Information-Entropic Deep Learning with Gaussian Process Regularisation for Uncertainty-Aware Quantitative Trading

Quantitative trading systems require predictive models that simultaneously deliver accurate forecasts, calibrated uncertainty quantification, and actionable risk measures. This paper proposes an information-theoretic semiparametric regression framework combining a convolutional neural network-Transformer (CNN-Transformer) network for nonlinear temporal dependencies with a Gaussian process (GP) prior for residual autocorrelation and calibrated predictive distributions. Three theoretical results are established: an identifiability theorem guarantees joint recoverability of the nonparametric and GP components; a consistency theorem showing that the penalised maximum likelihood estimator converges at a rate n-1/(2+deff); and a coverage theorem proving asymptotic nominal coverage of the GP's credible intervals. The framework enables an entropy-regulated trading module where predictive differential entropy informs position sizing via an uncertainty-penalised Kelly criterion, Kullback-Leibler divergence quantifies model uncertainty, and CVaR-constrained optimisation controls the tail risk. Simulations show the method outperforms the CNN, long short-term memory (LSTM), Transformer, XGBoost, random forest, least absolute shrinkage and selection operator (LASSO), and standard GP regression approaches. Backtesting on four Chinese A-share stocks yielded annualised returns of 15.9-22.4% with Sharpe ratios of 0.49-0.62, maximum drawdowns below 15%, and daily 95% CVaR reductions of 28-31% relative to a full-Kelly baseline, confirming both predictive accuracy and risk management effectiveness.