lacunary - Definition Index

Suppose $X$ is a set and $\mathcal{S}$ is the set of subsets of $X.$ Then $\mathcal{S}$ is called a $\sigma$-algebra on $X$ if the following three conditions are satisfied:

• $\mathrm{\varnothing }\subset \mathcal{S}.$
• if $E\in \mathcal{S},$ then $X\setminus E\in \mathcal{S}$ ($\mathcal{S}$ is closed under set difference.)
• if $E_1,E_2,\dots$ is a sequence of elements of $\mathcal{S},$ then $\bigcup _{k=1}^{\mathrm{\infty }}{E}_k\in \mathcal{S}$ ($\mathcal{S}$ is closed under countable unions.

An A/B test is a two-sample statistical inference problem. When we're running an A/B test, we have two groups of users, group A and group B.

We change some functionality for group A and leave it stable for group B. We want to see if there is a statistically significant difference between group A and group B for some metric.

TODO: finish this. Formalize the two-sample setup in terms of the two random variables $X$ and $Y$, and describe the CUPED variance-reduction technique the page is named for.

Abelian groups are groups whose operation is commutative. For $a,b\in G,a\ast b=b\ast a$.

For a real valued random variable

$$X:\mathrm{\Omega }\to \mathbb{R},$$

we call $X$ a continuous random variable if there exists a function

$$f_X:\mathbb{R}\to [0,\mathrm{\infty })$$

such that for every measurable set $A\subseteq \mathbb{R},$

$$P(\omega \in \mathrm{\Omega }:X(\omega )\in A)={\int }_A{f}_X(x)dx$$

where $f_X$ is the probability density function of $X.$

TODO: this definition excludes some continuous distributions like the Cantor distribution and in general isn't quite there yet in terms of measure theory. I need to revisit this when I've learned a bit more of measure theory.

TODO: define this.

The subgroup of $S_n$ consisting of all even permutations of $n$ letters is the alternating group $A_n$ on $n$ letters. If $n\ge 2$, then this set forms a subgroup of $S_n$ of order $n!/2$.

We say that a complex function $f$ is analytic in an open set $S$ if it has a complex derivative at every point of $S$. In other words, when studying complex functions, we consider differentiability on open sets rather than on specific points.

A complex function $f$ is said to be analytic at a point $z$ if it is analytic in some neighborhood of $z$. However, even here, the point being analytic depends on the neighborhood around the point being complex differentiable.

The Archimedean property is that given two positive numbers $x$ and $y,$ there is an integer $n$ such that

A set $A$ is said to be at most countable if $A$ is finite or countable.

Given the open or closed ball with center $\mathbf{x}$ and radius $r$ is defined as the set of points $\mathbf{y}$ such that or $|\mathbf{x}-\mathbf{y}|\le r,$ respectively.

A collection $V_{\alpha }$ of open subsets of $X$ is said to be a base for $X$ if the following is true: For every $x\in X$ and every open set $G\subset X$ such that $x\in G,$ we have $x\in V_{\alpha }\subset G$ for some $\alpha .$ In other words, every open set in $X$ is the union of a subcollection of $V_{\alpha }.$

A Bernoulli Process must possess the following properties:

1. The experiment consists of repeated trials.
2. Each trial results in a boolean outcome, which can be considered true or false, success or failure, etc.
3. The probability of success, $p$, remains constant across trials.
4. Repeated trials are independent.

In an estimation problem, the bias of an estimator $T(X)$ of a real-valued parameter $\vartheta$ of the unknown population is defined to be $b_T(P)=\mathbb{E}[T(X)]-\vartheta$ (which is denoted by $b_T(\theta )$ when $P$ is in a parametric family index by $\theta ).$

An estimator is said to be an unbiased estimator for $\vartheta$ if and only if $b_T(P)=0$ for any $P\in \mathcal{P}.$

In a @dynamical-system with a smoothly varying parameter $r,$ a sudden change in qualitative or topological behavior of the system, especially the creation, annihilation, or change of stability in its fixed points, is called a bifurcation.

The parameter values at which bifurcations occur are called bifurcation points.

Binary Classification is the task of putting things into one of two categories (each called a class.)

