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Many hold that Bayesian statistics "from a purely mathematical point of view" is entirely coextensive with probability (however it is that you want to define its boundaries as a mathematical disciplin …

This has always bothered me. "One should use the price of tea in China to obtain a better estimate of the chance of rain in Melbourne" is not a good characterization at all. One should use the price …

I enjoyed thinking about these answers and this is my attempt to put them into (nonrigorous) geometrical terms. Writing the joint density compositionally as $$p(\mathbf{x}_k \mid |\mathbf{x}| = 1)p(\ …

I'm trying to familiarize myself with the details of the proof that the Markov chains produce by the Metropolis-Hastings algorithm have a law of large numbers. I've found a half dozen or more referenc …

Consider a $p$ dimensional random variable with a discrete support. Consider a Markov transition kernel on the state space that is defined in terms of element-wise transition distributions. One can …

You can always write down the joint distribution compositionally. In terms of a density function: $$f(\theta_1, \dots, \theta_n) = f_1(\theta_1)f_2(\theta_2 \mid \theta_1)f_3(\theta_3 \mid \theta_1, \ …

Consider the joint Gaussian distribution of $(Y, Z, f(x))$. Observe that knowing both $Y$ and $Z$ together is equivalent to knowing $f(\mathbf{x})$ (the noiseless version of $Y$). Then we can compute …

In two statistics papers (linked inline below) I have come across two definitions of certain probability measures. I conjecture that for particular choices of the construction that they are equivalen …

The Borel-Kolmogorov paradox refers to situations where non-uniqueness in the notion of conditioning on a set of measure zero leads to apparent contradictions. As a formal matter, one requires instea …

Maybe something like this will work. Consider $U_1$ and $U_2$ drawn uniformly at random on the unit circle. Because they are uniformly distributed, we may rotate the circle until $U_1$ is at the ``n …

I think the difficulty is worse than just finding the right algorithm. The first matter of business is deciding which conditional distribution you want to draw from, because they are non-unique. I'm …

The intersection between computability theory and statistics is pretty interesting. From this paper by Vovk (2009): "It is widely accepted that advances in computing have brought about deep changes …

Cover and Thomas's Elements of Information Theory has a chapter on maximum entropy stochastic processes. The relevant quantity in that case is the entropy rate. See section 12.5, for example, which is …

This paper addresses a similar problem I think, although I believe they consider binary outcomes only: Ramsahai, R.R. (2007). Causal bounds and instruments. In Proceedings of the 23rd Annual Confere …

Suppose I am dead-set on using Bayesian inference on independent and identically distributed data, but I'm lazy and insist on using a parametric likelihood function come what may. I'd be reassured to …