Conditional expectation and probability
February 3, 2020 — September 21, 2022
Things I would like to re-derive for my own entertainment:
Conditioning in the sense of measure-theoretic probability. Kolmogorov formulation. Conditioning as Radon-Nikodym derivative. Clunkiness of definition due to niceties of Lebesgue integration.
H.H. Rugh’s answer is nice.
1 Conditional algebra
TBC
2 Nonparametric
Conditioning in full measure-theoretic glory for Bayesian nonparametrics. E.g. conditioning of Gaussian Processes is also fun.
3 Disintegration
4 BLUE in Gaussian conditioning
e.g. Wilson et al. (2021):
Let
be a probability space and denote by a pair of square integrable, centred random variables on . The conditional expectation is the unique random variable that minimises the optimization problem In words then, is the measurable function of that best predicts in the sense of minimizing the mean square error . Uncorrelated, jointly Gaussian random variables are independent. Consequently, when
and are jointly Gaussian, the optimal predictor manifests as the best unbiased linear estimator of