Questions tagged [notation]

For questions about statistical notation and mathematical notation used in statistics.

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Motivating notation that looks like the probability of differentials

The Elements of Statistical Learning exemplifies a notation (which I have occasionally seen elsewhere) that I would like clarification on. From page 18: $$\operatorname{EPE}(f) = \int [y - f(x)]^2 \...
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3 votes
1 answer
47 views

Einstein notation $-$ or another $-$ to denote constraints in high dimensional ILP problems

When discussing marginal sums of arrays in 3 dimensions or more, is it customary in the statistical and/or data science communities to use the Einstein summation convention? Is some other form ...
2 votes
2 answers
63 views

$L(\theta;x)=f(x;\theta)$ vs. $L(\theta;x)\propto f(x|\theta)$

My second-year notes in Statistical Inference and Modelling (unpublished) have a definition, The likelihood function of $X$, given the data $x$, is $L∶\Theta\rightarrow\mathbb{R}$ defined by $L(\...
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2 votes
1 answer
193 views

What is the difference between P(A|B,C) and P(A|B∩C)

Is it the same? If yes, which one is the more common/correct notation?
0 votes
0 answers
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Meaning of $\Psi$ and ]] in statistics [closed]

I am reading (first time) a statistics paper, and I have a few questions about the notations presented: $S^j_k$ = ... (page 5) I wonder if this is a typo or ]...] means something different. $\Psi(x^c;...
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2 votes
1 answer
30 views

how standard is R model formula notation

I like using R model formula notation such as $Y \sim X_1 + X_2$ when thinking of regression relationships at a high level. The same for lmer notation such as $Y \sim X | group$. I am wondering how ...
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0 votes
0 answers
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Summation signs in maximum likelihood (ridge regression example)

I can't find how to deal with summation signs ($\sum$) when performing maximum likelihood estimation. I've encountered it in ridge regression: $\frac{1}{n} \sum_{i=1}^n (y_i-\theta^Tx_i)^2 + \lambda\...
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5 votes
3 answers
232 views

Is the sample proportion ($\hat p$) a random variable?

Since it should vary from sample to sample, I suppose it should be a random variable. But if it is, when we write the variance of sample proportions, should we write an uppercase P-hat as the index ...
4 votes
4 answers
606 views

Why do we need importance sampling?

Let's say we want to calculate the following expectation: $$ \mathbb{E}_{z\sim p_z(z)}[f(z)] $$ One issue, is that the samples from $p_z(z)$ could be not very informative: We see here that $f(z)$ ...
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5 votes
2 answers
211 views

Is a data size in a binomial distribution random variable?

Supposing that there is a binomial distribution ${\rm Bin}(m|N, \mu)$, I think usually $N$ and $\mu$ are parameters and not random variables (or events), thus the notation here ${\rm Bin}(m|N, \mu)$ ...
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0 votes
1 answer
23 views

Correct notation for cross-level interaction

I am wondering if I am using the correct notation for a multilevel model with a cross-level interaction (i.e., a multilevel model with an interaction between level 1 and 2 covariates). Let $Y_{tj}$ be ...
5 votes
2 answers
189 views

What does the I operator stand for in the context of time series modeling?

I am reading a paper on parameter estimation for moving-averages, but I'm not exactly sure what the I operator underlined in red stands for. There is no explanation for it anywhere in the paper. Is it ...
0 votes
0 answers
29 views

What does $P[(X > 6)/(X > 5)]$ mean? (not | but /)

In a book, I'm in front of: $P[(X > 6) / (X > 5)]$ What does it mean? $X > 6$ knowing $X > 5$? I would have expected it like $P[(X > 6)|(X > 5)]$, then with a |, not a /, so I ...
0 votes
1 answer
33 views

Mathematical notation for number of samples of a predicted class that exceeds threshold of total number of samples

(The title may be somewhat confusing.) In my machine learning problem, I want to classify an individual based on the samples I have of each individual (i.e. each person has multiple samples). Once I ...
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1 vote
1 answer
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Understanding conditional notation

Which is the correct way to describe the conditional probability distribution of X conditioned upon Y where X = a, Y = b $$P_{X \mid Y}(a \mid b) \tag1$$ or $$P_{X \mid Y}(a,b) \tag2$$ What is the ...
3 votes
1 answer
54 views

What does capital D above an equal sign mean?

I was beginning to self study time series and came across this notation: $\stackrel{\mathcal{D}}{=}$. I wonder what it means? (I have limited access to the textbooks for the moment, so I have to post ...
1 vote
0 answers
34 views

PRML Book: Calculus of Variance

I am reading through Pattern Recognition and Machine Learning (PRML) Appendix D (page 705). Here is my question: what does the term $O(\epsilon ^ 2)$ in equation (D.1) and (D.2) stand for?
1 vote
1 answer
32 views

What is a name of regression?

