>  [156,] -0.005 0.802
#>  [157,]  0.010 0.546
#>  [158,] -0.150 0.699
#>  [159,] -0.063 0.626
#>  [160,] -0.144 0.501
#>  [161,] -0.123 0.507
#>  [162,] -0.442 0.677
#>  [163,] -0.080 0.569
#>  [164,] -0.008 0.626
#>  [165,] -0.131 0.699
#>  [166,] -0.069 0.762
#>  [167,] -0.076 0.940
#>  [168,] -0.001 0.588
#>  [169,] -0.380 0.769
#>  [170,] -0.148 0.606
#>  [171,] -0.203 0.686
#>  [172,] -0.231 0.717
#>  [173,] -0.405 0.935
#>  [174,] -0.310 0.869
#>  [175,] -0.068 0.915
#>  [176,] -0.160 0.918
#>  [177,] -0.069 0.530
#>  [178,]  0.118 0.604
#>  [179,]  0.018 0.516
#>  [180,] -0.188 0.640
#>  [181,] -0.061 0.420
#>  [182,] -0.046 0.586
#>  [183,] -0.191 0.449
#>  [184,]  0.031 0.605
#>  [185,] -0.218 0.590
#>  [186,] -0.226 0.560
#>  [187,] -0.292 0.611
#>  [188,] -0.069 0.558
#>  [189,] -0.301 0.637
#>  [190,] -0.118 0.451
#>  [191,] -0.204 0.464
#>  [192,] -0.092 0.534
#>  [193,] -0.044 0.627
#>  [194,] -0.060 0.479
#>  [195,] -0.192 0.608
#>  [196,] -0.257 0.891
#>  [197,] -0.136 0.658
#>  [198,] -0.196 0.629
#>  [199,] -0.074 0.629
#>  [200,] -0.215 0.466
#>  [201,] -0.307 0.431
#>  [202,] -0.410 0.546
#>  [203,] -0.208 0.686
#>  [204,] -0.073 0.637
#>  [205,] -0.292 0.455
#>  [206,] -0.089 0.406
#>  [207,] -0.215 0.507
#>  [208,] -0.315 0.645
#>  [209,] -0.175 0.531
#>  [210,] -0.244 0.477
#>  [211,] -0.126 0.458
#>  [212,] -0.162 0.673
#>  [213,] -0.193 0.583
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#>  [717,] -0.151 0.849
#>  [718,] -0.085 0.689
#>  [719,] -0.155 0.968
#>  [720,] -0.098 0.813
#>  [721,] -0.207 0.690
#>  [722,] -0.121 0.553
#>  [723,]  0.007 0.637
#>  [724,] -0.092 0.590
#>  [725,] -0.174 0.528
#>  [726,] -0.341 0.655
#>  [727,] -0.042 0.560
#>  [728,] -0.117 0.623
#>  [729,] -0.252 0.543
#>  [730,] -0.168 0.584
#>  [731,] -0.015 0.606
#>  [732,] -0.168 0.607
#>  [733,] -0.048 0.571
#>  [734,] -0.169 0.664
#>  [735,] -0.084 0.717
#>  [736,] -0.319 0.686
#>  [737,] -0.198 0.599
#>  [738,] -0.125 0.488
#>  [739,] -0.279 0.554
#>  [740,] -0.147 0.559
#>  [741,] -0.162 0.372
#>  [742,] -0.104 0.615
#>  [743,] -0.151 0.627
#>  [744,] -0.427 0.534
#>  [745,] -0.310 0.762
#>  [746,] -0.190 0.746
#>  [747,] -0.193 0.580
#>  [748,] -0.272 0.672
#>  [749,] -0.181 0.586
#>  [750,] -0.177 0.923
#>  [751,] -0.162 0.684
#>  [752,]  0.011 0.587
#>  [753,] -0.085 0.702
#>  [754,] -0.276 0.603
#>  [755,] -0.364 0.598
#>  [756,] -0.246 0.596
#>  [757,]  0.060 0.568
#>  [758,] -0.140 0.629
#>  [759,] -0.061 0.620
#>  [760,]  0.008 0.553
#>  [761,] -0.021 0.758
#>  [762,] -0.367 0.689
#>  [763,] -0.206 0.586
#>  [764,] -0.190 0.381
#>  [765,] -0.180 0.608
#>  [766,] -0.229 0.704
#>  [767,] -0.039 0.384
#>  [768,]  0.053 0.867
#>  [769,] -0.034 0.566
#>  [770,] -0.140 0.691
#>  [771,] -0.276 0.638
