Questions tagged [mathematical-statistics]

Mathematical theory of statistics, concerned with formal definitions and general results.

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172
votes
7answers
845k views

What's the difference between variance and standard deviation?

I was wondering what the difference between the variance and the standard deviation is. If you calculate the two values, it is clear that you get the standard deviation out of the variance, but what ...
159
votes
9answers
101k views

Bottom to top explanation of the Mahalanobis distance?

I'm studying pattern recognition and statistics and almost every book I open on the subject I bump into the concept of Mahalanobis distance. The books give sort of intuitive explanations, but still ...
137
votes
21answers
97k views

What's the difference between probability and statistics?

What's the difference between probability and statistics, and why are they studied together?
127
votes
10answers
91k views

Why does the Cauchy distribution have no mean?

From the distribution density function we could identify a mean (=0) for Cauchy distribution just like the graph below shows. But why do we say Cauchy distribution has no mean?
125
votes
9answers
94k views

Numerical example to understand Expectation-Maximization

I am trying to get a good grasp on the EM algorithm, to be able to implement and use it. I spent a full day reading the theory and a paper where EM is used to track an aircraft using the position ...
115
votes
12answers
69k views

Maximum Likelihood Estimation (MLE) in layman terms

Could anyone explain to me in detail about maximum likelihood estimation (MLE) in layman's terms? I would like to know the underlying concept before going into mathematical derivation or equation.
102
votes
10answers
166k views

Validation Error less than training error?

I found two questions here and here about this issue but there is no obvious answer or explanation yet.I enforce the same problem where the validation error is less than training error in my ...
102
votes
13answers
78k views

Simple algorithm for online outlier detection of a generic time series

I am working with a large amount of time series. These time series are basically network measurements coming every 10 minutes, and some of them are periodic (i.e. the bandwidth), while some other aren'...
97
votes
7answers
63k views

How does the reparameterization trick for VAEs work and why is it important?

How does the reparameterization trick for variational autoencoders (VAE) work? Is there an intuitive and easy explanation without simplifying the underlying math? And why do we need the 'trick'?
94
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8answers
32k views

If mean is so sensitive, why use it in the first place?

It is a known fact that median is resistant to outliers. If that is the case, when and why would we use the mean in the first place? One thing I can think of perhaps is to understand the presence of ...
94
votes
12answers
11k views

Who Are The Bayesians?

As one becomes interested in statistics, the dichotomy "Frequentist" vs. "Bayesian" soon becomes commonplace (and who hasn't read Nate Silver's The Signal and the Noise, anyway?). In talks and ...
90
votes
9answers
18k views

What is meant by a "random variable"?

What do they mean when they say "random variable"?
81
votes
5answers
40k views

Mutual information versus correlation

Why and when we should use Mutual Information over statistical correlation measurements such as "Pearson", "spearman", or "Kendall's tau" ?
78
votes
12answers
8k views

Famous statistical wins and horror stories for teaching purposes

I am designing a one year program in data analysis with a local community college. The program aims to prepare students to handle basic tasks in data analysis, visualization and summarization, ...
78
votes
5answers
45k views

How exactly did statisticians agree to using (n-1) as the unbiased estimator for population variance without simulation?

The formula for computing variance has $(n-1)$ in the denominator: $s^2 = \frac{\sum_{i=1}^N (x_i - \bar{x})^2}{n-1}$ I've always wondered why. However, reading and watching a few good videos about "...
72
votes
5answers
26k views

Central limit theorem for sample medians

If I calculate the median of a sufficiently large number of observations drawn from the same distribution, does the central limit theorem state that the distribution of medians will approximate a ...
72
votes
15answers
10k views

Why would parametric statistics ever be preferred over nonparametric?

Can someone explain to me why would anyone choose a parametric over a nonparametric statistical method for hypothesis testing or regression analysis? In my mind, it's like going for rafting and ...
67
votes
1answer
65k views

KL divergence between two multivariate Gaussians

I'm having trouble deriving the KL divergence formula assuming two multivariate normal distributions. I've done the univariate case fairly easily. However, it's been quite a while since I took math ...
65
votes
3answers
31k views

Variables are often adjusted (e.g. standardised) before making a model - when is this a good idea, and when is it a bad one?