Let's say there is a set of independent variables $x_1, x_2, ..., x_n$ and a target variable $y$. A transformation is applied to the initial set of variables: $z_1=f_1(x_1,x_2,..., x_k)$, $z_2=f_2(x_{...
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1 vote
0 answers
25 views

reproducing kernel hilbert space notation

I'm trying to understand reproducing kernel Hilbert spaces (RKHSs) from scientific papers, however I don't find any gentle introduction. However, my main problem, at the moment, seems to be to ...
1 vote
0 answers
24 views

How can we find a unique (suboptimal) solution to an optimisation problem with a very large search space?

A standard linear and unconstrained optimization problem has the following form: $\max_{x} f(x)$, For example $f(x) = cos(x)+sin(2x)$, with $-1 \leq x \leq 8$. The gradient $=0$ will allow finding all ...
5 votes
1 answer
88 views

What is meaning of such notations in general?

I would like to ask about the meaning of such notations in general. For example, $$ \mathbb{E}_{(\mathbf{u}, \mathbf{\sigma}, \mathbf{Y}_0)\sim N(0,1)\otimes \mu_D\otimes N(0,1)}\left[ e^{\mathbf{\...
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0 answers
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Relationship between Omega and X and notation

I have read that random variables X are a functions $ X: \Omega \rightarrow \mathbf{R} $. $ X(\omega) = ...$ E.g. $ X(\omega) = mean(\omega)$ (is this correct?) So then $x_1, x_2, ..., x_n$ are ...
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23 views

How should vector notation with multiple indices be interpret?

This is a question about the interpretation of mathematical notation in statistical models. Let's say that this equation represents a panel model: $y_{it} = \alpha + \boldsymbol{\beta}' \mathbf{X}_{it}...
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0 answers
15 views

Explanation of $X: (\Omega,\mathscr A,P)→(\mathbb R , \mathscr B',\mu)$ [duplicate]

A random variable $X$ is a function with a finite value so that: $X: (\Omega,\mathscr A,P)→(\mathbb R , \mathscr B',\mu)$ I've seen this line but I don't know how to understand it. Can you please ...
4 votes
2 answers
160 views

Mistake in Casella & Berger on page 207?

Page 28: A note on notation: Random variables will always be denoted with uppercase letters and the realized values of the variable (or its range) will be denoted by the corresponding lowercase ...
1 vote
0 answers
35 views

What does the (statistical) operator $\mathbb{D}$ usually mean? [closed]

Im reading and trying to understand the following paper on "Disentangled State Space Representations" (https://arxiv.org/pdf/1906.03255.pdf). In the derivation of the KL-loss term the ...
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2 votes
1 answer
51 views

Implied expressions with same probability

Given two variables $a$ and $b$ and two intervals $I$ and $J$, is the following affirmation true? $$ \Pr[a\in I]\geq p\ \land\ (a\in I\implies b\in J) \implies \Pr[b\in J]\geq p $$ Of course, $a\in I$ ...
1 vote
0 answers
13 views

Is there an established way to write conditional probability for whole expression?

I am writing a methodological paper and I have expression like this, which is slightly more complicated bayes rule, because everything is conditioned by c: Is ...
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1 vote
1 answer
29 views

Hierarchical mixed-effects models notation

I have a hierarchical mixed-effects model where observations i, from group S_2 (representing municipalities), are nested inside group S_1 (representing states), which in turn are nested inside N (...
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1 vote
1 answer
67 views

conditional expectation of random vectors

For a random vector $X=(X_1, ..., X_n)^\intercal$ the expectation value can be written as $\mathbb{E}[X] = (\mathbb{E}[X_1], ..., \mathbb{E}[X_n])^\intercal$ according to equation 2 in https://en....
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0 votes
1 answer
115 views

What do "Expectation" mean in cost function?

I want to know whether my understanding for expectation of loss function is correct or not. (x,y) means sample data-label distribution from true distribution. And function f which is respect to ...
4 votes
1 answer
471 views

Meaning of "$\stackrel{p}\longrightarrow$" in math notation (arrow with a p over it)

I have a problem with the concept of a symbol (an arrow with a P over it). Can anyone clarify this for me? For example what the symbol means in this relationship: $$\large Y_t^{(n)} \stackrel{p}\...
1 vote
1 answer
27 views

Basic Notation Help

Taking a robotic planning class and have no stats background. I am just trying to understand what this means: "What is the probability of the event $R=\{x||x-1|\le 1\}$?" Can someone tell me ...
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0 votes
0 answers
30 views

Notation for Percentile Range of Sample

Let $X$ be a random variable with unknown CDF $F_X$ and PDF $f_X$. What is the notation for a percentile range $[0,p]$ on a sample of $X$ that allows discussing sample statistics over that range? My ...
2 votes
1 answer
245 views

How to write the formulas for logistic & ordinal logistic regression models

In my thesis I am using logistic regression models (either binomial or ordinal, depending on the type of the dependent variable) to analyse the association between environmental quality and socio-...
0 votes
0 answers
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Is the expectation in the expression for Risk taken with repect to $\hat{\theta}$?