For $n,k\in \mathbb{N},n\ge k\ge 0,$ the binomial coefficient is the coefficient of the $x^k$ term in the polynomial expansion of the binomial power $(1+x{)}^n.$ It is denoted as $(\genfrac{}{}{0ex}{}{n}{k})$ and can be expressed through @factorial notation as

$$(\genfrac{}{}{0ex}{}{n}{k})=\frac{n!}{k!(n-k)!}.$$

A binomial distribution aggregates the outcomes of a Bernoulli process across multiple trials to tell you how likely it is to achieve a certain number of successes. The number $X$ of successes in $n$ Bernoulli trials is called a binomial random variable, and its probability distribution is what we call the binomial distribution.

If we let $n$ be the number of trials, $x$ be the number of successes, and $p$ be the probability of success on a given trial, and $q=1-p$ be the probability of failure on a given trial, then the probability distribution of the random variable $X$ is

$$b(x,n,p)=(\genfrac{}{}{0ex}{}{n}{x})p^x{q}^{n-x},x=0,1,2,\dots ,n.$$

To break this down some, there are $(\genfrac{}{}{0ex}{}{n}{x})$ ways to have $x$ successes out of $n$ trials (different orderings). For each, there are $x$ independent events that occur with a probability of $p$ and $n-x$ independent events that occur with a probability of $q=1-p.$

The mean and variance of the binomial distribution $b(x,n,p)$ are

$$\mu =np,\text{and}{\sigma }^2=npq.$$

We can use a summation over the formula above to find the probability of there being between $a$ and $b$ successes:

$$\sum _{x=a}^b(\genfrac{}{}{0ex}{}{n}{x})p^x{q}^{n-x}.$$

A vector with a fixed endpoint is called a bound vector.

The points on a manifold whose neighborhoods are homeomorphic to a neighborhood in a half $k$-ball form the boundary of the manifold. Formally,

$$\partial M=\{p\in M|\text{there exists a chart }(U,p)\text{ with }\phi (p)\in {\mathbb{R}}^{k-1}\times 0\subset {\mathbb{H}}^k\}.$$

That is, $p$ is a boundary point, if, in local coordinates, it maps to the edge of the half-space model.

$E$ is bounded if there is a real number $M$ and a point $q\in X$ such that for all $p\in E.$

The sequence $\left\{p_n\right\}$ is said to be bounded if its range (sequence) is bounded.

Let $E_0$ be the interval $[0,1].$ Remove the segment $(\frac{1}{3},\frac{2}{3}),$ and let

$$E_1=[0,\frac{1}{3}]\cup [\frac{2}{3},1].$$

Similarly, remove the middle thirds of these intervals, and let

$$E_2=[0,\frac{1}{9}]\cup [\frac{2}{9},\frac{3}{9}]\cup [\frac{6}{9},\frac{7}{9}]\cup [\frac{8}{9},1].$$

We can continue this forever, and we get a nested sequence $\left\{E_n\right\}$ of compact sets $E_n$ where:

(a) $E_{n+1}\subset E_n.$

(b) $E_n$ is the union of $2^n$ intervals, each of length $1/3^n.$

Finally, the set

$$P=\bigcap _{n=1}^{\mathrm{\infty }}{E}_n$$

is called the Cantor set.

The capacity $C$ of a discrete channel is given by

$$C=\underset{t\to \mathrm{\infty }}{lim}\frac{\log N(t)}{t},$$

where $N(t)$ is the number of allowed @signals of duration $t.$

Given two sets, $A$ and $B$, if there is a bijection (a one-to-one mapping of $A$ onto $B$) between $A$ and $B$, we say $A$ and $B$ have the same cardinal number, or that $A$ and $B$ are equivalent. We denote this as $A\sim B$.

Let's say we have an integral

$$\begin{array}{cc}{\int }_A^{B}f(x)dx& \text{(a)}\end{array}$$

whose integrand becomes infinite at a point $a$ in the interval of integration.

Then, (a) means, by definition

$${\int }_A^{B}f(x)dx=\underset{ϵ\to 0}{lim}{\int }_A^{a-ϵ}f(x)dx+\underset{\eta \to 0}{lim}{\int }_{a+\eta }^{B}f(x)dx.$$

However, it may be the case that neither limit exists when both $ϵ$ and $\eta$ approach $0$ independently, but that the limit

$$\underset{ϵ\to 0}{lim}[{\int }_A^{a-ϵ}f(x)dx+{\int }_{a+ϵ}^{B}f(x)dx]$$

does exist.

