Neural Quasiprobabilistic Likelihood Ratio Estimation

• Neural quasiprobabilistic likelihood ratio estimation generalizes classical density ratio methods to handle signed (negative) densities using modified loss functions and architectures.
• It employs convex loss functions, direct regression, and mixture modeling to maintain stability and consistency even when densities can assume negative values.
• The approach enhances inference tasks in domains like high-energy physics and Monte Carlo sampling by ensuring statistical efficiency and robust handling of implicit likelihoods.

Neural quasiprobabilistic likelihood ratio estimation refers to a broad family of neural methodologies that generalize classical likelihood ratio estimation to settings where probability densities can take negative values (quasiprobabilities) and where the likelihood is accessible only implicitly (e.g., via simulators or importance-weighted samples with negative weights). This problem arises in scientific inference tasks such as higher-order reweighting in high-energy physics, control-variates Monte Carlo, or any context where the underlying densities or importance weights are not strictly non-negative. These neural methods combine discriminative classification, convex regression, and mixture modeling to produce well-defined, consistent, and stable estimators for the density ratio $r(x) = p_1(x)/p_0(x)$ when $p_0$, $p_1$ may be sign-changing, and offer theoretical and practical tools for ensuring statistical efficiency, robustness, and applicability to real-world problems.

1. Foundations of Likelihood Ratio Estimation and Its Quasiprobabilistic Extension

Classical neural likelihood ratio estimation proceeds by formulating $r(x) = p_1(x)/p_0(x)$ as a binary classification problem between samples from $p_1$ and $p_0$. The Bayes-optimal classifier's output is a monotonic function of $r(x)$, and the ratio can be recovered by invertible transformations depending on the loss, e.g., for logistic loss the odds transform yields $r(x)=f/(1-f)$ (Rizvi et al., 2023, Moustakides et al., 2019). Formally, by choosing output parametrizations and appropriate loss pairs $(\phi, \psi)$, Fisher consistency ensures that the trained neural network recovers statistics such as $r(x)$, $p_0$0, or bounded transformations thereof.

In settings where $p_0$1 and/or $p_0$2 are not probability densities but rather quasiprobability densities—functions that integrate to one but may be negative on subsets of the domain—the classical constructions break down: the mixture $p_0$3 may not define a probability measure, resulting in ill-posed classifiers and optimization landscapes without bounded below loss (Drnevich et al., 2024, Drnevich et al., 22 Dec 2025). Moreover, standard divergence minimization and regression losses (e.g., Pearson, MSE, cross-entropy) become non-convex or divergent in the presence of negative densities.

Neural quasiprobabilistic likelihood ratio estimation thus generalizes the discriminative/statistical framework to signed and possibly indefinite measures, requiring new loss functions, novel architectures, and specialized evaluation metrics.

2. Loss Functions and Consistent Estimation in the Quasiprobabilistic Setting

Key challenges in the quasiprobability regime are ensuring convexity of the empirical loss and preserving a unique minimizer corresponding to the true ratio, even when data includes negative weights or signed samples. Several convex loss constructions have been introduced:

• The squared-loss (regression) form $p_0$4 with real-valued $p_0$5 and $p_0$6 permits the Bayes-optimal solution to take values throughout $p_0$7, directly linking $p_0$8 to $p_0$9; the ratio can then be mapped back to $p_1$0 via $p_1$1 (Drnevich et al., 22 Dec 2025). This bypasses the surjectivity limitations of the logistic classifier and remains stable for negative $p_1$2.
• The Signed Pearson loss (L_SP) is constructed for importance-weighted data:

$p_1$3

where $p_1$4 is a regularization term. This loss ensures coercivity and existence of a unique minimizer at $p_1$5 under mild assumptions. If both $p_1$6 and $p_1$7 are nonnegative, $p_1$8 reduces to the standard Pearson/uLSIF loss (Drnevich et al., 2024).

• General surrogate risk minimization with convex link functions $p_1$9 and score transformations allows for flexible invertibility between classifier output and target ratio, regardless of the sign of $r(x) = p_1(x)/p_0(x)$0 (Drnevich et al., 22 Dec 2025).

Additionally, these convex formulations guarantee stability and regularization without the need for special case handling, domain splitting, or output clipping.

3. Neural Architectures and Pseudoprobabilistic Modeling Approaches

Multiple neural strategies have been employed to realize consistent ratio estimation for quasiprobabilities:

• Direct regression networks: Multi-layer perceptrons map $r(x) = p_1(x)/p_0(x)$1 (and possibly $r(x) = p_1(x)/p_0(x)$2, in conditional or parameterized settings) to a scalar $r(x) = p_1(x)/p_0(x)$3, which is mapped to an estimated $r(x) = p_1(x)/p_0(x)$4 via the established link function (Drnevich et al., 22 Dec 2025, Drnevich et al., 2024).
• Signed Mixture Model (SMM) architectures: Each (potentially signed) density is modeled as a sum of strictly positive components $r(x) = p_1(x)/p_0(x)$5 (e.g., normalizing flows or Gaussian densities), with real weights $r(x) = p_1(x)/p_0(x)$6 carrying the sign structure. The ratio is then $r(x) = p_1(x)/p_0(x)$7, so differentiable and sign-flexible by construction. The SMM approach scales to higher dimensions via the expressive power of flows and supports direct generative interpretability (Drnevich et al., 2024).
• InferoStatic Networks (ISN): ISN parameterizes an “inferostatic potential” $r(x) = p_1(x)/p_0(x)$8, simultaneously enabling direct estimation of both the score vector $r(x) = p_1(x)/p_0(x)$9 and the parameterized likelihood ratio $p_1$0, with training based on KLRE and KSE for robust, local learning (Kong et al., 2022).
• Classifier-based architectures using signed labels and weights: The output is trained using modified losses and data pipelines to allow E[y|x] outside [-1,1], preserving the discrimination principle in the signed regime (Drnevich et al., 22 Dec 2025).

