model-evaluation

Model Evaluation and Comparison (ArviZ 1.0)

CRITICAL: ArviZ 1.0 replaces InferenceData with xarray.DataTree. az.waic is removed entirely — use PSIS-LOO-CV exclusively. Default credible interval is 0.89 ETI (not 0.94 HDI), controlled via ci_prob= and ci_kind= (replaces the old hdi_prob=).

For model building context, prior selection, and convergence diagnostics, see the pymc-modeling skill.

LOO-CV with ArviZ 1.0

Leave-one-out cross-validation via Pareto-smoothed importance sampling (PSIS).

import arviz as az
import arviz_stats  # registers the .azstats accessor

# dt is a DataTree from pm.sample()
loo_result = az.loo(dt)
print(loo_result)
# Returns: ELPDData with elpd_loo, se, p_loo, n_data_points, pareto_k

# Equivalent via the xarray accessor (DataArray / Dataset / DataTree all supported):
loo_result = dt.azstats.loo()

Pareto k Diagnostics

Pareto k values indicate reliability of PSIS approximation for each observation:

k value	Interpretation	Action
k < 0.5	Good	LOO estimate reliable
0.5 < k < 0.7	Marginal	Results usable but less accurate
0.7 < k < 1.0	Bad	Estimate unreliable — use moment matching or k-fold
k > 1.0	Very bad	PSIS fails entirely — must use k-fold CV

# Check Pareto k values
print(loo_result.pareto_k)

# Plot Pareto k diagnostics
az.plot_khat(loo_result)

# Count problematic observations
import numpy as np
k_values = loo_result.pareto_k.values
print(f"k > 0.7: {np.sum(k_values > 0.7)} observations")

What to Do When k > 0.7

1. Try moment matching first (fast, automatic)
2. If still bad, use k-fold cross-validation
3. Check if problematic observations are outliers — consider robust likelihood
4. Re-examine the model — high k often signals model misspecification

Moment Matching

Automatically refit problematic observations using moment matching:

# Requires log_likelihood in the DataTree
loo_mm = az.loo_moment_match(dt)

This importance-weights the posterior for each problematic observation, improving the PSIS approximation without refitting the model. Much faster than k-fold.

K-Fold Cross-Validation

When LOO is unreliable for many observations, use exact k-fold CV:

# Perform 10-fold cross-validation
kfold_result = az.loo_kfold(dt, K=10)
print(kfold_result)

This refits the model K times, so it is K times slower than LOO. Use only when LOO diagnostics indicate problems.

az.compare() — Full Workflow

Compare multiple models on predictive accuracy:

# dt1, dt2, dt3 are DataTree objects from pm.sample()
comparison = az.compare(
    {"linear": dt1, "quadratic": dt2, "spline": dt3},
    scale="log",        # log scale (default) or deviance
)
print(comparison)

Note: az.compare in ArviZ 1.0 only supports LOO, so the ic= argument has been dropped.

Interpreting the Comparison Table

Column	Meaning
rank	Model rank (0 = best)
elpd_loo	Expected log pointwise predictive density
p_loo	Effective number of parameters
d_loo	Difference in ELPD from best model
weight	Stacking weight (sums to 1)
se	Standard error of ELPD
dse	Standard error of the ELPD difference
warning	True if any Pareto k > 0.7

Decision Rules

• d_loo = 0: best model
• |d_loo| < 4: models are practically indistinguishable — prefer simpler one
• |d_loo| > 4 and |d_loo/dse| > 2: meaningful difference in predictive accuracy
• warning = True: LOO unreliable for this model — investigate Pareto k values

# Visualize comparison
az.plot_compare(comparison)

# Detailed forest plot of ELPD differences
az.plot_elpd({"linear": dt1, "quadratic": dt2, "spline": dt3})

See references/model_comparison.md for detailed usage.

