Maximum Entropy Principle

Boltzmann Price: Toward Understanding the Fair Price in High-Frequency Markets ArXiv ID: 2507.09734 “View on arXiv” Authors: Przemysław Rola Abstract In this paper, we introduce a parametrized family of prices derived from the Maximum Entropy Principle. The price is obtained from the distribution that minimizes bias, given the bid and ask volume imbalance at the top of the order book. Under specific parameter choices, it closely approximates the mid-price or the weighted mid-price. Using probabilities of bid and ask states, we propose a model of price dynamics in which both drift and volatility are driven by volume imbalance. Compared to standard models like Bachelier or Geometric Brownian Motion with constant volatility, our model can generate higher kurtosis and heavy-tailed distributions. Additionally, the drift term naturally emerges as a consequence of the order book imbalance. We validate the model through simulation and demonstrate its fit to historical equity data. The model provides a theoretical framework, integrating price, volume imbalance, and spread. ...

Statistics-Informed Parameterized Quantum Circuit via Maximum Entropy Principle for Data Science and Finance ArXiv ID: 2406.01335 “View on arXiv” Authors: Unknown Abstract Quantum machine learning has demonstrated significant potential in solving practical problems, particularly in statistics-focused areas such as data science and finance. However, challenges remain in preparing and learning statistical models on a quantum processor due to issues with trainability and interpretability. In this letter, we utilize the maximum entropy principle to design a statistics-informed parameterized quantum circuit (SI-PQC) for efficiently preparing and training of quantum computational statistical models, including arbitrary distributions and their weighted mixtures. The SI-PQC features a static structure with trainable parameters, enabling in-depth optimized circuit compilation, exponential reductions in resource and time consumption, and improved trainability and interpretability for learning quantum states and classical model parameters simultaneously. As an efficient subroutine for preparing and learning in various quantum algorithms, the SI-PQC addresses the input bottleneck and facilitates the injection of prior knowledge. ...