This paper provides a unified perspective for the Kullback-Leibler (KL)-divergence and the integral probability metrics (IPMs) from the perspective of maximum likelihood density-ratio estimation (DRE). Both the KL-divergence and the IPMs are widely used in various fields in applications such as generative modeling. However, a unified understanding of these concepts has still been unexplored. In this paper, we show that the KL-divergence and the IPMs can be represented as maximal likelihoods differing only by sampling schemes, and use this result to derive a unified form of the IPMs and a relaxed estimation method. To develop the estimation problem, we construct an unconstrained maximum likelihood estimator to perform DRE with a stratified sampling scheme. We further propose a novel class of probability divergences, called the Density Ratio Metrics (DRMs), that interpolates the KL-divergence and the IPMs. In addition to these findings, we also introduce some applications of the DRMs, such as DRE and generative adversarial networks. In experiments, we validate the effectiveness of our proposed methods.

Machine learning is an increasingly popular tool with some success in predicting stock prices. One promising method is the Trader-Company (TC) method, which takes into account the dynamism of the stock market and has both high predictive power and interpretability. Machine learning-based stock prediction methods, including the TC method, have been concentrating on point prediction. However, point prediction in the absence of uncertainty estimates lacks credibility quantification and raises concerns about safety. The challenge in this paper is to make an investment strategy that combines high predictive power and the ability to quantify uncertainty. We propose a novel approach called Uncertainty Aware Trader-Company Method (UTC) method. The core idea of this approach is to combine the strengths of both frameworks by merging the TC method with the probabilistic modeling, which provides probabilistic predictions and uncertainty estimations. We expect this to retain the predictive power and interpretability of the TC method while capturing the uncertainty. We theoretically prove that the proposed method estimates the posterior variance and does not introduce additional biases from the original TC method. We conduct a comprehensive evaluation of our approach based on the synthetic and real market datasets. We confirm with synthetic data that the UTC method can detect situations where the uncertainty increases and the prediction is difficult. We also confirmed that the UTC method could detect abrupt changes in data-generating distributions. We demonstrate with real market data that the UTC method can achieve higher returns and lower risks than baselines.

Recent developments in deep learning techniques have motivated intensive research in machine learning-aided stock trading strategies. However, since the financial market has a highly non-stationary nature hindering the application of typical data-hungry machine learning methods, leveraging financial inductive biases is important to ensure better sample efficiency and robustness. In this study, we propose a novel method of constructing a portfolio based on predicting the distribution of a financial quantity called residual factors, which is known to be generally useful for hedging the risk exposure to common market factors. The key technical ingredients are twofold. First, we introduce a computationally efficient extraction method for the residual information, which can be easily combined with various prediction algorithms. Second, we propose a novel neural network architecture that allows us to incorporate widely acknowledged financial inductive biases such as amplitude invariance and time-scale invariance. We demonstrate the efficacy of our method on U.S. and Japanese stock market data. Through ablation experiments, we also verify that each individual technique contributes to improving the performance of trading strategies. We anticipate our techniques may have wide applications in various financial problems.

The exponential mechanism is a general method to construct a randomized estimator that satisfies $(\varepsilon, 0)$-differential privacy. Recently, Wang et al. showed that the Gibbs posterior, which is a data-dependent probability distribution that contains the Bayesian posterior, is essentially equivalent to the exponential mechanism under certain boundedness conditions on the loss function. While the exponential mechanism provides a way to build an $(\varepsilon, 0)$-differential private algorithm, it requires boundedness of the loss function, which is quite stringent for some learning problems. In this paper, we focus on $(\varepsilon, \delta)$-differential privacy of Gibbs posteriors with convex and Lipschitz loss functions. Our result extends the classical exponential mechanism, allowing the loss functions to have an unbounded sensitivity.