In Chapter 13 of Larry Wasserman's All of Statistics A Concise Course in Statistical Inference is given: Definition 13.1 The risk of an estimator $\widehat{\theta}$ is $$ R(\theta, \widehat{\theta})=\...
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1 vote
1 answer
44 views

Weird Term of Log likelihood

Recall the setup of logistic regression: We assume that the posterior probability is of the form $p(Y=1|x) = \frac{1}{1+e^{\beta^Tx}}$ This assumes that Y|X is a Bernoulli Random variable. We now turn ...
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0 votes
1 answer
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What is the meaning of the tilde use in this probability expression? [duplicate]

$$X \sim \mathcal{N}(0,1)$$ $$Y|X = x \sim \mathcal{N}(x,1)$$ I'm familiar with the normal use of tilde in indicating a random variable follows a distribution (such as in the first line of the ...
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3 votes
1 answer
61 views

Regression Notation - does the distribution of $y_i$ refer to the distribution of $(Y|X)$?

The title question is the main focus of what I am trying to understand, but the reasoning behind my asking is more nuanced. I have recently made transition from learning about basic forms of ...
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0 votes
2 answers
51 views

Why is sample variance notated as $\frac{S_{x x}}{n-1}$?

I'm going through my class textbook and it defines sample variance as this: $$s^2 = \frac{\sum (x_i - \bar x)^2}{n-1} = \frac{S_{x x}}{n-1}$$ What exactly does $S_{x x}$ mean? Is it always synonymous ...
0 votes
0 answers
33 views

Notation used in time series

I found this wikipedia article about the concept of stationarity in stochastic processes. Let $\left\{X_t\right\}$ be a stochastic process and let $F_{X}(x_{t_1 + \tau}, \ldots, x_{t_n + \tau})$ ...
0 votes
2 answers
45 views

Likelihood notation - Bayesian models

I was reading up about variational Bayesian logistic regression here: https://www.inf.ed.ac.uk/teaching/courses/mlpr/2015/log/15b_lr_and_svi.pdf And the author first defines the model likelihood as $$ ...
0 votes
0 answers
85 views

How do I write out the components of my beta regression, and how do I interpret them afterwards?

I am estimating a beta-regression using the logit-link function with a continuous dependent variable bound between 0 and 1 and 5 continuous predictor variables. In R the equation is given by: betareg(...
1 vote
1 answer
20 views

Name of the formula which links an event with an alternatives' system

Given an event $B$ and a system of alternatives (I hope this is the right name in english, otherwise tell me the right one please) $(A_i)_{i=1}^n$, the following formula is valid: $$ P(B \, | \, \...
1 vote
0 answers
34 views

Matrix algebra with notation question on $z_{.2}$ [closed]

I am doing some matrix calculations and came across an expression which I don't really understand: $$(z_{i2}-z_{.2})(z_{j2}-z_{.2})$$ As I understand it, those are two integers which are multiplied by ...
0 votes
1 answer
30 views

How to show that $\mathbb E_{X,X'} \left[\sum_{i=1}^m\sum_{j=1}^mk(x_i,x_j) \right] = 2m\mathbb E_{x\sim p} [k(x,x)]$?

I posted a related question also to math.SE. Basically, I would like to know how to show the following, which is part of this paper: \begin{align} &\frac{1}{m} \mathbb E_{X,X'}\left[\sum_{i=1}^m\...
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0 votes
0 answers
37 views

Making predictions with support vector classification

I am reading this book (chapter 8.3 - 8.5) about SVM's, while also trying to work with the ksvm() function from the kernlab ...
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0 votes
1 answer
88 views

What does e+ mean in linear mixed model output? [duplicate]

In my mixed effect model output, the Estimate for Intercept is " 7.141e+00" I know e- but I am not sure what e+ is. Also, df has numbers like 5.461e+01, 1.052e+02, 5.396e+0. If that means ...
1 vote
1 answer
51 views

Notation for the summation in unbalanced panel data

Unbalanced panel data, where $i$ stands for sections and $t$ stands for time points, have different numbers of $t$ for every $i$ (or different numbers of $i$ for every $t$, but let me use the first ...
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2 votes
2 answers
358 views

What is the difference between a realisation of a random variable and random variable itself?

I am having a hard time distinguishing random variables from their realisations. (Please note that for the sake of simplicity of my question I use discrete values in my example below.) Usually, a ...
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