In what circumstances would you want to, or not want to scale or standardize a variable prior to model fitting? And what are the advantages / disadvantages of scaling a variable?
64
votes
14answers
10k views

What is the most surprising characterization of the Gaussian (normal) distribution?

A standardized Gaussian distribution on $\mathbb{R}$ can be defined by giving explicitly its density: $$ \frac{1}{\sqrt{2\pi}}e^{-x^2/2}$$ or its characteristic function. As recalled in this ...
64
votes
8answers
6k views

Are bayesians slaves of the likelihood function?

In his book "All of Statistics", Prof. Larry Wasserman presents the following Example (11.10, page 188). Suppose that we have a density $f$ such that $f(x)=c\,g(x)$, where $g$ is a known (nonnegative, ...
64
votes
4answers
93k views

Why is it that natural log changes are percentage changes? What is about logs that makes this so?

Can somebody explain how the properties of logs make it so you can do log linear regressions where the coefficients are interpreted as percentage changes?
61
votes
4answers
17k views

What is so cool about de Finetti's representation theorem?

From Theory of Statistics by Mark J. Schervish (page 12): Although DeFinetti's representation theorem 1.49 is central to motivating parametric models, it is not actually used in their ...
56
votes
13answers
5k views

What are the breakthroughs in Statistics of the past 15 years?

I still remember the Annals of Statistics paper on Boosting by Friedman-Hastie-Tibshirani, and the comments on that same issues by other authors (including Freund and Schapire). At that time, clearly ...
56
votes
9answers
3k views

Are we exaggerating importance of model assumption and evaluation in an era when analyses are often carried out by laymen

Bottom line, the more I learn about statistics, the less I trust published papers in my field; I simply believe that researchers are not doing their statistics well enough. I'm a layman, so to speak. ...
55
votes
19answers
9k views

Mathematical Statistics Videos

A question previously sought recommendations for textbooks on mathematical statistics Does anyone know of any good online video lectures on mathematical statistics? The closest that I've found are: ...
53
votes
6answers
6k views

What are the main theorems in Machine (Deep) Learning?

Al Rahimi has recently given a very provocative talk in NIPS 2017 comparing current Machine Learning to Alchemy. One of his claims is that we need to get back to theoretical developments, to have ...
51
votes
3answers
12k views

Consider the sum of $n$ uniform distributions on $[0,1]$, or $Z_n$. Why does the cusp in the PDF of $Z_n$ disappear for $n \geq 3$?

I've been wondering about this one for a while; I find it a little weird how abruptly it happens. Basically, why do we need just three uniforms for $Z_n$ to smooth out like it does? And why does the ...
50
votes
3answers
8k views

How can I calculate $\int^{\infty}_{-\infty}\Phi\left(\frac{w-a}{b}\right)\phi(w)\,\mathrm dw$

Suppose $\phi(\cdot)$ and $\Phi(\cdot)$ are density function and distribution function of the standard normal distribution. How can one calculate the integral: $$\int^{\infty}_{-\infty}\Phi\left(\...
50
votes
4answers
16k views

When is a biased estimator preferable to unbiased one?

It's obvious many times why one prefers an unbiased estimator. But, are there any circumstances under which we might actually prefer a biased estimator over an unbiased one?
49
votes
2answers
46k views

What is the distribution of the sum of non i.i.d. gaussian variates?

If $X$ is distributed $N(\mu_X, \sigma^2_X)$, $Y$ is distributed $N(\mu_Y, \sigma^2_Y)$ and $Z = X + Y$, I know that $Z$ is distributed $N(\mu_X + \mu_Y, \sigma^2_X + \sigma^2_Y)$ if X and Y are ...
49
votes
6answers
13k views

Motivation for Kolmogorov distance between distributions

There are many ways to measure how similar two probability distributions are. Among methods which are popular (in different circles) are: the Kolmogorov distance: the sup-distance between the ...
48
votes
3answers
90k views

Derive Variance of regression coefficient in simple linear regression

In simple linear regression, we have $y = \beta_0 + \beta_1 x + u$, where $u \sim iid\;\mathcal N(0,\sigma^2)$. I derived the estimator: $$ \hat{\beta_1} = \frac{\sum_i (x_i - \bar{x})(y_i - \bar{y})}...
47
votes
3answers
9k views

How does saddlepoint approximation work?