This limit is called the Cauchy principal value of the integral and is written as

$$\text{pr. v. }{\int }_A^{B}f(x)dx.$$

A sequence $\left\{p_n\right\}$ in a metric space $X$ is said to be a Cauchy sequence if for every there is an integer $N$ such that is $n\ge N$ and $m\ge N.$

The center of a group $G$ is all the elements that commute with all elements of $G$:

$$Z(G)=\{z\in G|zg=gz\text{ for all }g\in G\}.$$

The characteristic equation tells us how to find the @eigenvalues and is given by:

$${\lambda }^2-\tau \lambda +\mathrm{\Delta }=0.$$

The circle in which the Taylor series converges to the function is called the circle of convergence for the Taylor series.

A class is a collection of individuals or individual objects. A class can be defined either by extension (specifying members) or by intension (specifying conditions).

Classification is the activity of assigning objects to some pre-existing classes or categories.

A set $E$ is closed if every limit point of $E$ is a point of $E.$

If $X$ is a metric space, $E\subset X,$ and $E^{\prime }$ denotes the set of all limit points of $E$ in $X,$ then the closure of $E$ is the set $\bar{E}=E\cup E^{\prime }.$

Real numbers $a_1,\dots ,a_n$ are commensurable if there exists a common measure such that each $a_i$ is an integer multiple of it: $$a_i=k_{i}m,k_i\in \mathbb{Z},i=1,\dots ,n.$$ Equivalently, all pairwise ratios are rational: $a_i/a_j\in \mathbb{Q}$ for all $i,j$ (with $a_j\ne 0$).

A ring in which multiplication is commutative is called a commutative ring.

The commutator subgroup of $G$ is the group $C$ generated by all elements of the set

$$\{ab{a}^{-1}{b}^{-1}|a,b\in G\}.$$

A subset $K$ of a metric space $X$ is said to be compact if every open cover of $K$ contains a finite subcover. More explicitly, the requirement is that if $\left\{G_{\alpha }\right\}$ is an open cover of $K,$ then there are finitely many indicies ${\alpha }_1,\dots ,{\alpha }_n$ such that $$K\subset G_{{\alpha }_1}\cup \cdots \cup G_{{\alpha }_n}.$$

The complement of $E$ (denoted by $E^c$) is the set of all points $p\in X$ such that $p\notin E.$

A metric space in which every cauchy sequence converges is said to be complete.

The derivative of a complex function $f$ with respect to $z$ at $z_0$ is defined as

provided the limit exists.

In the above definition of complex derivative, if the limit exists, $f$ is said to be differentiable at the point $z_0$.

A complex-valued function $f$ of a complex variable $z$ is a rule that assigns a complex number $f(z)$ to each complex number $z$ in a set $S$. We call such a function a complex function.

A complex number is one which can be written in the form $a+bi$, where $a$ and $b$ are real numbers. The set of all complex numbers is denoted $\mathbb{C}$.

Let $\mathbf{f}$ be a function that maps an open set $E\subset R^n$ into $R^m.$ Let $\{{\mathbf{e}}_1,\dots ,{\mathbf{e}}_n\}$ and $\{{\mathbf{u}}_1,\dots ,{\mathbf{u}}_n\}$ be the standard bases of $R^n$ and $R^m.$ The component functions of $\mathbf{f}$ are the real functions $f_1,\dots ,f_m$ defined by

$$\mathbf{f}(\mathbf{x})=\sum _{i=1}^m{f}_i(\mathbf{x}){\mathbf{u}}_i(\mathbf{x}\in E),$$

or, equivalently, by $f_i(\mathbf{x})=\mathbf{f}(\mathbf{x})\cdot {\mathbf{u}}_i,1\le i\le m.$

Suppose $X,Y,Z$ are metric spaces, $E\subset X,$ $f:E\to Y,$ $g:f(E)\to Z,$ $h:E\to Z$ with

$$h(x)=g(f(x))(x\in E).$$

The function $h$ is called the composition or the composite of $f$ and $g.$ The notation

$$h=g\circ f$$

is frequently used.

A function $\phi$ is said to be concave if $-\phi$ is a convex function.