#>  [772,] -0.320 0.554
#>  [773,] -0.496 0.656
#>  [774,] -0.193 0.689
#>  [775,] -0.147 0.770
#>  [776,] -0.247 0.806
#>  [777,] -0.215 0.545
#>  [778,] -0.291 0.712
#>  [779,] -0.067 0.569
#>  [780,] -0.198 0.501
#>  [781,] -0.207 0.459
#>  [782,] -0.067 0.508
#>  [783,]  0.055 0.478
#>  [784,] -0.270 0.614
#>  [785,] -0.110 0.526
#>  [786,]  0.071 0.575
#>  [787,] -0.079 0.601
#>  [788,] -0.048 0.696
#>  [789,] -0.244 0.520
#>  [790,] -0.299 0.437
#>  [791,] -0.227 0.660
#>  [792,] -0.187 0.719
#>  [793,] -0.235 0.743
#>  [794,] -0.282 0.714
#>  [795,] -0.134 0.800
#>  [796,] -0.236 0.614
#>  [797,] -0.269 0.612
#>  [798,] -0.052 0.843
#>  [799,] -0.175 0.672
#>  [800,] -0.002 0.511
#>  [801,] -0.135 0.645
#>  [802,] -0.141 0.579
#>  [803,] -0.163 0.737
#>  [804,] -0.229 0.806
#>  [805,] -0.414 0.512
#>  [806,]  0.003 0.329
#>  [807,] -0.061 0.536
#>  [808,] -0.272 0.544
#>  [809,] -0.271 0.349
#>  [810,] -0.162 0.602
#>  [811,] -0.140 0.630
#>  [812,]  0.047 0.556
#>  [813,] -0.235 0.602
#>  [814,] -0.201 0.701
#>  [815,] -0.117 0.522
#>  [816,] -0.109 0.709
#>  [817,] -0.181 1.015
#>  [818,] -0.053 0.659
#>  [819,] -0.023 0.639
#>  [820,] -0.224 0.525
#>  [821,] -0.212 0.489
#>  [822,] -0.187 0.701
#>  [823,] -0.104 0.677
#>  [824,] -0.210 0.505
#>  [825,] -0.233 0.498
#>  [826,] -0.278 0.724
#>  [827,] -0.103 0.763
#>  [828,] -0.143 0.599
#>  [829,] -0.043 0.639
#>  [830,] -0.334 0.616
#>  [831,] -0.154 0.573
#>  [832,] -0.322 0.664
#>  [833,] -0.150 0.644
#>  [834,] -0.065 0.605
#>  [835,] -0.213 0.515
#>  [836,] -0.218 0.802
#>  [837,] -0.207 0.640
#>  [838,] -0.173 0.854
#>  [839,] -0.234 0.667
#>  [840,] -0.167 0.764
#>  [841,] -0.279 0.522
#>  [842,] -0.083 0.631
#>  [843,] -0.242 0.559
#>  [844,] -0.057 0.576
#>  [845,] -0.359 0.640
#>  [846,] -0.291 0.965
#>  [847,] -0.417 0.773
#>  [848,] -0.147 0.900
#>  [849,]  0.181 0.655
#>  [850,] -0.089 0.612
#>  [851,] -0.017 0.429
#>  [852,] -0.255 0.750
#>  [853,] -0.067 0.593
#>  [854,]  0.072 0.555
#>  [855,] -0.246 0.587
#>  [856,] -0.386 0.502
#>  [857,] -0.148 0.778
#>  [858,] -0.122 0.702
#>  [859,] -0.227 0.685
#>  [860,] -0.213 0.647
#>  [861,] -0.215 0.499
#>  [862,] -0.170 0.745
#>  [863,]  0.067 0.472
#>  [864,] -0.226 0.505
#>  [865,] -0.169 0.636
#>  [866,] -0.053 0.636
#>  [867,] -0.015 0.493
#>  [868,] -0.178 0.677
#>  [869,] -0.069 0.417
#>  [870,] -0.209 0.722
#>  [871,] -0.271 0.684
#>  [872,] -0.192 0.655
#>  [873,] -0.417 0.718
#>  [874,] -0.167 0.685
#>  [875,] -0.201 0.707
#>  [876,] -0.063 0.888
#>  [877,] -0.155 0.540
#>  [878,] -0.278 0.700
#>  [879,] -0.154 0.479
#>  [880,] -0.204 0.653
#>  [881,]  0.083 0.633
#>  [882,] -0.103 0.562
#>  [883,] -0.105 0.641
#>  [884,] -0.067 0.476
#>  [885,]  0.053 0.738
