Search Results

As I read it, Bishop is asserting that for each maximal clique $C$ we may define the potential as $\psi_C(x_C) = \prod_S U_S(x_S)$ where $S$ denotes cliques which are subsets of $C$ (and $U$ is the en …

If you are interested in constructive distributions that you can simulate from (as opposed to something simpler with a known expression for the density) you have a lot of flexibility in constructing p …

Here is one approach to consider. Treating the data as points on the surface of the unit sphere, consider the collection of convex subsets on this surface that contain all of your observations. Then, …

The quantile alone is insufficient to define a maximum entropy density. Intuitively this is because the quantile is a single point and is not enough to prescribe an entire density; you must specify ad …

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 …

Others have hit on this, but I thought I'd contribute how I'd write the problem down (briefly): Let $x_i \sim N(m, 1)$ for $i \in \lbrace 1, \dots, N\rbrace$ and define $y_i \equiv x_i$ if $x_i > 0 …

I think there are two separate things going on here. One is the issue of a maximum entropy distribution. The other is of whether or not distributions are invariant under different parameterizations. …

It is possible to develop probability theory taking conditional probability as one of the basic definitions; see section 3.2 in this book and the references mention there. Renyi was one of the first …

EDIT: This is wrong -- careless mistake as noted in the comments. I thought I had deleted it, but here it still is. Working with the RHS of your inequality we have \begin{eqnarray}\sum_i (P_i - …

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(\ …

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 …

This is not an answer, but maybe worth thinking about (and I cannot yet leave comments). My intuition about the Stein phenomenon is that while the individual coordinates of the Gaussian random variab …

The ridge estimator corresponds to the posterior mean under a Normal linear regression model with a conjugate Normal-inverse-gamma prior on the regression coefficients: $\beta \mid \sigma^2, \lambda …

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, \ …