Sufficient Statistic Parameterization (SSP)

• SSP is a framework that uses low-dimensional sufficient statistics to encapsulate all information in data for probabilistic model representation.
• It supports applications like differential privacy, data thinning, and model expansion by enabling noise addition and synthetic data generation.
• SSP highlights the tradeoff between statistical optimality and computational tractability, especially in high-dimensional models with intractable partition functions.

Sufficient Statistic Parameterization (SSP) describes a principled methodology for representing probabilistic models, estimation strategies, reduction procedures, or computational pipelines by explicitly leveraging sufficient statistics. An SSP expresses statistical or probabilistic inference as a function of (often low-dimensional) statistics that retain all information about parameters of interest present in the original data. This framework provides the foundational structure for numerous domains, including exponential families, differential privacy, algorithmic reductions, data thinning, and diagrammatic probability.

1. Foundational Principles of Sufficient Statistic Parameterization

In classical statistical theory, a sufficient statistic $T(X)$ for parameter $\theta$ is a function such that the conditional distribution of the data $X$ given $T(X)$ is independent of $\theta$; equivalently, the likelihood $p(x \mid \theta)$ factors as $g(T(x), \theta) h(x)$. In exponential families, this factorization is canonical: $$p(x \mid \theta) = h(x) \exp\left\{ \eta(\theta)^\top T(x) - A(\theta) \right\}$$ For i.i.d. data $D = \{x^{(i)}\}_{i=1}^n$, the joint likelihood depends on $D$ only through $T(D) = \sum_{i=1}^n T(x^{(i)})$.

SSP formalizes the representation of a statistical problem, model, or algorithm as a function—or optimization over functions—of sufficient statistics. Lehmann-Scheffé theory ensures that, for risk minimization, no statistical information about $\theta$ is lost when passing to $T(D)$ in well-behaved models (Montanari, 2014). SSP also provides a setting for model expansions, synthetic data generation, structure-preserving data transformations, and categorical abstractions (Dharamshi et al., 2023, Jacobs, 2022).

2. SSP in Differential Privacy and Private Regression

A key application of SSP is in designing differentially private machine learning algorithms, particularly for linear and logistic regression. Here, private estimation is often reduced to privatizing the sufficient statistics (e.g., $S_1=X^\top X$, $S_2=X^\top y$ in least squares regression) under appropriate noise mechanisms (Ferrando et al., 23 May 2024).

Classic (data-independent) SSP applies calibrated Gaussian noise to each $S_k$: $$\widetilde{S}_k = S_k + \eta_k, \qquad \eta_k \sim \mathcal{N}(0,\,\sigma_k^2 I)$$ where $\sigma_k$ is chosen according to the global sensitivity of $S_k$ and differential privacy $(\epsilon,\delta)$ constraints.

Recent advances introduce data-dependent SSP (DD-SSP), exploiting the fact that sufficient statistics can often be rewritten as linear queries (pairwise marginals over discretized features). By first running a private mechanism for all pairwise marginals (e.g., AIM), then post-processing to reconstruct the privatized sufficient statistics, DD-SSP achieves tighter estimates and lower empirical error, with provably equivalent privacy guarantees via post-processing (Ferrando et al., 23 May 2024).

For example, in logistic regression where no finite-dimensional sufficient statistic exists, DD-SSP employs a Chebyshev polynomial approximation of the log-likelihood, reducing the problem to privatizing empirical moments (again linear queries). Empirically, DD-SSP and synthetic-data approaches using the same privatized queries yield almost identical utility, demonstrating that query-based sufficient-statistic estimation determines overall performance.

3. Algorithmic, Computational, and Complexity Aspects

Although SSP provides statistically lossless reductions, a fundamental computational caveat arises: reducing data to sufficient statistics can convert tractable estimation tasks into computationally hard problems. In many high-dimensional exponential families (notably those whose normalization constants correspond to #P-hard partition functions), inverting the moment map $T^*(\theta)$ to recover $\theta$ from $T$ is intractable (Montanari, 2014).

Montanari demonstrates that, under mild regularity,

• If there exists a polynomial-time consistent estimator $\Phi$ mapping $T$ to $\theta$ (satisfying $\|\Phi(T,\varepsilon)-\theta\| \le \varepsilon$ whenever $T\approx T^*(\theta)$), then there must exist a FPRAS for the partition function $Z(\theta)$.
• For antiferromagnetic Ising models on $k$-regular graphs with $\beta$ above a critical threshold, such approximation is not possible unless $\mathrm{RP} = \mathrm{NP}$ (Montanari, 2014).

Thus, while SSP enables information-theoretically optimal reduction, it may destroy computational tractability in general graphical or latent-variable models, particularly where partition function approximation is hard.