#>  [886,] -0.300 0.558
#>  [887,] -0.125 0.584
#>  [888,] -0.133 0.627
#>  [889,] -0.292 0.751
#>  [890,] -0.158 0.655
#>  [891,] -0.365 0.472
#>  [892,] -0.202 0.638
#>  [893,] -0.227 0.649
#>  [894,] -0.127 0.492
#>  [895,] -0.037 0.548
#>  [896,] -0.140 0.527
#>  [897,] -0.201 0.631
#>  [898,] -0.082 0.720
#>  [899,] -0.154 0.920
#>  [900,] -0.233 0.620
#>  [901,] -0.048 0.650
#>  [902,]  0.014 0.475
#>  [903,] -0.194 0.583
#>  [904,] -0.264 0.773
#>  [905,] -0.055 0.588
#>  [906,] -0.037 0.946
#>  [907,] -0.145 0.877
#>  [908,] -0.169 0.703
#>  [909,] -0.398 0.531
#>  [910,] -0.229 0.519
#>  [911,] -0.149 0.709
#>  [912,] -0.263 0.541
#>  [913,] -0.010 0.634
#>  [914,] -0.107 0.668
#>  [915,] -0.073 0.802
#>  [916,] -0.183 0.549
#>  [917,] -0.198 0.687
#>  [918,] -0.020 0.612
#>  [919,] -0.344 0.733
#>  [920,] -0.258 0.726
#>  [921,] -0.178 0.588
#>  [922,] -0.194 0.644
#>  [923,] -0.197 0.552
#>  [924,] -0.157 0.702
#>  [925,] -0.186 0.456
#>  [926,] -0.301 0.793
#>  [927,] -0.293 0.793
#>  [928,] -0.071 0.605
#>  [929,] -0.177 0.758
#>  [930,] -0.276 0.803
#>  [931,]  0.004 0.607
#>  [932,]  0.128 0.737
#>  [933,]  0.031 0.695
#>  [934,] -0.243 0.647
#>  [935,] -0.057 0.735
#>  [936,] -0.100 0.849
#>  [937,] -0.258 0.595
#>  [938,] -0.337 0.575
#>  [939,] -0.218 0.669
#>  [940,] -0.403 0.624
#>  [941,] -0.240 0.629
#>  [942,] -0.009 0.583
#>  [943,] -0.052 0.702
#>  [944,] -0.081 0.389
#>  [945,] -0.236 0.481
#>  [946,] -0.328 0.654
#>  [947,] -0.268 0.739
#>  [948,] -0.129 0.845
#>  [949,] -0.258 0.627
#>  [950,] -0.193 0.337
#>  [951,] -0.182 0.339
#>  [952,] -0.401 0.359
#>  [953,] -0.205 0.471
#>  [954,] -0.019 0.565
#>  [955,] -0.096 0.568
#>  [956,] -0.131 0.572
#>  [957,] -0.048 0.633
#>  [958,] -0.164 0.547
#>  [959,] -0.098 0.693
#>  [960,] -0.186 0.586
#>  [961,] -0.040 0.755
#>  [962,] -0.040 0.692
#>  [963,] -0.159 0.574
#>  [964,] -0.317 0.539
#>  [965,] -0.289 0.960
#>  [966,] -0.122 0.721
#>  [967,] -0.089 0.727
#>  [968,] -0.061 0.629
#>  [969,]  0.080 0.506
#>  [970,] -0.222 0.825
#>  [971,] -0.132 0.481
#>  [972,] -0.329 0.740
#>  [973,] -0.448 0.418
#>  [974,] -0.038 0.388
#>  [975,] -0.298 0.189
#>  [976,] -0.166 0.622
#>  [977,] -0.202 0.672
#>  [978,] -0.085 0.651
#>  [979,]  0.041 0.919
#>  [980,] -0.104 0.908
#>  [981,] -0.352 0.661
#>  [982,] -0.047 0.640
#>  [983,] -0.138 0.650
#>  [984,] -0.231 0.788
#>  [985,] -0.079 0.994
#>  [986,] -0.219 0.721
#>  [987,] -0.080 0.540
#>  [988,] -0.116 0.540
#>  [989,] -0.188 0.585
#>  [990,] -0.332 0.661
#>  [991,] -0.102 0.675
#>  [992,] -0.019 0.670
#>  [993,] -0.192 0.431
#>  [994,]  0.039 0.568
#>  [995,] -0.183 0.863
#>  [996,] -0.277 0.797
#>  [997,] -0.007 0.796
#>  [998,] -0.110 0.479
#>  [999,] -0.241 0.551
#> [1000,] -0.114 0.528

