Questions tagged [notation]
For questions about statistical notation and mathematical notation used in statistics.
639
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Partial derivative notation extreme gradient boosting
What is meant by this notation regarding gradient boosting source:
$$
g_i = \partial_{\hat{y}_i^{(t-1)}} l(y_i, \hat{y}_i^{(t-1)})
$$
Is it the partial derivative of $l$ w.r.t. $\hat{y}_i^{(t-1)}$ , i....
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Conditional volatility and notation to express it
I'm taking a risk modelling class, and we're discussing conditional vs unconditional volatility.
We have standard volatility $\sigma = \sqrt{\mathbb{E}[r_t - \mathbb{E}(r_t)]^2}$
But what is ...
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How to interpret the subscript of this equation?
In this paper the authors in equation 18 present $\hat{\sigma_t^2}$ as
$$ \hat{\sigma}_{t}^2 = \frac{[X, X]_{t_{i M_n}} - [X, X]_{t_{(i-1)M_n}}}{t_{iM_n} - t_{(i-1)M_n}}. \tag{18} $$
How to read the ...
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Distribution of the relative range ($R/\sigma$) [duplicate]
Let us assume a $n$-sized random sample $\{ X_1 , \dots, X_n \}$ from a random variable $X \sim N(\mu, \sigma)$.
Let
$X_\mathrm{min} = \mathrm{min}\{ X_1 , \dots, X_n \}$
$X_\mathrm{max} = \mathrm{...
2
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1
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27
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Understanding notation for input space
I'm taking the free Caltech machine learning course. I'm having trouble understanding the notation on one of the problems:
In this problem, you will create your own target function f and data set D ...
2
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1
answer
26
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The "Multiple Error Terms" notation for hierarchical models
I'm seeking clarification regarding notation for Bayesian hierarchical models, specifically the mixed effects model. Consider the following hierarchical model for the outcome of unit $i \in N$ in ...
5
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1
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328
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Conditional variance notation
Why does the formula $\operatorname{Var} (Y\mid X)= E \left (\left(Y- E (Y\mid X)\right )^2\mid X\right )$ condition $EY$, but not $Y$, on $X$?
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Is the semicolon (;) notation used to indicate operations are performed concurrently in backpropagation algorithm by Bengio?
I am trying to understand the backpropagation algorithm in a multi-layer perceptron environment.
Algorithm 6.4 Backward computation for the deep neural network of
algorithm 6.3, which uses, in ...
1
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0
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59
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How borked is my attempted notation for a multilevel gamma regression with AR(1) autocorrelated residuals?
I have no math background and am in way over my head trying to figure out the correct notation for my model: a multilevel random-slopes gamma regression with log link and AR(1) autocorrelated ...
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How to mark vector of row sums of a matrix?
We are submitting a paper documenting new statistical model. We have a matrix Fi,t and a vector Fi, defined like this:
Thus, the vector Fi contains row sums of matrix Fi,t. Since the definition of Fi ...
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Notation for an ordered pair of stochastic processes
There are two stochastic processes, $\{ Y_{1,t} \}$ and $\{ Y_{2,t}\}$. If I take them as an ordered pair, what notation do I use:
$\{(Y_{1,t},Y_{2,t})\}$,
$\{(Y_1,Y_2)_t\}$,
$(\{Y_{1,t}\},\{Y_{2,t}\}...
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3
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315
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What is the cross product of two probability distributions $P \times Q$?
In many papers on machine learning and statistics, I encounter the following notation
Let $P$ be a distribution, and let $Q$ be another distribution.
Then the author creates an object $P \times Q$
...
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2
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Purpose of distribution "over" a variable notation
The deep learning book (Goodfellow, Bengio, and Courville, 2016)'s notation section says $\mathcal{N}(\mathbf{x}; \mathbf{\mu}, \mathbf{\Sigma})$ represents a normal distribution over $\mathbf{x}$, ...
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Mathematical/statistical notation of a linear regression model with three-way interaction
I am unsure how to present the equation of a three way interaction linear model (without expanding all the interaction effects).
Let me clarify:
$y$ is a continuous outcome measured at time $i$ (for $...
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41
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Semantics: Conditional Distribution vs Marginal Distribution
This is something which has always confused me.
Suppose we take a standard statistical regression model:
\begin{equation}
Y_i = \beta_0 + \beta_1 X_i + \epsilon_i
\end{equation}
where:
$Y_i$ is the ...
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4
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iid data (Bayesian) vs iid random variables (Frequentist)?