Common network design elements include standard feed-forward architectures (typically three hidden layers, 64–128 units, suitable activations), weighted normalization, and explicit sign-handling in label and sample management.

4. Practical Training Strategies and Implementation Details

The quasiprobabilistic setting introduces several distinctive implementation considerations:

• Batch construction and weighting: Sampling draws from both target and reference densities (using signed importance weights for quasiprobabilities), requiring careful normalization of gradient magnitudes to balance positive and negative sample contributions (Drnevich et al., 2024).
• Initialization and regularization: Weight initialization for mixture/flow architectures often uses k-means cluster means to cover data support; weights are split between positive and negative. Regularization by weight decay ($p_1$1 or $p_1$2) and explicit clipping of mixture weights control pathological growth. Early stopping is monitored using the validation Signed Pearson loss (Drnevich et al., 2024).
• Optimizer recommendations: Adam or AdamW with learning-rate scheduling (warmup and cosine decay) is used to ensure stable convergence, along with standard dropout and early stopping for regularization in ordinary MLPs (Rizvi et al., 2023, Drnevich et al., 2024).
• Evaluation metrics: Standard pointwise metrics (MSE, MAE) are complimented by the use of extended Sliced-Wasserstein (SW) distances for signed measures to faithfully quantify the match between signed predicted and true distributions (Drnevich et al., 22 Dec 2025).

Typical pseudocode for the loss-based ratio estimator involves (i) sampling batches from the signed mixture, (ii) assigning signed labels, (iii) computing convex loss (squared or generalized), (iv) backpropagating with signed weight normalization, and (v) validating performance via extended Wasserstein metrics and reweighted histograms (Drnevich et al., 2024, Drnevich et al., 22 Dec 2025).

5. Empirical Performance and Application Domains

Empirical evaluation across both toy and real-world benchmarks demonstrates the superiority of convex loss and mixture-based neural estimators in settings with negative densities or weights:

Setting	Standard Methods Fail	Signed Losses Succeed	SMM Approach
1D Gaussian toy	Oscillatory, diverging ratio	Accurate zero crossing, MSE $p_1$3	Precise local fidelity, MSE $p_1$4
HEP SMEFT reweighting	Baseline $p_1$5–15.2	Loss-based L_SP $p_1$6	SMM $p_1$7

The loss-based and SMM methods enable robust recovery of negative interference patterns, low maximum bias, and substantial improvement in signed Wasserstein distance compared to classical ratio classification (e.g., a factor-of-2 better in SW metric for di-Higgs SMEFT, with $p_1$8 vs $p_1$9) (Drnevich et al., 22 Dec 2025).

Applications include parameter inference with negative-weighted simulation data, importance-weighted off-policy learning where correction terms are negative, and effective field theory analyses where destructive interference dominates parts of the phase space (Drnevich et al., 2024, Drnevich et al., 22 Dec 2025).

6. Limitations, Open Directions, and Extensions

Limitations of current neural quasiprobabilistic likelihood ratio estimation include:

• The lack of theoretical convergence guarantees in fully sign-changing density settings; explicit nonasymptotic rates remain undetermined.
• The SMM approach can exhibit component cancellation when large positive and negative mixture weights offset, potentially degrading gradient quality and interpretability.
• Extension to multi-class or continuous-parameter quasiprobabilities, including amortized inference over entire families $p_0$0, is a plausible direction, with SMM and KLRE/ISN approaches providing structural advantages (Drnevich et al., 2024, Kong et al., 2022).
• Adaptive pruning for SMM and more robust training heuristics for high-dimensional settings are under active investigation.

Possible extensions include constructing quasiprobabilistic Bayesian inference pipelines, enhanced control variate development in Monte Carlo, and direct importance-weighted learning for RL and off-policy correction tasks with negative weights (Drnevich et al., 2024, Drnevich et al., 22 Dec 2025).

7. Conceptual Integration and Quasiprobabilistic Perspective

All successful neural approaches to quasiprobabilistic ratio estimation share a common strategy: they avoid direct estimation of densities or normalization constants, instead approximating the likelihood ratio or its proper transformation via convex, Fisher-consistent risk minimization or flexible generative flows. The resulting estimators are termed “quasiprobabilistic” because the output may not be a proper probability but still provides the correct parametrization for downstream inference or decision tasks. This property enables integration with MCMC, importance sampling, and probabilistic inference methods without requiring explicit normalization or non-negativity (Drnevich et al., 22 Dec 2025, Cobb et al., 2023, Kong et al., 2022).

By unifying the developments in convex loss functions, discriminative signed mixture modeling, kernel-based learning, and potential-based amortized architectures, neural quasiprobabilistic likelihood ratio estimation provides a foundation for inference in cases where classical density-based and standard classifier-based approaches are either unstable or undefined. Empirically, these methods have established new state-of-the-art results in contexts with negative densities, enabling a broad class of simulation-based inference tasks that were previously inaccessible to classical tools (Drnevich et al., 2024, Drnevich et al., 22 Dec 2025).