Likelihood-Preserving Embeddings

• Likelihood-Preserving Embeddings are representations that retain the statistical structure of data by controlling the distortion of log-likelihood ratios in compressed spaces.
• They are applied across exponential families, graph models, kernel methods, and diffusion architectures to support exact or approximate inference and link prediction.
• Current research addresses challenges in balancing compression and fidelity, with promising extensions for nonlinear, adaptive, and distributed inference frameworks.

Likelihood-Preserving Embeddings are representations designed to enable effective statistical inference, model selection, and hypothesis testing directly within compressed or embedded data spaces, such that classical likelihood-based conclusions are provably retained. Contrary to conventional dimensionality reduction methods that obscure the data's probabilistic structure, likelihood-preserving embeddings seek to tightly control or manipulate the distortion of log-likelihood ratios, sometimes approaching the information-theoretic ideal of sufficient statistics. These methods extend across multivariate, network, kernel, meta-embedding, and deep learning regimes, unifying modern advances in surrogate likelihood estimation and robust distributed inference.

1. Foundations: Likelihood-Ratio Distortion and the Hinge Theorem

The central metric formalizing likelihood preservation is the Likelihood-Ratio Distortion $\Delta_n$ (Akdemir, 27 Dec 2025). For any embedding pair $(T_\phi, h_\psi)$ mapping data $X_{1:n}$ to compressed statistics and a surrogate log-likelihood $\widetilde{L}_n(\theta)$, $\Delta_n$ captures the supremum error between true and embedded log-likelihood ratios: $$\Delta_n = \sup_{\theta,\theta'} \big| [L_n(\theta) - L_n(\theta')] - [\widetilde L_n(\theta) - \widetilde L_n(\theta')] \big|$$ The Hinge Theorem establishes necessary and sufficient conditions for all likelihood-based inferential procedures to be preserved:

• If the pointwise error $$\varepsilon_n = \sup_\theta |\frac{1}{n} L_n(\theta) - h_\psi(\theta,S_\phi)|$$ satisfies $\varepsilon_n = o_p(1/n)$, then $\Delta_n = o_p(1)$, and all likelihood-ratio tests, Bayes factors, MLEs, and AIC/BIC criteria are asymptotically preserved.
• In exponential families, exact preservation implies recovery of sufficient statistics and a minimal embedding dimension at least as large as the model's parameter count.

Thus, likelihood preservation in embedding is fundamentally a problem of controlling $\Delta_n$, not merely matching moments or distributions (Akdemir, 27 Dec 2025).

2. Explicit Likelihood-Preserving Constructions

2.1 Direct Learning of Sufficient Statistics

For exponential family models, likelihood-preserving embeddings can be constructed by learning or extracting the canonical sufficient statistic, which guarantees $\Delta_n = 0$ (Akdemir, 27 Dec 2025). Neural architectures implementing $T_\phi : X \to \mathbb{R}^m$ and $h_\psi : \Theta \times \mathbb{R}^m \to \mathbb{R}$ can be jointly optimized via pointwise likelihood-matching losses,

$$\mathcal L_{\rm point}(\phi,\psi) = \mathbb{E}_{\theta, X_{1:n}} \Bigl[ \bigl(\tfrac{1}{n} L_n(\theta) - h_\psi(\theta, S)\bigr)^2 \Bigr]$$

with sharp bounds connecting empirical error to $\Delta_n$ and downstream inferential loss. For non-exponential family models (e.g., Cauchy), these embeddings approximate likelihood ratios only up to the finite-dimensional information retained.

2.2 Graph Likelihood Embeddings

In directed and undirected graphs, graph-likelihood objectives are used to encode edge probabilities as functions of node embeddings, using logistic models with negative sampling (Abu-el-haija et al., 2017). Edge existence is contrasted with soft pseudo-negative pairs from random walks: $$\operatorname{Pr}(G) = \prod_{u,v \in V} \sigma(g(u,v))^{\mathcal{D}_{uv}} (1-\sigma(g(u,v)))^{\mathbb{1}[(u,v) \notin E_\text{train}]}$$ Asymmetric low-rank decompositions $g(u,v) = \langle L^T f(Y_u), R f(Y_v) \rangle$ yield directed edge modeling, and embedding dimension $b$ regularizes geometric complexity. This approach produces concise, likelihood-preserving embeddings suitable for link prediction, with strong generalization and memory efficiency.

2.3 Kernel Mean Embedding Likelihoods

Inlikelihood-free settings, the Kernel Embedding Likelihood-Free Inference (KELFI) framework constructs surrogate likelihoods by embedding data and parameter spaces in RKHS and approximating expectations through conditional mean embeddings (Hsu et al., 2019): $$q(y|\theta) = \langle \kappa_\epsilon(y,\cdot),\hat\mu_{X|\Theta=\theta} \rangle_{\mathcal{H}_k}$$ These surrogate densities converge uniformly to the true soft likelihood at rate $O_p((m\lambda)^{-1/2}+\lambda^{1/2})$ and can be directly sampled for posterior inference, with automatic hyperparameter selection via the surrogate marginal likelihood.

