Questions tagged [notation]
For questions about statistical notation and mathematical notation used in statistics.
639
questions
1
vote
0
answers
22
views
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....
- 165
0
votes
0
answers
23
views
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 ...
- 5
1
vote
0
answers
56
views
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 ...
- 83
0
votes
0
answers
21
views
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{...
- 769
2
votes
1
answer
27
views
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 ...
- 153
2
votes
1
answer
26
views
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 ...
- 731
5
votes
1
answer
328
views
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 $X$?
- 85
1
vote
0
answers
14
views
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
vote
0
answers
59
views
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 ...
- 21
1
vote
0
answers
36
views
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 ...
- 6,133
4
votes
1
answer
62
views
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}\}...
- 64.6k
5
votes
3
answers
315
views
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 $P \times Q$
...
- 1,440
-1
votes
2
answers
74
views
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 $\mathbf{x}$, ...
- 11
1
vote
1
answer
79
views
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 $...
- 495
1
vote
0
answers
41
views
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 ...
6
votes
4
answers
204
views
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 ...
- 355
1
vote
1
answer
43
views
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 ...
- 2,646
0
votes
0
answers
24
views
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 ...
- 143
0
votes
1
answer
26
views
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 $\sum\limits_{i_1=1}^D\sum\limits_{i_2=1}^{i_1}...\sum\limits_{i_M=1}^{i_{M-1}}w_{i_1i_2...i_M}x_{i_1}x_{i_2}...x_{i_M}$...
0
votes
0
answers
35
views
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 ...
- 907
2
votes
0
answers
27
views
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 ...
- 143
8
votes
2
answers
1k
views
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\...
- 77
0
votes
0
answers
6
views
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 ...
0
votes
0
answers
35
views
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 ...
- 101
1
vote
1
answer
78
views
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 ...
- 7,118
2
votes
1
answer
74
views
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{...
- 29
2
votes
1
answer
74
views
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 ...
- 41
2
votes
1
answer
524
views
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 ...
- 223
13
votes
4
answers
1k
views
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....
- 233
2
votes
1
answer
118
views
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:
- 565
9
votes
2
answers
924
views
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 \...
- 565
2
votes
0
answers
53
views
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 \...
- 21
0
votes
0
answers
73
views
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 ...
0
votes
1
answer
66
views
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) ...
0
votes
1
answer
42
views
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 ...
- 412
2
votes
2
answers
62
views
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 &...
- 665
0
votes
0
answers
21
views
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, ...
- 115
3
votes
2
answers
58
views
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 ...
- 498
3
votes
1
answer
317
views
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 ...
- 1,246
4
votes
3
answers
290
views
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 ...
- 211
2
votes
1
answer
76
views
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)?
0
votes
1
answer
29
views
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 $t$ ...
- 103
0
votes
0
answers
155
views
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:
...
1
vote
1
answer
65
views
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 ...
- 11
1
vote
1
answer
59
views
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 ...
- 87
2
votes
0
answers
77
views
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 ...
- 193
1
vote
1
answer
300
views
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
votes
1
answer
258
views
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 ...
- 173
0
votes
0
answers
32
views
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 ...
- 5,364
2
votes
1
answer
41
views
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 &...
