We present a complete mathematical treatment of the reliability-weighted product-of-experts (RW-PoE) framework for multimodal Bayesian classification with provable guarantees. The modality posteriors are fused via a log-linear pool with weights $ {\mathit{\boldsymbol{w}}}\in\Delta_{M-1} $ fitted by maximum likelihood on the simplex. We prove the following: (i) Strict convexity of the weight objective if and only if the modality log-posteriors are independent in the simplex-tangent sense of Definition 4.2, which guarantees a unique global maximum likelihood estimate (MLE); (ii) Additive decomposition of $ H(Y)-H(Y\!\mid\! X) $ across a hierarchical cascade; (iii) An explicit non-asymptotic contraction bound on an interior simplex region and the exact $ \gamma $-dependent local linearization of the finite-step entropic mirror update; (iv) Finite-sample marginal conformal coverage at least $ 1-\alpha $, with the upper discretization bound $ 1-\alpha+1/(n_{\mathrm{cal}}+1) $ under the no-ties condition. A Hellinger–Bayes criterion yields a distribution-specific floor $ R^\star_{a, \mathrm{root}}\ge0.235 $ on the acoustic-only error at the root node for the observed distributions, showing that fusion is necessary in this setting (Remark 3.3 specifies the scope of this claim). A clinical study on $ n = 1105 $ infant cry episodes (CHU Sidi Bel-Abbès) reports $ \widehat{{\mathit{\boldsymbol{w}}}} = (0.42\pm0.03, \, 0.31\pm0.04, \, 0.27\pm0.03) $, realized coverage $ 0.910 $ (target $ 0.900 $; no-ties theoretical band $ [0.900, 0.905] $), accuracy $ 0.916 $, $ \kappa = 0.889 $ (95% CI: $ 0.844 $–$ 0.934 $).
Citation: Safia Benarbia, Fatimah Alshahrani, Mohammed Fethi Khalfi, Wahiba Bouabsa. A reliability-weighted product-of-experts framework for multimodal Bayesian classification: convexity, conformal coverage, and an information-theoretic necessity criterion[J]. AIMS Mathematics, 2026, 11(9): 31134-31184. doi: 10.3934/math.20261231
We present a complete mathematical treatment of the reliability-weighted product-of-experts (RW-PoE) framework for multimodal Bayesian classification with provable guarantees. The modality posteriors are fused via a log-linear pool with weights $ {\mathit{\boldsymbol{w}}}\in\Delta_{M-1} $ fitted by maximum likelihood on the simplex. We prove the following: (i) Strict convexity of the weight objective if and only if the modality log-posteriors are independent in the simplex-tangent sense of Definition 4.2, which guarantees a unique global maximum likelihood estimate (MLE); (ii) Additive decomposition of $ H(Y)-H(Y\!\mid\! X) $ across a hierarchical cascade; (iii) An explicit non-asymptotic contraction bound on an interior simplex region and the exact $ \gamma $-dependent local linearization of the finite-step entropic mirror update; (iv) Finite-sample marginal conformal coverage at least $ 1-\alpha $, with the upper discretization bound $ 1-\alpha+1/(n_{\mathrm{cal}}+1) $ under the no-ties condition. A Hellinger–Bayes criterion yields a distribution-specific floor $ R^\star_{a, \mathrm{root}}\ge0.235 $ on the acoustic-only error at the root node for the observed distributions, showing that fusion is necessary in this setting (Remark 3.3 specifies the scope of this claim). A clinical study on $ n = 1105 $ infant cry episodes (CHU Sidi Bel-Abbès) reports $ \widehat{{\mathit{\boldsymbol{w}}}} = (0.42\pm0.03, \, 0.31\pm0.04, \, 0.27\pm0.03) $, realized coverage $ 0.910 $ (target $ 0.900 $; no-ties theoretical band $ [0.900, 0.905] $), accuracy $ 0.916 $, $ \kappa = 0.889 $ (95% CI: $ 0.844 $–$ 0.934 $).
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Safia Benarbia, Fatimah Alshahrani, Mohammed Fethi Khalfi, Wahiba Bouabsa. A reliability-weighted product-of-experts framework for multimodal Bayesian classification: convexity, conformal coverage, and an information-theoretic necessity criterion[J]. AIMS Mathematics, 2026, 11(9): 31134-31184. doi: 10.3934/math.20261231
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