arXiv subjects

Dmitry Lesnik

Publications and source records attributed to Dmitry Lesnik.

8 recordsLinked to original sources

arXiv2026

State Algebra for Probabilistic Logic

This paper presents a Probabilistic State Algebra as an extension of deterministic propositional logic, providing a computational framework for constructing Markov Random Fields (MRFs) through pure linear algebra. By mapping logical states to real-valued coordinates interpreted as energy potentials, we define an energy-based model where global probability distributions emerge from coordinate-wise Hadamard products. This approach bypasses the traditional reliance on graph-traversal algorithms and compiled circuits, utilising $t$-objects and wildcards to embed logical reduction natively within matrix operations. We demonstrate that this algebra constructs formal Gibbs distributions, offering a rigorous mathematical link between symbolic constraints and statistical inference. A central application of this framework is the development of Probabilistic Rule Models (PRMs), which are uniquely capable of incorporating both probabilistic associations and deterministic logical constraints simultaneously. These models are designed to be inherently interpretable, supporting a human-in-the-loop approach to decisioning in high-stakes environments such as healthcare and finance. By representing decision logic as a modular summation of rules within a vector space, the framework ensures that complex probabilistic systems remain auditable and maintainable without compromising the rigour of the underlying configuration space.

cs.AI↗

arXiv2025

Principled Frequentist Estimation of Racial Disparity in Credit Approval under Unobserved Race

Estimating racial disparities in loan-approval probabilities when race is unobserved is routinely required for fair lending compliance. In such cases, race probabilities-typically from Bayesian Improved Surname Geocoding (BISG)-stand in for true race. Prior work shows that common heuristic approaches, including the Threshold and Weighting estimators, are inconsistent under valid identification assumptions, compromising internal validity. A recent Bayesian approach demonstrates consistency under assumptions reasonable in many fair lending contexts. This approach hinges on the insight that identification requires the race predictors to be exogenous with respect to loan approval, essentially an instrumental-variables design. We present a frequentist counterpart to this solution via Ordinary Least Squares (OLS) and Maximum Likelihood Estimation (MLE) under a similar exogeneity assumption. To satisfy these assumptions in practice, we introduce (i) a surname-only proxy analogous to BISG and (ii) an income-stratified prior for race probabilities. Monte Carlo simulations and an application to 2023 Los Angeles HMDA data confirm superior performance: this method reduces RMSE in the LA Black/White adverse-impact ratio by 79.7% (from 10.639pp to 2.158pp) compared to a Weighting estimator with the standard prior.

stat.AP↗

arXiv2025

Probabilistic Rule Models as Diagnostic Layers: Interpreting Structural Concept Drift in Post-Crisis Finance

Machine learning models used for high-stakes predictions in domains like credit risk face critical degradation due to concept drift, requiring robust and transparent adaptation mechanisms. We propose an architecture, where a dedicated correction layer is employed to efficiently capture systematic shifts in predictive scores when a model becomes outdated. The key element of this architecture is the design of a correction layer using Probabilistic Rule Models (PRMs) based on Markov Logic Networks, which guarantees intrinsic interpretability through symbolic, auditable rules. This structure transforms the correction layer from a simple scoring mechanism into a powerful diagnostic tool capable of isolating and explaining the fundamental changes in borrower riskiness. We illustrate this diagnostic capability using Fannie Mae mortgage data, demonstrating how the interpretable rules extracted by the correction layer successfully explain the structural impact of the 2008 financial crisis on specific population segments, providing essential insights for portfolio risk management and regulatory compliance.

q-fin.RM↗

arXiv2025

State Algebra for Propositional Logic

This paper presents State Algebra, a novel framework designed to represent and manipulate propositional logic using algebraic methods. The framework is structured as a hierarchy of three representations: Set, Coordinate, and Row Decomposition. These representations anchor the system in well-known semantics while facilitating the computation using a powerful algebraic engine. A key aspect of State Algebra is its flexibility in representation. We show that although the default reduction of a state vector is not canonical, a unique canonical form can be obtained by applying a fixed variable order during the reduction process. This highlights a trade-off: by foregoing guaranteed canonicity, the framework gains increased flexibility, potentially leading to more compact representations of certain classes of problems. We explore how this framework provides tools to articulate both search-based and knowledge compilation algorithms and discuss its natural extension to probabilistic logic and Weighted Model Counting.

cs.AI↗

arXiv2016

Retarded action principle and self-financing portfolio dynamics

We derive a consistent differential representation for the dynamics of a self-financing portfolio for different hedging strategies. In the basis of the derivation there is the so called "retarded action principle", which represents the causality in the evolution of dependent stochastic variables. We demonstrate this principle on example of a vanilla and a storage option.

q-fin.MF↗

arXiv2015

A state vector algebra for algorithmic implementation of second-order logic

We present a mathematical framework for mapping second-order logic relations onto a simple state vector algebra. Using this algebra, basic theorems of set theory can be proven in an algorithmic way, hence by an expert system. We illustrate the use of the algebra with simple examples and show that, in principle, all theorems of basic set theory can be recovered in an elementary way. The developed technique can be used for an automated theorem proving in the 1st and 2nd order logic.

cs.AI↗

arXiv2015

Intrinsic Storage Valuation by Variational Analysis

The mathematical problem concerning intrinsic storage optimisation is formulated and solved by means of variational analysis. The solution, though obtained in implicit form, still sheds light on many important features of the optimal exercise strategy. It is shown how the solution depends on different constraint types including carry cost and cycle constraint. Additionally, the relationship between intrinsic and stochastic solutions is investigated. In particular, we show that the optimal stochastic exercise decision is always close to the intrinsic one.

math.OC↗

arXiv2012

Storage option an Analytic approach

The mathematical problem of the static storage optimisation is formulated and solved by means of a variational analysis. The solution obtained in implicit form is shedding light on the most important features of the optimal exercise strategy. We show how the solution depends on different constraint types including carry cost and cycling constraint. We investigate the relation between intrinsic and stochastic solutions. In particular we give another proof that the stochastic problem has a "bang-bang" optimal exercise strategy. We also show why the optimal stochastic exercise decision is always close to the intrinsic one. In the second half we develop a perturbation analysis to solve the stochastic optimisation problem. The obtained approximate solution allows us to estimate the time value of the storage option. In particular we find an answer to rather academic question of asymptotic time value for the mean reversion parameter approaching zero or infinity. We also investigate the differences between swing and storage problems. The analytical results are compared with numerical valuations and found to be in a good agreement.

q-fin.CP↗