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It means the random variables are statistically independent.

Here are two examples. $x$ is normally distributed around mean $\theta$. The likelihood for $\theta$ is a normal distribution with mean $x$. $x$ is Gamma distributed with rate $\theta$. The likeli …

Suppose Y=0 with probability $p$ and Y=1 with probability $(1-p)$. Since the distribution of X is known and $P(Y>X)$ is known, you can solve for $p$. Similarly, suppose Z=0 with probability $q$ and …

The posterior distribution need not be proper even if the prior is proper. For example, suppose $v$ has a Gamma prior with shape 0.25 (which is proper), and we model our datum $x$ as drawn from a Gau …

You are right that discriminative models have two sets of parameters. You are also right that in practice only one set of parameters is used. This is not a contradiction. The paper is about having …

Also see my answer to Approximating distributions in expectation propagation. Your question also mentions multidimensional normal distributions. However, those are not fully factorized. …

Later in that section, there is an example where the posterior mean using the inferential prior is larger than the posterior mean using the true prior, and this is said to be an example of positive mi …

You can solve this problem with a hierarchical model where the number of colors in the bag is first drawn from a distribution over integers ranging from 1 to infinity (say, a Poisson distribution), th …

You need to substitute the particular $t(x)$ that you are trying to approximate, get the formula for $Z$, then take derivatives. See the examples in A family of algorithms for approximate Bayesian in …

Let $c(x|y)$ be your censoring function. Then $$ f(x_1,...,x_T) = \int_{y_1} \cdots \int_{y_T} f(y_1,...,y_T) \prod_{i=1}^T c(x_i|y_i) dy_i $$ Note that if any $x_i > 0$ then $c(x_i|y_i)$ forces $y_i …

It looks like (3) is just a typo. The authors meant to write $$ p(\tilde{w}_{qm},s_{qm}) = \mathcal{N}(\tilde{w}_{qm}|0,\sigma_w²)[\pi^{s_{qm}}(1-\pi)^{1-s_{qm}}] $$ This is evident from later equati …

I dispute your claim that "In either case, you can't really tell how common or rare billionaires are". Let $f$ be the unknown fraction of billionaires in the population. With a uniform prior on $f$, …