Model Averaging

Stacking Weights (Default)

Stacking minimizes KL divergence from the true predictive distribution to the weighted mixture. This is the recommended default.

comparison = az.compare({"m1": dt1, "m2": dt2, "m3": dt3})
# Stacking weights are in the "weight" column by default
print(comparison["weight"])

Pseudo-BMA+ Weights

Alternative weighting based on Bayesian bootstrap of ELPD:

comparison = az.compare(
    {"m1": dt1, "m2": dt2, "m3": dt3},
    method="BB-pseudo-BMA",
)

When to Use Which

Method	Use When
Stacking	Default. Best for prediction when true model is not in the set
Pseudo-BMA+	Want Bayesian uncertainty over weights
Equal weights	Models represent different scientific hypotheses to average over

Generating Averaged Predictions

weights = comparison["weight"].values
# Manually mix posterior predictive samples
# weighted by stacking weights

See references/stacking.md for detailed averaging workflows.

Bayes Factors via Bridge Sampling

Bayes factors compare marginal likelihoods. Conceptually different from LOO (predictive accuracy vs. evidence).

# Bayes factors are difficult to compute reliably
# Bridge sampling is the most reliable method but requires specialized setup
# For most applied work, LOO-CV is preferred

# Approximate Bayes factor from LOO (rough):
# BF ~ exp(elpd_loo_m1 - elpd_loo_m2)
# This is a very rough approximation — use with caution

Limitations of Bayes Factors

• Highly sensitive to prior specification (unlike LOO)
• Numerically unstable for complex models
• Penalize model complexity differently than LOO
• Not recommended for routine model comparison — prefer LOO

LOO-PIT Calibration

LOO probability integral transform checks if the model is calibrated:

az.plot_loo_pit(dt, var_names=["observed_data_name"])

Interpretation

• Uniform histogram: model is well-calibrated
• U-shaped: underdispersed predictions (too narrow)
• Inverted U: overdispersed predictions (too wide)
• Skewed: systematic bias in predictions

This is a powerful diagnostic that LOO uniquely provides — it checks calibration without held-out data.

New ArviZ 1.0 Functions

loo_expectations()

Compute LOO-weighted posterior expectations (mean, variance, quantile) for each observation. Requires both posterior_predictive and log_likelihood groups on the DataTree:

# LOO-weighted posterior predictive mean for each observation
loo_mean = az.loo_expectations(dt, kind="mean")
loo_var = az.loo_expectations(dt, kind="var")
loo_q = az.loo_expectations(dt, kind="quantile", probs=[0.055, 0.945])

loo_metrics()

Compute common LOO-based predictive metrics (RMSE, MAE, etc.) from posterior_predictive and log_likelihood:

metrics = az.loo_metrics(dt, kind="rmse")

.azstats xarray accessor

import arviz_stats registers an .azstats accessor on DataArray, Dataset, and DataTree. This gives a fluent xarray-native interface alongside the top-level az.* functions (which remain available after import arviz as az):

import arviz_stats  # registers accessor; required even if you already imported arviz

dt.azstats.loo()                       # same as az.loo(dt)
dt["posterior"].azstats.rhat()         # on a Dataset
dt["posterior"].azstats.ess()
dt["posterior"].azstats.summary()
dt["posterior"].azstats.hdi()
dt["posterior"].azstats.eti()

loo_r2()

Bayesian R-squared via LOO:

r2 = az.loo_r2(dt)
print(f"LOO-R2: {r2.mean():.3f} [{r2.quantile(0.055):.3f}, {r2.quantile(0.945):.3f}]")

loo_score()

Compute LOO-based scoring rules (CRPS, log score):

score = az.loo_score(dt, score_func="crps")

loo_subsample()

LOO with subsampling for large datasets:

# When n > 10000, subsample for speed
loo_sub = az.loo_subsample(dt, observations=1000)

reloo()

Exact refit LOO for observations with high Pareto k:

# Refits the model for problematic observations
loo_exact = az.reloo(dt, loo_result, model=model)

Standard Evaluation Workflow

import arviz as az

# 1. Compute LOO
loo = az.loo(dt)
print(loo)

# 2. Check Pareto k
az.plot_khat(loo)

# 3. If k > 0.7, try moment matching
if (loo.pareto_k > 0.7).any():
    loo = az.loo_moment_match(dt)

# 4. LOO-PIT calibration
az.plot_loo_pit(dt, var_names=["y"])

# 5. Compare models
comparison = az.compare({"model_a": dt_a, "model_b": dt_b})
az.plot_compare(comparison)
print(comparison)

# 6. Predictive R2
r2 = az.loo_r2(dt)