Gibbs is the right call when the family is conjugate-friendly under the Polya-Gamma scheme and you want an exact fit that samples the variance rather than fixing it. For a binomial areal field, auto takes this path on its own.

re_cov_nested and agq (Tier 2)

Two further Tier 2 backends serve narrower roles. re_cov_nested integrates a random-effect covariance matrix rather than conditioning on a fixed standard deviation. A random-slope term such as (1 + x | g) has no scalar sigma_re to condition on, so when a slope term is present the Laplace path redirects to this backend automatically and integrates the covariance with a CCD design and a PC or LKJ prior. You reach it not by name but by writing a slope term and fitting at mode = "laplace".

set.seed(20260531)
ng <- 60; ni <- 15
g   <- rep(seq_len(ng), each = ni)
xg  <- rnorm(ng * ni)
Sig <- matrix(c(0.9^2,            0.5 * 0.9 * 0.6,
                0.5 * 0.9 * 0.6,  0.6^2), 2)
b   <- matrix(rnorm(ng * 2), ng) %*% chol(Sig)
eta <- -0.2 + 0.7 * xg + b[g, 1] + b[g, 2] * xg
dsl <- data.frame(y = rbinom(ng * ni, 1, plogis(eta)),
                  x = xg, g = factor(g))

fit_rc <- tulpa(y ~ x + (1 + x | g), data = dsl,
                family = "binomial", mode = "laplace")
fit_rc$backend
#> [1] "re_cov_nested"

The slope term routed the fit to re_cov_nested without being named. What comes back is the integrated covariance, not a point estimate of one: a 2x2 matrix carrying the intercept and slope variances on its diagonal and their covariance off it.

round(fit_rc$Sigma_mean, 3)
#>       [,1]  [,2]
#> [1,] 1.052 0.339
#> [2,] 0.339 0.627

