Questions tagged [intuition]

Questions that seek a conceptual or non-mathematical understanding of statistics.

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Motivation behind definition of PMF of function of $2$ variables

I am really curious to understand what motivates the definition $$p_{g(X,Y)} (g(X,Y)=z) = \sum_{(x,y)\in g^{-1}(\{z\})} p_{X,Y} (x,y)$$ where $g$ is a two variable function, and $X,Y$ are random ...
niobium's user avatar
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Why intuitively is standard deviation the correct thing to scale to get Central Limit Theorem?

Let me start off by saying I already know all the rigorous formulas, but let me explain why I still feel like something is missing in my understanding. There is no need for any answer going over e.g. ...
D.R's user avatar
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32 views

Looking for an intuitive explanation of D-Criterion for Optimal Design Problem

I know only a little about Fisher information and optimal experimental design, but I'm trying to better understand the subject. If I have an experiment composed of a single detector and my detector ...
David G.'s user avatar
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Question Intuition behind mathematics of activation function in a neural network.

Does this intuition behind why an activation function is used in a neural network make sense mathematically : For this example lets consider a fully connected (NOT CONVOLUTIONAL) network that ...
Stef's user avatar
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19 votes
8 answers
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What is the intuition behind the idea that for linear regression, the number of observations should exceed the number of parameters?

If a population model has k independent variables and 1 intercept, why are k+1 observations required to perform OLS estimates? What is the intuition behind this?
user57623's user avatar
  • 309
7 votes
1 answer
650 views

What is a *likelihood ratio test* for a specific distribution, and how does it relate to hypothesis tests?

I'm just now being introduced to likelihood-ratio tests (LRT), and I am having trouble following the concept and terminology. For example, I posed a question about determining whether two samples {x} ...
feetwet's user avatar
  • 1,108
7 votes
2 answers
299 views

Intuition and reasoning why LASSO can only select $n$ features when $n \ll p$

I'm struggling to grasp the intuition behind why LASSO can only select at most $n$ features when $n << p$, where $n$ is the number of samples and $p$ is the number of features. I've read through ...
Jacob Bumgarner's user avatar
1 vote
1 answer
45 views

Is my interpretation of "the probability over data $X_1, ... X_N$ correct?

This may seem like a pretty simple question, but I want to make sure I am getting this right because it seems pretty foundational. I'm reading this note on conformal prediction. In the very first ...
paul's user avatar
  • 405
31 votes
13 answers
7k views

How to explain 1.5 children?

I teach undergrad stats and every year one student asks "You can't have 1.5 children" (the mean for the dataset). I am flummoxed every time to create a sensical answer. I've tried: "no ...
RLDavis's user avatar
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3 votes
1 answer
113 views

How to explain unbiasedness in basic terms?

If I take some estimator X. Lets say that X is unbiased. Suppose I have 100 samples and each sample has 5 points. I now calculate the value of X on each sample. Because X is unbiased, this means that ...
stats_noob's user avatar
2 votes
3 answers
130 views

Shouldn't we consider the difference in variance between population and a sample while calculating confidence intervals?

To comprehend the concept of confidence intervals, I came up with an example. I want to share it here for your better understanding what my question is all about. Suppose, we want to figure out what ...
Davie Blain's user avatar
13 votes
1 answer
855 views

Intuitive explanation for the fat tails of the t-distribution

Given some standard assumptions, the test statistic $$ \frac{\Delta\bar{X}}{\sigma/\sqrt{N}} $$ is normally distributed if $\sigma$ is known and t-distributed if $\sigma$ has to be estimated from the ...
monade's user avatar
  • 509
4 votes
1 answer
125 views

Markov's inequality intuitions

Can someone explain intuitively how Markov's inequality was derived? It seems plausible, but looking a it, I can't 'see' how it's true.
jbuddy_13's user avatar
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2 votes
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Intuition behind testing seasonality hypothesis

In this post to prove the statistical significance of a statement about a seasonality of a timeseries (every april returns are high) the author simulates alternative paths using the Monte Carlo method ...
gournge's user avatar
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0 answers
46 views

Reference about the comparison between covariance matrices

Suppose we have 2 symmetric matrices $A$ and $B$. Then, we say that $A \succeq B$ if $A - B$ is a positive semi-definite matrix. I was wondering about the intuition and interpretation of $A \succeq B$,...
Yonatan Kurniawan's user avatar
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0 answers
52 views

How should I interpret parameters of the SARIMA model in time series analysis?

I am a bit confused as to why the SARIMA model requires four parameters beyond the ARIMA model just to remove the seasonal component from a time series. Obviously $m$ is required to specify the ...
user10478's user avatar
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0 answers
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Are non-constant polynomial means a special case of seasonality?

