Bayesian Model Averaging¶

Model comparison selects the best model. Bayesian Model Averaging (BMA) goes further: instead of committing to one model, it combines all models' predictions weighted by their posterior probability. The result is a predictive distribution that accounts for model uncertainty, not just parameter uncertainty.

When to use BMA¶

The evidence is not decisive between two or more models (Bayes factor < 10:1)
You want predictions that are robust to the choice of model structure
You are comparing models that are all physically plausible (e.g. different boundary conditions, different DOF counts)

When evidence is decisive (\(w_{\text{best}} \approx 1\)), BMA collapses to the best model — it gives the same answer but with no extra cost.

Model weights¶

After running ModelComparison, call weights() to get posterior model probabilities:

comp.run()
w = comp.weights()
# array([0.97, 0.03])  — M2 is overwhelmingly preferred

With informative prior model probabilities (e.g. expert knowledge):

# sorted_names matches summary() order (best → worst)
w = comp.weights(prior_weights=[0.3, 0.7])  # favour M2 a priori

BMA prediction¶

bma = comp.predict(
    forward_models={
        "M1: single k":  lambda theta: natural_frequencies(theta[0], theta[0]),
        "M2: k1 + k2":   lambda theta: natural_frequencies(theta[0], theta[1]),
    },
    n_eval=2000,
)

print(bma)
# BMAResult(n_models=2, n_obs=2, weights=[M2: k1+k2: 0.973, M1: single k: 0.027])

bma.mean()   # BMA mean prediction, shape (n_obs,)
bma.std()    # BMA std, shape (n_obs,)
bma.quantile([0.05, 0.95])  # credible interval

bma.plot_bands(x=freq_indices, y_obs=observed_freqs)

Per-model decomposition¶

dec = bma.decompose()
# {'M2: k1+k2': {'weight': 0.973, 'mean': array([...]), 'std': array([...])},
#  'M1: single k': {'weight': 0.027, ...}}

bma.plot_decompose()   # line width ∝ model weight

This is useful to verify that the dominant model drives the BMA prediction, or to identify when two models make noticeably different predictions.

Example output¶

Running the scenario in examples/model_averaging.py (2-DOF frame, M1 vs M2):

Model                   log p(y|M)    log10 BF  Evidence vs best
M2: k1 + k2                  4.12        0.00  Barely worth mentioning   <-- best
M1: single k                 2.87       -0.54  Substantial

Posterior model weights:
  M2: k1 + k2          w = 0.947
  M1: single k         w = 0.053

M2 is strongly preferred but not decisively — the BMA prediction will be mostly M2 with a small M1 contribution, slightly widening the credible band.

API reference¶

See the full runnable script at examples/model_averaging.py.