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When $a \ne 0$, there are two solutions to \(ax^2 + bx + c = 0\) and they are $$x = {-b \pm \sqrt{b^2-4ac} \over 2a}.$$

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Well Posed Problems

According to Hadamard, a problem is well-posed  if a. it has a solution b. the solution is unique c. the solution depends continuously on data and parameters.

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Density Estimation

The problem of model the probability distribution p(x) of a random variable x given a finite set x1, ..., xn, of observations is know as density estimation 

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Expectation

The average value of some function f(x) under a probability distribution p(x) is called the expectation of f(x) and is denoted by E[f]

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v-structure

A v-structure is an ordered tuple (X, Y, Z) such that there is an arc from X to Y,  and from Z to Y but not from X to Z

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Marginal likelihood

Marginal Likelihood is a likelihood function in which some parameters have been marginalized. In Bayesian Statistics is referred as the evidence.

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Stochastic

A stochastic system is a no-deterministic system whose state is randomly determined

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Posterior probability

The probability function after the evidence is incorporated

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Monte Carlo

Monte Carlo methods are methods based on computer simulations that contains random variables and repeat a process over and over to estimate mostly a probability or expected values.

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Proper scoring rule

A proper  scoring rule is one which maximally rewards the true probability distribution

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Scoring Rule

A scoring rule S assigns a score to a classifier’s prediction of the value of a target variable for a partially observed instance based upon the posterior probability distribution.

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Marginal Distribution

Let X and Y be random variables, It’s distribution is called the joint distributions of X and Y. Individual distributions of X any Y are then called the margianl distributions. The term marginal is used because the used to be found by summing values in a table along rows or columns, and writing the sum in the margins of the table

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Sufficient Statistics

A statistic is sufficient for a family of probability distributions if the sample from which it is calculated gives no additional information than does the statistic, as to which of those probability distributions is that of the population from which the sample was taken. Given a set o statistics T₁(x)...  Tₘ(x), Let X be a random variable whose distribution is parametrized by θ ε Θ, let p(x|θ) denote the density mass function. A statistic T(x) is sufficient for  θ  if the distribution of x given T(x) is independent of θ, i.e. p(x|T,θ) = p(x|T)