How does saddlepoint approximation work? What sort of problem is it good for? (Feel free to use a particular example or examples by way of illustration) Are there any drawbacks, difficulties, things ...
46
votes
4answers
9k views

Taking the expectation of Taylor series (especially the remainder)

My question concerns trying to justify a widely-used method, namely taking the expected value of Taylor Series. Assume we have a random variable $X$ with positive mean $\mu$ and variance $\sigma^2$. ...
46
votes
3answers
19k views

Empirical relationship between mean, median and mode

For a unimodal distribution that is moderately skewed, we have the following empirical relationship between the mean, median and mode: $$ \text{(Mean - Mode)}\sim 3\,\text{(Mean - Median)} $$ How ...
45
votes
6answers
9k views

Why don't linear regression assumptions matter in machine learning?

When I learned linear regression in my statistics class, we are asked to check for a few assumptions which need to be true for linear regression to make sense. I won't delve deep into those ...
44
votes
5answers
15k views

What is the purpose of characteristic functions?

I'm hoping that someone can explain, in layman's terms, what a characteristic function is and how it is used in practice. I've read that it is the Fourier transform of the pdf, so I guess I know what ...
43
votes
9answers
7k views

Correlation does not imply causation; but what about when one of the variables is time?

I know this question has been asked a billion times, so, after looking online, I am fully convinced that correlation between 2 variables does not imply causation. In one of my stats lectures today, we ...
43
votes
2answers
15k views

Differences between Bhattacharyya distance and KL divergence

I'm looking for an intuitive explanation for the following questions: In statistics and information theory, what's the difference between Bhattacharyya distance and KL divergence, as measures of the ...
41
votes
7answers
17k views

Are there any examples of where the central limit theorem does not hold?

Wikipedia says - In probability theory, the central limit theorem (CLT) establishes that, in most situations, when independent random variables are added, their properly normalized sum tends ...
39
votes
8answers
94k views

Looking for a good and complete probability and statistics book

I never had the opportunity to visit a stats course from a math faculty. I am looking for a probability theory and statistics book that is complete and self-sufficient. By complete I mean that it ...
38
votes
8answers
21k views

Is it possible to prove a null hypothesis?

As the question states - Is it possible to prove the null hypothesis? From my (limited) understanding of hypothesis, the answer is no but I can't come up with a rigorous explanation for it. Does the ...
38
votes
3answers
15k views

Distribution of scalar products of two random unit vectors in $D$ dimensions

If $\mathbf{x}$ and $\mathbf{y}$ are two independent random unit vectors in $\mathbb{R}^D$ (uniformly distributed on a unit sphere), what is the distribution of their scalar product (dot product) $\...
38
votes
4answers
49k views

How do you Interpret RMSLE (Root Mean Squared Logarithmic Error)?

I've been doing a machine learning competition where they use RMSLE (Root Mean Squared Logarithmic Error) to evaluate the performance predicting the sale price of a category of equipment. The problem ...
38
votes
4answers
27k views

Is a strong background in maths a total requisite for ML?

I'm starting to want to advance my own skillset and I've always been fascinated by machine learning. However, six years ago instead of pursuing this I decided to take a completely unrelated degree to ...
37
votes
8answers
15k views

What is Bayes' theorem all about?

What are the main ideas, that is, concepts related to Bayes' theorem? I am not asking for any derivations of complex mathematical notation.
37
votes
2answers
2k views

Probability inequalities

I am looking for some probability inequalities for sums of unbounded random variables. I would really appreciate it if anyone can provide me some thoughts. My problem is to find an exponential upper ...
36
votes
6answers
34k views

Why does the variance of the Random walk increase?

The random walk that is defined as $Y_{t} = Y_{t-1} + e_t$, where $e_t$ is white noise. Denotes that the current position is the sum of the previous position + an unpredicted term. You can prove that ...
36
votes
6answers
3k views

How can I analytically prove that randomly dividing an amount results in an exponential distribution (of e.g. income and wealth)?

In this current article in SCIENCE the following is being proposed: Suppose you randomly divide 500 million in income among 10,000 people. There's only one way to give everyone an equal, 50,000 ...

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