Conditional expectation and probability

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.

Conditional algebra



Conditioning in full measure-theoretic glory for Bayesian nonparametrics. E.g. conditioning of Gaussian Processes is also fun.


Chang and Pollard (1997), Kallenberg (2002).

BLUE in Gaussian conditioning

e.g. Wilson et al. (2021):

Let \((\Omega, \mathcal{F}, \mathbb{P})\) be a probability space and denote by \((\boldsymbol{a}, \boldsymbol{b})\) a pair of square integrable, centered random variables on \(\mathbb{R}^{n_{a}} \times \mathbb{R}^{n_{b}}\). The conditional expectation is the unique random variable that minimizes the optimization problem \[ \mathbb{E}(\boldsymbol{a} \mid \boldsymbol{b})=\underset{\hat{\boldsymbol{a}}=f(\boldsymbol{b})}{\arg \min } \mathbb{E}(\hat{\boldsymbol{a}}-\boldsymbol{a})^{2} \] In words then, \(\mathbb{E}(\boldsymbol{a} \mid \boldsymbol{b})\) is the measurable function of \(\boldsymbol{b}\) that best predicts \(\boldsymbol{a}\) in the sense of minimizing the mean square error \((6)\).

Uncorrelated, jointly Gaussian random variables are independent. Consequently, when \(\boldsymbol{a}\) and \(\boldsymbol{b}\) are jointly Gaussian, the optimal predictor \(\mathbb{E}(\boldsymbol{a} \mid \boldsymbol{b})\) manifests as the best unbiased linear estimator \(\hat{\boldsymbol{a}}=\mathbf{S} \boldsymbol{b}\) of \(\boldsymbol{a}\)


Chang, J. T., and D. Pollard. 1997. Conditioning as Disintegration.” Statistica Neerlandica 51 (3): 287–317.
Kallenberg, Olav. 2002. Foundations of Modern Probability. 2nd ed. Probability and Its Applications. New York: Springer-Verlag.
Schervish, Mark J. 2012. Theory of Statistics. Springer Series in Statistics. New York, NY: Springer Science & Business Media.
Wilson, James T, Viacheslav Borovitskiy, Alexander Terenin, Peter Mostowsky, and Marc Peter Deisenroth. 2021. Pathwise Conditioning of Gaussian Processes.” Journal of Machine Learning Research 22 (105): 1–47.

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