Learning about probabilistic inference and forecasting by playing with multivariate normal distributions

TL;DR

This paper explores probabilistic inference and forecasting using multivariate normal distributions, illustrating how their properties facilitate understanding uncertainties, correlations, and predictions in measurement systems through numerical examples and R-based analysis.

Contribution

It provides a comprehensive, practical guide to applying multivariate normal properties for inference and forecasting, emphasizing covariance matrix construction and evidence propagation.

Findings

Illustrates covariance matrix construction in measurement scenarios

Demonstrates effects of systematics and correlations on inference

Shows how to perform predictions and fits under linear models

Abstract

The properties of the normal distribution under linear transformation, as well the easy way to compute the covariance matrix of marginals and conditionals, offer a unique opportunity to get an insight about several aspects of uncertainties in measurements. The way to build the overall covariance matrix in a few, but conceptually relevant cases is illustrated: several observations made with (possibly) different instruments measuring the same quantity; effect of systematics (although limited to offset, in order to stick to linear models) on the determination of the 'true value', as well in the prediction of future observations; correlations which arise when different quantities are measured with the same instrument affected by an offset uncertainty; inferences and predictions based on averages; inference about constrained values; fits under some assumptions (linear models with known standard deviations). Many numerical examples are provided, exploiting the ability of the R language to handle large matrices and to produce high quality plots. Some of the results are framed in the general problem of 'propagation of evidence', crucial in analyzing graphical models of knowledge.