In this video, it is said that an otherwise-stationary time series with non-constant linear mean is analyzed by taking the first difference of the time series to produce a new, stationary time series. ...
user10478's user avatar
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1 vote
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what is the Bernoulli product measure's Radon-Nikodym derivative wrt Lebesgue measure? [closed]

The Bernoulli product measure $\mu$ can be defined for each $p\in (0,1)$ on $\Omega = \{0,1\}^\mathbb N=\{\omega=(\omega_i)|\omega_i\in\{0,1\}, i\in\mathbb N\}=\Pi_{i=1}^\infty \{0,1\}$. The measure $...
fromscratch's user avatar
2 votes
0 answers
56 views

Intuition behind occurence of non central chi squared distribution in conditional coordinates of a random walk

Description of background Consider a 2d random walk with drift: $$X(t) = \sum_{k=1}^t X_k \\ Y(t) = \sum_{k=1}^t Y_k$$ where each $X_k$ and $Y_k$ are independently exponentially distributed with rate ...
Sextus Empiricus's user avatar
5 votes
4 answers
606 views

Provide an intuitive example of the linearity of expectation

Can anyone explain the linearity of expectation in an intuitive way? I have been trying to understand this for far too long now. Please don't use any equations and such, try to use real world examples ...
Jeygopi's user avatar
  • 99
6 votes
1 answer
2k views

Intuitive explanation of conformal prediction

I have recently started learning about conformal prediction. I am a programmer without a strong mathematical background, but with a strong intuitive, applied background in statistics. I am trying to ...
Tripartio's user avatar
  • 2,176
2 votes
0 answers
115 views

Intuition behind rank of covariance matrix and testing hypotheses

I am trying to acquire some intuition about testing multivariate hypotheses where the test statistic involves inverse covariance matrix. As an example, suppose we have a $p$-variate random vector that ...
Richard Hardy's user avatar
3 votes
0 answers
82 views

Geometric intuition for how ridge ($L_2$) regularization helps under multicollinearity

We have some nice posts (1, 2 and likely more) illustrating multicollinearity geometrically. Now, ridge regression ($L_2$ regularization) is known to be a remedy of multicollinearity. What is the ...
Richard Hardy's user avatar
2 votes
0 answers
170 views

Understanding intuitive difference between KL divergence and Cross entropy

I know there are related questions already asked, for example this one. I also know the following: KL divergence $D_{KL}(P\Vert Q)$ is given as: $$\begin{align} D_{KL}(P\Vert Q) & = -\sum_xP(x)\...
Mahesha999's user avatar
0 votes
1 answer
62 views

Real-World Example of Correlation of Random Variables

I'm encountering a result in research that is counter-intuitive to me. Specifically, I have two matrics, $X, Y$, where $X_i$ is the ith column of matrix $X$. In my research: $\Large{\rho}$$ (\sum X_i, ...
3 votes
1 answer
283 views

Can you explain bootstrapping like I’m 5?

I think I have a handle on what bootstrapping is and why we need to use it. Please confirm if my understanding is correct: Goal of bootstrapping: To find the SE of a feature’s coefficient that you ...
Katsu's user avatar
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1 vote
0 answers
123 views

Concrete example of what Sufficient Statistics is [closed]

Having read articles to try to understand Sufficient Statistics. Sufficient statistics for layman A sufficient statistic summarizes all the information contained in a sample so that you would make ...
mon's user avatar
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0 answers
150 views

Intuitive statistics book [duplicate]

I am looking for a statistics book that not only gives formulas or proofs but also gives intuitive explanations. For example, the standard deviation is defined by ${\sigma_x} = \sqrt{\frac{1}{n}{\sum\...
8 votes
2 answers
1k views

Is variance the area under the curve of the distribution of a population?

I am trying to understand what variance is, I already know the "official" definition "Variance is the average squared deviations from the mean" But I am trying to give it a visual ...
RodParedes's user avatar
4 votes
0 answers
320 views

What's the intuition behind the fact that sample mean and sample variance are independent when sampling from a normal population?

Let $X_1, \dotsc,X_n$ be i.i.d. from $N(\mu,\sigma^2)$, then we know that sample mean $\bar X\equiv \frac{1}{n}\sum_{i=1}^nX_i$ and $S^2=\frac{1}{n-1}(X_i-\bar X)^2$ are independent. Obviously, they ...
ExcitedSnail's user avatar
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3 votes
1 answer
267 views

Difference between likelihood functions for pmf vs pdf

Can someone explain the intuition behind how the likelihood function for a specific value of $\theta$ is different if $f_\theta$ is a pmf vs a pdf? I thought that it was simply the probability that a ...
Eunice Lo's user avatar
1 vote
0 answers
38 views

How to connect the intuitions to the math of adaptive processes?

Formal Definition Wikipedia gives the following definition of a process adapted to a filtration: Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a probability space; $I$ be an index set with total order $\...
Galen's user avatar
  • 8,402
3 votes
0 answers
160 views

Is there an outer product counterpart for the Covariance?