2.4 Volume-Preserving Map Embeddings

Multiscale cascaded diffusion models can achieve likelihood preservation by embedding images through hierarchical volume-preserving maps (e.g., Laplacian pyramid, orthonormal wavelets) (Li et al., 13 Jan 2025). These transforms are invertible and have unit Jacobian determinants: $$\log p_\theta(x) = \log p_\theta( h(x) ) = \sum_{s=1}^S \log p_\theta( z^{(s)} | z^{(<s)} )$$ This design enables exact likelihood computation over multi-scale representations, facilitating perceptually aligned density estimation and out-of-distribution detection.

3. Kernel Gaussian Embedding and Likelihood-Based Hypothesis Testing

Kernel Gaussian embedding combines kernel mean and covariance embeddings to map probability measures $P$, $Q$ to mutually singular Gaussians in RKHS, enabling nonparametric two-sample testing by the relative entropy (KL divergence) between the embedded distributions (Santoro et al., 11 Aug 2025): $$T_\gamma(P, Q) = \mathrm{KL}( \mathcal{N}(\mu_P, \Sigma_P + \gamma I), \mathcal{N}(\mu_Q, \Sigma_Q + \gamma I) )$$ The resulting test statistic exhibits a $0/\infty$ dichotomy: vanishing under the null ($P=Q$) and diverging under the alternative ($P \neq Q$), providing consistency and uniform power guarantees.

4. Spectral and Meta-Embedding Approaches

4.1 Spectral Dimensionality Reduction under Maximum Likelihood

Maximum Entropy Unfolding (MEU), Acyclic LLE (ALLE), and related spectral methods are recast as exact or approximate maximum likelihood embeddings under the Gaussian Markov random field formalism (Lawrence, 2010). The log-likelihood of the embedding is

$$\log p(X) = -\tfrac{1}{2} \operatorname{Tr}[X^T L X] - \text{const}$$

where $L$ encodes the neighborhood graph. Variants (e.g., graphical lasso) yield embeddings that preserve likelihood by enforcing global or local constraints.

4.2 Meta-Embeddings and Likelihood Ratio Preservation

Gaussian meta-embeddings (GMEs) formalize likelihood ratio scoring in speaker recognition; each input $y$ is mapped to the likelihood function $f(y)(z) \propto P(y|z)$ over latent speaker identity $z$ (Brummer et al., 2018). Scoring is performed by Hilbert inner products between such functions, propagating uncertainty and handling heavy-tailed models.

5. Applications: Distributed Inference, Image Likelihoods, and Practical Compression

5.1 Distributed Clinical Inference

Likelihood-preserving embeddings facilitate federated statistical inference with privacy constraints by transmitting only summary vectors that allow centralized exact likelihood evaluation (Akdemir, 27 Dec 2025). For linear models, sufficient summary statistics reduce data transmission by $>100\times$ without loss of inferential performance.

5.2 Whitened CLIP Likelihoods

Whitened CLIP (W-CLIP) is an invertible linear transformation that renders CLIP vision-language embeddings isotropic Gaussian, enabling closed-form log-likelihood scoring for images and captions (Betser et al., 11 May 2025): $$L(x) = -\frac{1}{2} \| W(x - \mu) \|^2 - \frac{d}{2} \log(2\pi)$$ Statistical tests confirm high normality, and empirical applications to artifact detection, domain shift, and out-of-distribution classification demonstrate practical efficacy.

5.3 Perceptual Likelihoods in Image Modeling

Hierarchical likelihood-preserving embeddings, via Laplacian or wavelet decompositions, facilitate joint training of cascaded generative models capable of robust density estimation and perceptually meaningful anomaly detection (Li et al., 13 Jan 2025). The training objective aligns with weighted sums of Wasserstein distances, connecting likelihood maximization to Earth Mover's Distance.

6. Limitations and Controversies

The impossibility theorem demonstrates that universal likelihood preservation demands injectivity—essentially no information loss—relegating strong guarantees to model-class-specific or approximately sufficient approaches (Akdemir, 27 Dec 2025). For non-exponential families, trade-offs between compression and inferential fidelity are unavoidable. Linear whitening (e.g., W-CLIP) achieves only approximate likelihood preservation, and volume-preserving transformations must be computationally tractable and invertible.

7. Future Directions

Future work involves:

• Nonlinear, adaptive, or mixture-model extensions of likelihood-preserving embedding frameworks.
• Domain-adaptive or multimodal whitening strategies to better match complex distributions.
• Formal statistical guarantees for metric-aligned score matching and perceptual likelihood surrogates.
• Efficient algorithms for distributed and federated inference under stringent communication and privacy constraints.

Likelihood-preserving embedding remains an active area of research, linking statistical theory, representation learning, and practical inference in high-dimensional settings.