I've been pondering the differences in notation / language used in some of the resources I've read for statistics / machine learning.
Warning: this might be embarrassingly obvious to any decent ...
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1
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How to define & mathematically denote "true individual probability"?
The notion of an individual's "true probability" or "true risk" of a certain outcome $Y$ is contested [1, 2, 3]. Nevertheless, it is often discussed and useful for certain kinds of ...
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Random variable notation in Gaussian Process
In a number of online articles (e.g. 1, 2) I see the $x$ values over which a Gaussian Process is used written as $X$, e.g. the training points are $X$, test points are $X_*$, and (in the 1D case) the ...
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How to rewrite a multivariate polynomial term without redundancies?
This is from an exercise found here on page 60. My question is: what is meant by $\sum\limits_{i_1=1}^D\sum\limits_{i_2=1}^{i_1}...\sum\limits_{i_M=1}^{i_{M-1}}w_{i_1i_2...i_M}x_{i_1}x_{i_2}...x_{i_M}$...
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Terminology / notation for the set of variables associated with a distribution
Suppose I have a conditional (or any) distribution as so:
$$
p(A \mid B,C,D)
$$
and in the text, I want to refer to the variables $\{A,B,C,D\}$ associated with that density (or mass). Is there a ...
2
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Notation for calculating $p(v)$ marginalising over $h$ in an Restricted Boltzmann Machine
I am working through equation 22 in Introduction to Boltzmann Machines
I am a little confused with the notation, in particular in the line:
As I understand it, we want the probability of a specific ...
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What does Φ mean? [closed]
In the documentation of METAL, a software for meta-analyzing GWAS results, Φ is in the formulas.
Intermediate Statistics $Z_i = \Phi^{−1}(P_i/2) * \textrm{sign}(\Delta_i$) Overall P-value
$P=2\...
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Is a structural VAR method ideal to analyze the effects of the German 9Euro Public transport ticket on CPI?
I am trying to analyse the effects that the German policy to introduce a 9Euro public transport ticket had on German CPI using a VAR model (which, according to my reading, is the most appropriate ...
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35
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What does this notation mean? $\mathbb{P}_{x\sim \mathcal{D}}$: Clarification in the definition of generalization error $R(h)$
I was reading the book foundations of machine learning by Mehryar Mohri, Afshin Rostamizadeh and Ameet Talwalkar, and I came up with this definition of generalization error:
I dont understand what ...
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1
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78
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Notational confusion about conditional independence in Pearl 2009
First I read this definition which introduces $X$, $Y$ and $Z$ as sets of random variables.
Definition (Pearl 2009)
Let $V = \{V_1, V_2, \ldots \}$ be a finite set of variables. Let $P(\cdot)$ be a ...
2
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1
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74
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How to write a Kullback–Leibler divergence in mathematically correct notation?
Let we have random variables $X$, $Y$, $Z$ and a function $q: Z \to \mathbb{R}$. How to write a Kullback–Leibler divergence between $q(Z)$ and $p(Z|X)$ mathematically correct?
If we write $D_{\text{...
2
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1
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Confused by the notation in DAG depicting structural causal model and corresponding functional equations
Statistical Notations confuses me a lot and I get lost easily in following when the authors are talking about random variables vs observations, probabilities, probability density functions or ...
2
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1
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524
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What are balanced and orthogonal designed experiments?
I have recently started studying Designed Experiments and have have come across non-rigorous definitions of "balanced" and "orthogonal" experiments and would be interested in ...
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How should variables in statistics be understood?
I had a slightly basic question about variables (particularly in statistics).
We would say that “height” is a variable. In my mind, “height” does not just represent different values, like 2, 45, or 75....
2
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Why do I sometimes see the expectation E, probability P and distributions written in blackboard bold?
For example,
$\mathbb{P}(X = 1)$
rather than
$P(X=1)$
And for example:
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How would you write mathematically that a random variable follows some unknown distribution?
For example, if I had a random variable X and knew it followed a normal distribution, I might write:
$X \sim N(5, 10)$
What would I write if I'm trying to say that the distribution is unknown?
$X \in \...
2
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0
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What does $\times_i \Sigma_i$ mean?
I cannot for the life of me figure this out. The context is from game theory (source: Game Theory by Fudenberg, Tirole):
... the space of mixed strategy profiles is denoted $\Sigma = \times_i \...
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In the Bernoulli distribution $P(x^{(i)} ; \theta)=\theta^\color{red}{x^{(i)}}(1-\theta)^{(1-\color{red}{x^{(i)}})}$ what does $x^{(i)}$ stand for?