A point $p$ in a metric space $X$ is said to be a condensation point of a set $E\subset X$ if every neighborhood of $p$ contains uncountably many points of $E.$

Let $X$ and $Y$ be random variables, not necessarily independent. The probability that $Y$ takes the value $y$ when $X$ takes the value $x$ (for any specific $x\in \mathcal{X},y\in \mathcal{Y}$) is called the conditional probability $p(y|x)$ and is given by:

$$p(y|x)=\frac{p(x,y)}{p(x)},p(x)=\sum _{y}p(x,y).$$

(Note that $p(x,y)$ is the joint probability of $X=x,Y=y.$)

We define the conditional entropy of $Y$ given $X,$ $H(Y|X),$ as the average of the entropy of $Y$ for each value $x$ of $X$ weighted according to the probability of getting that particular $x.$ That is,

$$H(Y|X)=-\sum _{x\in \mathcal{X},y\in \mathcal{Y}}p(x,y)\log p(y|x).$$

The conditional mutual information of random variables $X$ and $Y$ given $Z$ is defined by

It is the uncertainty about $X$ after learning $Z$ minus the uncertainty about $X$ after learning $Y$ and $Z$.

From this, we can give the formula for conditional probability. The probability of $A$ given $B$ is

$$P(A|B)=\frac{P(A\text{and}B)}{P(B)}.$$

That is to say, the probability of $A$ occurring given $B$ has occurred is the portion of times $B$ occurs that $A$ also occurs.

If $P(A)=P(A|B)$, then $A$ and $B$ are independent events, and $P(A)\cdot P(B)=P(A\text{and}B).$

TODO: define $f_{Y|X}(y|x)$ for the discrete and continuous cases; note it is a probability distribution in $y$ for each fixed $x$ with Relate to conditional probability, which states the event-level version.

For joint probability mass functions $p(x,y)$ and $q(x,y),$ the conditional relative entropy

$$D(p(y|x)\Vert q(y|x))$$

is the @average of the relative-entropies between the conditional probability mass functions $p(y|x)$ and $q(y|x)$ averaged over the probability mass function $p(x).$ More formally,

Let $A,B$ and $C$ be events. $A$ and $B$ are said to be conditionally independent given $C$ if and only if and $P(A|B,C)=P(A|C).$

Equivalently, conditional independence may be stated as $P(A,B|C)=P(A|C)P(B|C),$ where $P(A,B|C)$ is the joint probability of $A$ and $B$ given $C.$

A confidence interval tells us how likely it is that a population parameter falls within a specified range, based on a statistic calculated from a sample of data. This interval provides a range of values that, with a certain level of confidence (usually expressed as a percentage like $95\mathrm{\%}$ or $99\mathrm{\%}$), is believed to encompass the true parameter value. The width of the interval gives an idea of the precision of our estimate, with narrower intervals representing more precise estimates.

To say that the range of values has a certain level of confidence means that if we were to repeat the experiment or sampling process many times (theoretically an infinite number of times), the true population parameter would fall within that interval in the stated percentage of all trials. For example, a $95\mathrm{\%}$ confidence level means that $95$ out of $100$ such confidence intervals would contain the true population parameter (which is unknown).

If you were to repeat the sampling process an infinite number of times, each time calculating a new $95\mathrm{\%}$ confidence interval using the same method, about $95\mathrm{\%}$ of these intervals would contain the true population parameter. Each interval is calculated from a different sample and might be different in range, but the method of calculation ensures that $95\mathrm{\%}$ of these intervals will capture the true parameter value.

If $X$ is a metric space, a set $E\subset X$ is said to be connected if $E$ is not a union of two nonempty separated sets.

A set $S$ is said to be connected if every pair of points in $S$ can be joined by a finite number of line segments joined end to end that lie entirely within $S$.

Systems for which a conserved quantity exist are called conservative systems.

Given a system $\dot{\mathbf{x}}=\mathbf{f}\mathbf{(}\mathbf{x}\mathbf{)},$ a conserved quantity is a @real-valued continuous function $E(\mathbf{x})$ that is constant on trajectories, i.e. $dE/dt=0.$ We also require that $E(\mathbf{x})$ is nonconstant on every open set, to avoid trivial examples such as $E(\mathbf{x})=0.$

Suppose $X$ and $Y$ are metric spaces, $E\subset X,p\in E,$ and $f:E\to Y.$ Then $f$ is said to be continuous at $p$ if for every there exists a such that

for all points $x\in E$ for which

If $f$ is continuous at every point of $E,$ then $f$ is said to be continuous on $E$.

If a sample space contains an infinite number of possibilities equal to the number of points on a line segment, it is called a continuous sample space.

A differentiable function $\mathbf{f}$ of an open set $E\subset R^n$ into $R^m$ is said to be continuously differentiable in $E$ if ${\mathbf{f}}^{\prime }$ is a continuous function of $E$ into $L(R^n,R^m).$ More explicitly, it is required that to every $\mathbf{x}\in E$ and to every corresponds a such that