Integrating the covariance through a Gaussian grid over the hyperparameters means the fit carries a Pareto-k-hat accuracy diagnostic, the nested-approximation counterpart to the Rhat a sampler reports.

round(fit_rc$pareto_k, 2)
#> [1] 0.55

Here k-hat sits below the 0.7 threshold, so the Gaussian grid fits the covariance posterior well and the integrated matrix can be trusted. With smaller groups or sparser binary data that posterior turns skewed and k-hat climbs past 0.7 – not a defect but the signal to escalate to the exact debias, control = list(re_cov = "gibbs"), which replaces the deterministic integration with a Metropolis-within-Gibbs sweep and a conjugate inverse-Wishart draw for the covariance.

The adaptive Gauss-Hermite quadrature backend, agq, integrates a random-intercept variance by quadrature and is callable through its fitter agq_fit() for the single-grouping case rather than through tulpa().

ep (Tier 2)

Expectation Propagation approximates the posterior of a fixed-effect GLM by a Gaussian whose per-observation sites match the moments of the tilted distribution, rather than the mode curvature Laplace uses. It is exact when the likelihood is Gaussian and is typically more accurate than Laplace on a skewed GLM likelihood. EP fits fixed effects only (no random-effect, spatial, or latent structure); reach it with mode = "ep" or call tulpa_ep() directly.

fit_ep <- tulpa(y ~ x, data = d, family = "binomial", mode = "ep")
coef(fit_ep)

The R-callable backends span both exact tiers and the structured tier. The next section puts three of them on the same model so the trade between cost and guarantee is visible in numbers.

Cost versus guarantee

The clearest way to see what a tier buys is to fit one model several ways and lay the results side by side. Take the binomial random-intercept model from above and fit it at Laplace, MALA, and Pathfinder, timing each with system.time().

t_lap  <- system.time(
  f_lap  <- tulpa(y ~ x + (1 | g), data = db, family = "binomial",
                  mode = "laplace", sigma_re = 0.6))[["elapsed"]]
t_mala <- system.time(
  f_mala <- tulpa(y ~ x + (1 | g), data = db, family = "binomial",
                  mode = "mala", sigma_re = 0.6,
                  control = list(n_iter = 450, warmup = 150)))[["elapsed"]]
t_pf   <- system.time(
  f_pf   <- tulpa(y ~ x + (1 | g), data = db, family = "binomial",
                  mode = "pathfinder", sigma_re = 0.6,
                  control = list(n_draws = 450)))[["elapsed"]]

Collect the slope estimate, its standard error, the tier, and the elapsed time into one table.

slope_se <- function(f) summary(f)["x", "std.error"]
data.frame(
  backend  = c(f_lap$backend, f_mala$backend, f_pf$backend),
  tier     = c(f_lap$inference_tier, f_mala$inference_tier,
               f_pf$inference_tier),
  slope    = round(c(coef(f_lap)["x"], coef(f_mala)["x"],
                     coef(f_pf)["x"]), 3),
  slope_se = round(c(slope_se(f_lap), slope_se(f_mala),
                     slope_se(f_pf)), 3),
  seconds  = round(c(t_lap, t_mala, t_pf), 3)
)
#>      backend tier slope slope_se seconds
#> 1    laplace    2 0.802    0.128    0.00
#> 2       mala    1 0.823    0.138    0.07
#> 3 pathfinder    2 0.802    0.130    0.05

Three patterns sit in this table. The slope estimates agree across all three backends to within their standard errors. That is expected: the posterior mean is the easy quantity, and every method finds it. The standard errors agree too, because for this model the posterior is near-Gaussian and the Tier 2 approximation is accurate. That agreement is the signal that Laplace was safe here, confirmed independently by the high IMH acceptance earlier. Timing is where the methods part ways. The deterministic Laplace fit returns fastest. Pathfinder costs an optimisation plus a draw step. MALA pays for its full chain of gradient evaluations, and that cost is the visible price of the exact guarantee.