I am trying to understand the Bernoulli example for the biais given in Deep Learning by Bengo et. al. in 5.4.2:
Consider a set of samples $\{\color{red}{x^{(1)}, . . . , x^{(m)}}\}$ that are ...
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1
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Help resolving apparent ambiguity: P(A|B,C) = P(A|(B,C)) or P((A|B), C)?
Per title, it always confuses me than not, specially when more random variables are used. There is more than one way to get confused imho:
(As in title) P(A|B,C) = P(A|(B,C)) or P((A|B), C)?
P(A,B|C) ...
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1
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What does $R(G:X)$ mean in Rao's book?
I was reading Rao's chapter 4 of his Linear Statistical Inference, 2nd ed. He uses the notation $R(G:X)$ in section 4.i (p. 294, formula (e), p. 296 formula (4i.1.21)) and that notation appears again ...
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Does a random sample $X_1, \dots, X_n$ imply $n > 1$? [closed]
I am calculating the variance for an estimator which is a function of a random sample $X_1, \dots, X_n$. However, if $n=1$, then $E(\theta^2) = \infty$ and the variance is infinite. Can I assume $n &...
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Notation for uncorrelated random structure in mixed model
I will use the hsb dataset inside descriptr in R as a toy example. In my original dataset variables are different of course, ...
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Statistical notation question: How do I represent sorting variables, individually and by each other, symbolically?
I'd like to write a formula for a correlation coefficient that involves sorting continuous observations, both within a variable and by another variable. For example, I'd like to say that $r$ is ...
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317
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Notation confusion regarding Expectation in Kullback-Leibler divergence definition
KL-divergence is often defined as
$$D_{KL}(P||Q) = E_{x\sim P} \left[ \log\left(\frac{P(X)}{Q(X)}\right) \right] = \int_{-\infty}^{\infty} P(x) \log\left(\frac{P(X)}{Q(X)}\right)dx$$
I don't quite ...
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3
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Difference between Greek and Latin letters in OLS
Currently I am learning about the OLS method. In the literature I see different notations for the to be explained variable $y_i$. The different notations have to do with the letters used. In one ...
2
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1
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76
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Is Z a random variable of X?
Is Z a random variable of X if it is transformed from X? If Z is a random variable, and therefore a function, of X, could one denoted Z as Zₓ(x)?
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What does the $t|y(x, \mathbf{w})$ in ${\cal N}(t|y(x, \mathbf{w}), \beta^{-1}) $ mean?
I'm starting to work through Bishop's "Pattern Recognition and Machine Learning" book, and have run into unfamiliar statistics notation. In Eq. 160, Bishop writes the PDF for a quantity $t$ ...
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155
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Mathematical formula for logistic regression mixed effects model
I am unsure how to write these models in mathematical notation when it is a model with nested random effects. This is how the models were created in R:
...
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65
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What is the meaning of this normal distribution?
Following is a normal distribution equation:
$$q(\mathbf x_t\mid\mathbf x_{t-1})=\mathcal N(\mathbf x_t;\sqrt{1-\beta_t}\mathbf x_{t-1},\beta_t\mathbf I).$$
I understand there should be mean and ...
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1
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59
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Issue with the mathematical equation of a Gamma glm with a random intercept
I wanted to write the mathematical equation for my model below. It is a glm that tries to explain the relationship between the sum of observed biomass of several replicas at different observation ...
2
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0
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77
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Lost in Fisher information notation
I believe I understand the gist of what Fisher Information is, but I want to be rigorous and I am confused by notation. Also I believe I am masquerading mistakes by ...
1
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1
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300
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Accuracy without true negatives
Are there any named metric that calculates TP/(TP+FN+FP)?
It is like the accuracy, but disregarding TN
In my problem, we have thousand of possible labels. However, the models return a small list of ...
4
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258
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Proper notation for Random Matrix and observation
My question is simple, but I could not find any references for this particular case.
When we have a random vector, we usually use $\boldsymbol{X}$ to denote the random vector and $\boldsymbol{x}$ to ...
0
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0
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Looking for help unpacking a NHST notation using probabilities
In brief, I am used to a definition of the P-value as: the probability of observing our sample data (or something more extreme) if the null hypothesis is true. Consequently, seeing null hypotheses of ...
2
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1
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41
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Meaning of $\uparrow$ in below d-separation algorithm from Koller
In Probabilistic Graphical Models by Koller and Friedman there is an algorithm to find the nodes reachable from node $X$ via trails that are active, given conditioning set $Z$. What is the meaning of &...