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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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 ...
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$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My second-year notes in Statistical Inference and Modelling (unpublished) have a definition,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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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?
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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;...
moyourmo
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1
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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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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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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 ...
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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)]
$$
One issue, is that the samples from $p_z(z)$ could be not very informative:
We see here that $f(z)$ ...
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Is a data size in a binomial distribution random variable?
Supposing that there is a binomial distribution ${\rm Bin}(m|N, \mu)$, I think usually $N$ and $\mu$ are parameters and not random variables (or events), thus the notation here ${\rm Bin}(m|N, \mu)$ ...
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Correct notation for cross-level interaction
I am wondering if I am using the correct notation for a multilevel model with a cross-level interaction (i.e., a multilevel model with an interaction between level 1 and 2 covariates).
Let $Y_{tj}$ be ...
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What does the I operator stand for in the context of time series modeling?
I am reading a paper on parameter estimation for moving-averages, but I'm not exactly sure what the I operator underlined in red stands for. There is no explanation for it anywhere in the paper. Is it ...
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What does $P[(X > 6)/(X > 5)]$ mean? (not | but /)
In a book, I'm in front of:
$P[(X > 6) / (X > 5)]$
What does it mean?
$X > 6$ knowing $X > 5$?
I would have expected it like $P[(X > 6)|(X > 5)]$, then
with a |, not a /, so I ...
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Mathematical notation for number of samples of a predicted class that exceeds threshold of total number of samples
(The title may be somewhat confusing.)
In my machine learning problem, I want to classify an individual based on the samples I have of each individual (i.e. each person has multiple samples).
Once I ...
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Understanding conditional notation
Which is the correct way to describe the conditional probability distribution of X conditioned upon Y where X = a, Y = b
$$P_{X \mid Y}(a \mid b) \tag1$$ or $$P_{X \mid Y}(a,b) \tag2$$
What is the ...
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What does capital D above an equal sign mean?
I was beginning to self study time series and came across this notation: $\stackrel{\mathcal{D}}{=}$. I wonder what it means? (I have limited access to the textbooks for the moment, so I have to post ...
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PRML Book: Calculus of Variance
I am reading through Pattern Recognition and Machine Learning (PRML) Appendix D (page 705).
Here is my question: what does the term $O(\epsilon ^ 2)$ in equation (D.1) and (D.2) stand for?
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1
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What is a name of regression?
Let's say there is a set of independent variables $x_1, x_2, ..., x_n$ and a target variable $y$. A transformation is applied to the initial set of variables: $z_1=f_1(x_1,x_2,..., x_k)$, $z_2=f_2(x_{...
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reproducing kernel hilbert space notation
I'm trying to understand reproducing kernel Hilbert spaces (RKHSs) from scientific papers, however I don't find any gentle introduction. However, my main problem, at the moment, seems to be to ...
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How can we find a unique (suboptimal) solution to an optimisation problem with a very large search space?
A standard linear and unconstrained optimization problem has the following form:
$\max_{x} f(x)$,
For example $f(x) = cos(x)+sin(2x)$, with $-1 \leq x \leq 8$.
The gradient $=0$ will allow finding all ...
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What is meaning of such notations in general?
I would like to ask about the meaning of such notations in general. For example,
$$
\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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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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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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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 ...
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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 ...
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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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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, $a\in I$ ...
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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
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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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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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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 ...
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1
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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:
$$\large Y_t^{(n)} \stackrel{p}\...
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Basic Notation Help
Taking a robotic planning class and have no stats background. I am just trying to understand what this means:
"What is the probability of the event $R=\{x||x-1|\le 1\}$?"
Can someone tell me ...
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Notation for Percentile Range of Sample
Let $X$ be a random variable with unknown CDF $F_X$ and PDF $f_X$. What is the notation for a percentile range $[0,p]$ on a sample of $X$ that allows discussing sample statistics over that range? My ...
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How to write the formulas for logistic & ordinal logistic regression models
In my thesis I am using logistic regression models (either binomial or ordinal, depending on the type of the dependent variable) to analyse the association between environmental quality and socio-...
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Is the expectation in the expression for Risk taken with repect to $\hat{\theta}$?
In Chapter 13 of Larry Wasserman's All of Statistics A Concise Course in Statistical Inference is given:
Definition 13.1 The risk of an estimator $\widehat{\theta}$ is
$$
R(\theta, \widehat{\theta})=\...
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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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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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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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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 ...
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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 $F_{X}(x_{t_1 + \tau}, \ldots, x_{t_n + \tau})$ ...
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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
$$ ...
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How do I write out the components of my beta regression, and how do I interpret them afterwards?
I am estimating a beta-regression using the logit-link function with a continuous dependent variable bound between 0 and 1 and 5 continuous predictor variables.
In R the equation is given by: betareg(...
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Name of the formula which links an event with an alternatives' system
Given an event $B$ and a system of alternatives (I hope this is the right name in english, otherwise tell me the right one please) $(A_i)_{i=1}^n$, the following formula is valid:
$$
P(B \, | \, \...
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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 ...
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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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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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1
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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 ...
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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
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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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