4. Generalizations: Data Thinning, Model Expansion, Categorical SSP

Generalizations of SSP provide new methodologies for data decomposition, hypothesis testing, and categorical abstraction:

• Generalized Data Thinning: SSP unifies sample splitting and convolution-based thinning in exponential families. For a random variable $X$, SSP provides a joint distribution over independent $(X^{(1)},...,X^{(K)})$ and a mapping $T$ such that $X=T(X^{(1)},...,X^{(K)})$, ensuring no information loss about $\theta$ and full preservation of Fisher information (Dharamshi et al., 2023).
• Parameter Expansion: Embedding a base model $p(x;\theta)$ into a larger family $p(x;\theta,\eta)$ that "activates" additional sufficient components can strictly improve statistical testing power and accelerate EM convergence. The reduction in error bounds is quantified by a measure $R$ based on differences of Hellinger distances before and after expansion. This formalizes how parameter expansions activate new data-relevant summary statistics (Yatracos, 2015).
• Categorical/Diagrammatic SSP: In categorical probability, every discrete probabilistic channel factors through a unique (up to isomorphism) sufficient statistic, corresponding to the splitting of a self-adjoint idempotent in the Kleisli category of finite sets and Markov kernels. The Fisher-Neyman factorization appears as a split idempotent with retraction (the sufficient statistic) and section (the residual) (Jacobs, 2022).

5. SSP in Algorithmic and Applied Domains

• Differentially Private ML: SSP-based algorithms, including data-dependent variants, form the foundation for modern query-answering and synthetic-data generation under differential privacy. Empirical studies confirm significant accuracy improvements of DD-SSP over data-independent alternatives (Ferrando et al., 23 May 2024).
• Games and Control: For finite-horizon two-player zero-sum stochastic Bayesian games, optimal or suboptimal strategies can be parametrized entirely by sufficient statistics (belief states, dual parameters), allowing for dynamic programming or recursive LP formulations. Windowed LP strategies yield provable near-optimality in large games (Orpa et al., 2020).
• Bayesian/Hybrid Analysis Pipelines: In gravitational wave background detection, cross-correlation statistics and variances constructed segmentwise serve as approximate sufficient statistics. Reducing petabyte-scale strain data to summary statistics enables tractable posterior inference with no loss of scientific information in the weak-signal regime (Matas et al., 2020).
• Amortized Inference: In deep latent-variable models, neural networks parameterize sufficient statistics ("neural sufficient statistics") to construct scalable, adaptive importance samplers or amortized proposals, directly generalizing exponential-family conjugacy to non-conjugate and high-dimensional settings (Wu et al., 2019).
• Information Theory: For feedback Gaussian channels, SSP enables full parameterization of optimal encoding processes in terms of two sequentially updated sufficient statistics (Kalman filter innovations), leading to explicit Riccati equations for capacity computation (Charalambous et al., 2021).

6. Theoretical Guarantees and Limitations

Guarantees

• Statistical optimality: SSP ensures maximum data reduction without information loss in models admitting sufficient statistics (Lehmann-Scheffé).
• Preservation of Fisher information: In thinning/generalized SSP, the sum of Fisher informations of independent folds equals that of the original variable (Dharamshi et al., 2023).
• Equivalence in synthetic data and query-SSP: ML on synthetic datasets constructed to match DP-released marginals achieves the same utility as direct estimation from privatized sufficient statistics (Ferrando et al., 23 May 2024).
• Categorical existence/uniqueness: Every channel in a positive Markov category admits a unique split (up to isomorphism) by a sufficient statistic (Jacobs, 2022).

Limitations

• Computational intractability: For general exponential families with hard partition functions (e.g., non-attractive Ising models), SSP-based parameter recovery is infeasible (Montanari, 2014).
• Nonexistence in non-exponential families: No nontrivial SSP is available in many models lacking low-dimensional sufficient statistics (e.g., Bernoulli, Cauchy).
• Dependence on exact model specification: In real data, model misspecification ruins the conditional independence and sufficiency guarantees needed by general SSP decompositions (Dharamshi et al., 2023).
• Approximation errors in high-dimensional regimes: Approximating sufficient statistics (e.g., for logistic regression via Chebyshev expansions) introduces a bias-variance tradeoff dependent on the accuracy of the approximation (Ferrando et al., 23 May 2024).

7. Summary Table: Applications and Properties of SSP

Domain/Method	Role of SSP	Guarantee/Challenge
Differential Privacy	Privatize $T(D)$, reconstruct	Utility, privacy; DP postproc.
Data Thinning/Splitting	Decompose $X$, retain info	Fisher information preserved
Exponential Families	Reduce data to $T$	Statistically optimal
Graphical Models	Use global $T$ or local stats	May be intractable
Categorical Probability	Diagrammatic split idempotent	Universal existence/uniqueness
Game Theory	Strategy parametrization	Recursive LP, DP parametrized
Deep Learning/Amortized	Neural $T_\phi$ param.	Blockwise/proposal efficiency

SSP is a unifying concept in modern statistical methodology, subsuming classical reduction, privacy, optimal control, learning theory, and categorical probability, while delineating the tradeoff between information-theoretic sufficiency and computational feasibility. Its ongoing development continues to motivate foundational work in privacy-preserving analysis, model expansion, algorithmic complexity, and formal probability.

References: (Ferrando et al., 23 May 2024, Montanari, 2014, Jacobs, 2022, Dharamshi et al., 2023, Yatracos, 2015, Matas et al., 2020, Orpa et al., 2020, Wu et al., 2019, Charalambous et al., 2021)