Covariance The covariance of two quantities $X$ and $Y$ within a population, $Cov(X,Y)$, is symmetric and bilinear. It is also true that $Cov(X,X) \ge 0$. So, clearly $Cov(X,Y)$ qualifies as an inner ...
Fermion's user avatar
  • 31
1 vote
1 answer
35 views

Linear models when responses have no link

I am studying normal linear regression and wanted to ask a question about its utility when working with independent RV. Suppose that we have for $k \in [1,\dots,n]$, $$Y_k = \beta_0 + \beta_1x_{k1} + \...
Kilkik's user avatar
  • 345
0 votes
0 answers
61 views

Intuition on expected value of an estimator [duplicate]

When we don't know what's the mean of a normal distribution we try to estimate it and after a time we get lucky and have the true mean (in a magical way). What does it mean the expected value of the ...
Abderrahmen Hamdi's user avatar
1 vote
0 answers
40 views

Interpretation of covariance and linear dependency [duplicate]

What is the best interpretation of covariance you can give ? I know that if $X$ and $Y$ are random variables, then if $Cov(X,Y)>0$, then if realizations of $X$ are higher than expected, then ...
Kilkik's user avatar
  • 345
3 votes
1 answer
113 views

Mean of geometric distribution is odds?

Context: I mean the $P(X=k)=(1-p)^k p$ not the $P(Y=k)=(1-p)^{k-1} p$. Apparently the mean of the 1st kind of geometric is $\frac{1-p}{p}$ instead of $\frac{1}{p}$ for the 2nd kind of geometric. I ...
BCLC's user avatar
  • 2,414
3 votes
1 answer
103 views

What does conditional independence mean semantically?

I've just spent the last 3 hours reading every post, question, Medium article, and textbook entry on conditional independence, and I still don't really understand it. Can somebody explain what it ...
NaiveBae's user avatar
  • 257
0 votes
0 answers
344 views

What exactly does the Box-Cox transformation do to a time series?

If I were to try and rephrase the argument in the original Box-Cox paper in my own words, I would say something like the following: given a model $$ y = x \beta , $$ if the residuals do not appear to ...
Anthony's user avatar
  • 500
2 votes
1 answer
422 views

Why do we need a smaller sample size to detect a smaller proportion?

The plot below shows the sample size needed to detect a proportion with a precision 0.01 for various true proportions: This assumes an infinite population size, and the confidence intervals are fixed ...
Mhairi McNeill's user avatar
1 vote
0 answers
503 views

Why PCA is invariant under rotation

Lets say that we have a matrix of variables (the columns are variables and rows are the observations) called X whenre X = [x1, x2, ...., xp] where ...
Amir Jalilifard's user avatar
1 vote
0 answers
198 views

Does the number of samples, as opposed to the sample size in each sample, matter for the Central Limit Theorem? [closed]

(1)So here is a formula that describes CLT I found at https://en.wikipedia.org/wiki/Central_limit_theorem. According to the first part of the explanation, n as in Xn describes the number of samples(i....
e1phy's user avatar
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1 vote
0 answers
81 views

Why is my intuition about probability in this regard so flawed? [closed]

Take the following example: Take a sample from 100 people and measure their height. Assume that we know that height is approximately normally distributed, with a sample mean of 175 cm and sample ...
Sen90's user avatar
  • 111
13 votes
5 answers
2k views

What's complicated about regression to the mean?

Note: I am a bit of a novice when it comes to statistics and data analysis. Reading the chapter on regression to the mean in Kahneman's Thinking Fast and Slow, I came across the following passage: ...
ciru_4011's user avatar
  • 133
2 votes
0 answers
149 views

Intuition behind log in kl distance

So, let's start stating that I already read both Why KL-Divergence uses "ln" in its formula? and What is the role of the logarithm in Shannon's entropy? ... However, I still have no ...
Alberto's user avatar
  • 1,071
1 vote
0 answers
257 views

Intuition about the relation between joint distribution, marginal distribution, and conditional distribution

The wording "intuition" might be a bit imprecise. I want to discuss how we visualize in our head going from one to another among the joint PDF, marginal PDF, and conditional PDF. To make the ...
whoknowsnot's user avatar
12 votes
5 answers
2k views

Is the exact value of any likelihood meaningless?

While reading about likelihood, I have heard that "the exact value of any likelihood is meaningless" why? So, because of that we may use the likelihood ratio. So, my question is, why the ...
Alice's user avatar
  • 640
5 votes
1 answer
166 views

What is the intuition behind the odds scale?

What is an intuitive explanation of the odds scale? In a logistic regression such as $$logit(p) = \beta_0 + \beta_1 x$$ we often interpret $\beta_1$ by looking at the odds ratio, $e^{\beta_1}$, which ...
thomaskeefe's user avatar
1 vote
1 answer
222 views

MLE of the Uniform Distribution

In a uniform distribution where $0\leq X \leq \theta$, the pdf is represented as $f(X|\theta) = \frac{1}{\theta}I(0\leq X \leq \theta)$, and the likelihood is $L(\theta) = \prod\frac{1}{\theta}I(0\leq ...
Dan W's user avatar
  • 183
4 votes
2 answers
874 views

Intuition for why mean of lognormal distribution depends on variance of normally distributed rv

Let $X\sim\mathcal{N}(\mu,\sigma^2)$, which is a normal distribution. Then, $\text{exp}(X)\sim\text{Lognormal}(\mu,\sigma^2)$, and its mean is $$ \mathbb{E}[\text{exp}(X)]=\text{exp}\left(\mu+\dfrac{\...
T_T's user avatar
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