Computing Probability Density Values for Multivariate Normal Distributions

The multivariate normal distribution is a fundamental probability distribution widely used in statistics and machine learning, particularly when dealing with high-dimensional data. Computing its probability density requires understanding several key concepts and implementation steps.

First, the multivariate normal distribution is determined by a mean vector and covariance matrix. The mean vector describes the central position in each dimension, while the covariance matrix contains information about correlations between dimensions. The probability density function calculation depends not only on the distance from the data point to the mean but also on the structure of the covariance matrix.

Specifically, the computational procedure involves: Calculating the deviation vector: Given a data point, first compute its difference from the mean vector. Covariance matrix inversion: Since covariance matrices are typically symmetric and positive-definite, efficient linear algebra methods (such as Cholesky decomposition) can be used for inversion to enhance numerical stability. In code implementation, this can be achieved using numpy.linalg.cholesky() followed by solving triangular systems. Quadratic form computation: Using the deviation vector and the inverse covariance matrix, compute the quadratic form expression, which measures the "distance" between the data point and the distribution center. This can be implemented as dev.T @ inv_cov @ dev in matrix operations. Normalization term: The multivariate normal probability density includes a determinant normalization term to ensure the integral equals 1. The complete formula combines exponential of the quadratic form with the normalization constant (2π)^{-k/2} |Σ|^{-1/2}.

In practical applications, logarithmic probability density is often computed to avoid numerical underflow, which is particularly common in machine learning optimization problems. Furthermore, if the covariance matrix is singular (e.g., in dimensionality-reduced data), pseudoinverse or other regularization methods can be employed for stable computation, such as using numpy.linalg.pinv() with a tolerance parameter.

Understanding multivariate normal distribution probability density computation not only aids statistical modeling but also forms the foundation for advanced methods like Gaussian processes and Bayesian optimization, where efficient density evaluation is crucial for performance.