Read the table as a guarantee you bought, not effort you wasted. The sampler purchased a promise this model did not need and another model would. The agreement you see is itself the product of a near-Gaussian posterior. Push the model toward a skewed or heavy-tailed posterior, by shrinking the data to a few dozen rows, moving to a sparse binomial with most outcomes zero, or adding a poorly identified variance component, and the Tier 2 standard error would start to drift away from the Tier 1 one. The samplers would track the true posterior spread through that drift, while the Gaussian approximation would keep reporting the symmetric interval its assumption forces on it. That is the moment the price of the sampler turns into a reason to pay it.

The marginal likelihood ties the comparison back to model choice. compare_models() reads logLik() from each fit, which on the Laplace tier is the approximate log marginal likelihood.

compare_models(laplace = f_lap, pathfinder = f_pf, criterion = "loglik")
#>        model n_params    logLik
#> 1    laplace        2 -259.8379
#> 2 pathfinder       14 -255.1685

The glance() accessor surfaces the per-fit diagnostics that matter for the sampler tiers: the number of post-warmup draws, the acceptance rate, and the divergence count.

glance(f_mala)[c("n_samples", "mean_accept", "n_divergent")]
#>   n_samples mean_accept n_divergent
#> 1       375        0.72          NA

An acceptance rate inside the healthy band and no divergences say the MALA chain mixed well, which is the prerequisite for trusting its intervals. Without that check, a Tier 1 fit is only nominally exact: a stuck chain reports a posterior it never explored.

Choosing a mode

The tier system gives a default worth following and a small set of rules for when to depart from it.

Start at mode = "laplace". The fit is deterministic, carries no Monte Carlo error, and returns in well under a second on the models in this vignette and in a fraction of a second per thousand rows on much larger ones. Use it for the first fit and for model comparison by marginal likelihood, and keep it as the baseline even when you intend to finish with a sampler.
Check Laplace before you trust its intervals. The cheapest check is mode = "imh_laplace" on the same model: a high acceptance rate, above roughly 0.5, says the posterior is close to the Laplace Gaussian and the intervals are safe. An acceptance rate below about 0.1 says the posterior is far from Gaussian and the Laplace intervals are biased. The fit earlier accepted at a high rate, which is a clean pass.
Move to mode = "mala" for exact moments at moderate dimension. When the posterior is visibly non-Gaussian, when imh_laplace acceptance is low, or when you simply need intervals you can quote without the Gaussian caveat, the gradient sampler is the next step. Budget a few thousand iterations with warmup at a third to a half of them, and watch the acceptance rate settle near 0.574 and the divergence count stay at zero.
Use mode = "pathfinder" when you want draws cheaply. It costs more than Laplace and far less than MALA, returns a sample you can feed to downstream code, flags a poor Gaussian fit through its ELBO, and doubles as a warm start for a sampler.
Let mode = "auto" choose for structured models. For a latent block or a spatial field, auto routes to the designed path: nested Laplace for latent and most spatial fields, exact Polya-Gamma Gibbs for binomial areal models, and the Laplace path for very large plain models past tens of thousands of rows. Read selection_reason to confirm the branch. On a small plain model with no spatial or latent structure auto defaults to the MALA gradient sampler (Tier 1); name mode = "laplace" instead when you want the fast deterministic fit.
Treat mode = "optimized" as a deliberate exception. Tier 3 is opt-in because its uncertainty is unreliable and its failures are silent. It fits a narrow case: a fast point estimate when you have an independent handle on the uncertainty. Keep it off the default path, and off any interval you intend to report.

The thread through all of it is that the tier is visible and the choice is recorded, so you are never reasoning about uncertainty you cannot account for. Read $backend and $inference_tier off any fit to see what ran and what its intervals promise, then read $selection_reason to see why that backend was chosen; when the promise on a fit is not the one your analysis needs, move up a tier and pay for the stronger guarantee on purpose.