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Gaussian Processes: Non-Parametric Regression and Optimization

Central bankers vs robots

March 17, 2025

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  • A recent paper titled "The Emotions of Monetary Policy" explores how central bankers' non-verbal cues, such as facial expressions and tone, influence financial markets using machine learning techniques.

  • The study analyzed press conferences by Mario Draghi and Christine Lagarde, finding that non-verbal information can impact financial decision-making, though causation remains uncertain.

  • As ML-driven analysis becomes mainstream, real-time decoding of non-verbal cues may become a tool for market watchers.

  • This could lead to an arms race between algorithmic interpreters and moderated speakers, potentially extending to other areas like fiscal policy announcements and corporate forecasts.

🩺 AI failed to detect critical health conditions: study. Link

  • A study published in Nature's Communications Medicine journal found that AI systems predicting patient mortality often miss critical health conditions.

  • These machine learning models, increasingly used in hospitals, failed to identify about 66% of injuries that could lead to death.

  • The research examined commonly cited models using data from ICU and cancer patients, emphasizing the need to understand specific situations where these models perform well.

  • The study suggests that large language models, similar to ChatGPT, might be more effective in medical contexts if trained on relevant medical literature, but further research is necessary to ensure reliability in clinical settings.

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🧠 Gaussian Processes: Non-Parametric Regression and Optimization

Gaussian Processes (GPs) are a powerful tool for non-parametric regression and optimization, widely used in machine learning and statistics. Unlike parametric models that assume a fixed functional form, GPs define a distribution over functions, allowing flexibility in capturing complex patterns in data.

Gaussian Processes for Regression

In regression tasks, GPs model the relationship between inputs and outputs as a probability distribution over possible functions. Given observed data points, a GP provides a probabilistic prediction for new inputs. The core components of a GP are:

  • Mean function m(x)m(x)m(x): Represents the expected value of the function at any given point.

  • Covariance function k(x,x′)k(x, x')k(x,x′): Defines the relationship between different input points, determining the smoothness and variability of the function.

  • Training and inference: The GP prior is updated with observed data to obtain a posterior distribution, which gives both mean predictions and uncertainty estimates.

Covariance Functions and Hyperparameters

The choice of the covariance function is crucial, as it defines the smoothness and structure of the functions a GP can model. Common kernels include the squared exponential (RBF), Matérn, and periodic kernels. The hyperparameters of the kernel, such as length scale and variance, are typically learned from data using methods like maximum likelihood estimation or Bayesian inference.

Gaussian Processes for Optimization

GPs are widely used in Bayesian optimization, a technique for optimizing expensive-to-evaluate functions. The GP serves as a surrogate model for the objective function, capturing its shape while incorporating uncertainty. Acquisition functions, such as expected improvement or upper confidence bound, guide the selection of new evaluation points to efficiently find the optimum.

Advantages and Limitations

  • Advantages:

    • Provides uncertainty estimates.

    • Works well with small to medium-sized datasets.

    • Flexible and non-parametric, adapting to complex functions.

  • Limitations:

    • Computational cost grows cubically with the number of data points.

    • Requires careful selection of kernel functions.

    • Not suitable for large-scale datasets without approximations.

Conclusion

Gaussian Processes offer a flexible and powerful approach to non-parametric regression and optimization. Their ability to provide uncertainty estimates makes them valuable in applications where data is limited or expensive to obtain. Despite their computational challenges, GPs remain a cornerstone of probabilistic modeling in machine learning.

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