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](../pymc-modeling/SKILL.md). ## LOO-CV with ArviZ 1.0 Leave-one-out cross-validation via Pareto-smoothed importance sampling (PSIS). ```python 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 | ```python # 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: ```python # 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: ```python # 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: ```python # 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 ```python # 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. ```python 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: ```python 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 ```python 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). ```python # 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: ```python 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: ```python # 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`: ```python 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`): ```python 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: ```python 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): ```python score = az.loo_score(dt, score_func="crps") ``` ### loo_subsample() LOO with subsampling for large datasets: ```python # When n > 10000, subsample for speed loo_sub = az.loo_subsample(dt, observations=1000) ``` ### reloo() Exact refit LOO for observations with high Pareto k: ```python # Refits the model for problematic observations loo_exact = az.reloo(dt, loo_result, model=model) ``` ## Standard Evaluation Workflow ```python 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) ```