Questions tagged [statistics]

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Naïve array sampling algorithm: the possibility of a item being chosen and its time complexity

This naïve sampling algorithm I am talking about is fairly simple: create a set for storing chosen items first, randomly select an item from the array, and examminate if it is in the set. If it isn't, ...
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28 views

Tight bounds for expected maximum of k binomial(n,p) IIDs

What is the tightest lower and upper bound for the expected maximum value of k IID Binomial(n, p) random variables I tried to derive it : $$Pr[max \leq C] = (\sum_{i = 0}^C {n \choose i}p^i(1 - p)^i)^...
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1 vote
1 answer
30 views

Distribution maximizing ratio of expected maximum over the mean

I’m looking for a distribution that is non-negative, or has good tail bounds (so non-negative with high probability) and maximizes the ratio between the expected maximum of $n$ iid samples and the ...
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2 votes
0 answers
44 views

Algorithm for half-sample mode better than O(N^2)

The half-sample mode is a mode estimator that homes in on the highest density region of a set of samples in search of the mode. It is one of the better mode estimators, though it fails for J-shaped ...
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1 vote
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Precise definition of Universal Learner in Machine Learning

It is surprising to me that I cannot find a precise definition of universal learner on the internet. I can guess what it should bebut I don't want to make a mistake, therefore I have come here. Here's ...
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Hoeffding's inequality applicability for sample complexity

I am trying to determine some bounds for sample complexity. Suppose we have a bounded loss function $\ell$ and target function $f:\mathcal{X}\to\mathcal{Y}$. Hypothesis $h$ is learned, then the ...
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Computational complexity of median filter in image processing

I found this paper http://nomis80.org/ctmf.pdf, that describes an efficient algorithm for applying a median filter to an image. Though this works for non-HDR images, I am not sure how well it scales ...
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4 votes
2 answers
825 views

What empirical evidence do we have for or against a correlation between fault density and LOC?

LOC = lines of code KLOC = Thousand lines of code Fault (or defect) density = number of reported bugs per line of code. Software artifact = function, class, module Reading research papers on fault ...
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Efficient random sampling from large discrete distribution

I have a random variable $X$ that can take finite values in $\{X_1, ..., X_n\}$ with probabilities $\{p_1,..., p_n\}$. Is there a computationally efficient way to sample a number from this set? My ...
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1 vote
0 answers
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Distributed Graph Consensus to fit a distribution?

$G$ is a strongly connected graph with nodes $V$ and edges $E$. Each node $v_i$ receives a sample $x_i$ from a Gaussian $\mathcal{N}(\mu,\sigma^2)$ with unknown mean and variance. The objective is for ...
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Distribution independence property testing

I have been reading the proof in the following paper, and I am unable to understand some parts in the proof. This paper shows that a distribution $A$ over $[n]\times[m]$, $n\geq m$, can be $\epsilon$-...
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2 answers
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Sliding window max/min algorithm without dynamic allocations

I'm working on a suite of DSP tools in Rust, and one of the features of the library is a collection of windowed statistics (mean, RMS, min, max, median, etc) on streams of floating-point samples. For ...
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1 answer
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EM algorithm - What happens with the standard deviation?

What have you tried? So I watched this video. According to the video, we've to calculate the variance $\sigma^2$ as follows: $$ \sigma_{k}^{2} = \frac { p_{1} \left(x_{1} -...
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PCA applicability

I understand that PCA takes a data set with input size, with output labels, and reduces the inputs to a set of principal components size r, where r < n. My question is whether or not this can be ...
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Most popular path in weighted cylic directed graph

Context I have a graph $G=(V,E)$ with weighted edges, all weights are positive integers $w(e)\in\mathbb{N}\setminus\{0\}$. The weights represent the popularity/count of each edge, for example $w(e) = ...
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1 answer
190 views

Approximate max weight path in directed graph

Context This question is related to the fact one can't use Bellman-Ford to find max weight paths in directed graphs with cycles. The reason is that giving a new graph $\tilde{G}$ with negative weights ...
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1 answer
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Chebyshev’s inequality problem in one exercises I can't understand if I did it right or not

This is what do I have to solve: Byron Book: Exercise 8.3 chapter 8 Verify the use of Chebyshev’s inequality in (8.6) of Example 8.16. Show that if the population mean is indeed 48.2333 and the ...
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Separating labelled points with a plane, minimizing label variance

Suppose we have observations with associated labels $\{({\bf x}_1, y_1), ({\bf x}_2, y_2), \dots, ({\bf x}_n, y_n)\}$ where ${\bf x}_i \in \mathbb{R}^d$ and $y_i \in \mathbb{R}$. Can we efficiently ...
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Tracking approximate values of specific percentiles

At the moment I use HdrHistogram to track approximate distribution of a stream of values. But I don't really need all percentiles, just three of them. Is there some algorithm to track specific ...
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  • 111
1 vote
0 answers
51 views

Differential Privacy with only positive noise

The Laplace mechanism is a standard way of making the output of a function $f$ differentially private. More concretely, let $\Delta_f$ be the sensitivity of $f$, i.e. the maximum value by which the ...
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1 answer
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How to implement conditional probability distribution on set-valued Random Variables

I'm trying to implement conditional probability distribution when the events of two RVs are sets. If I try to extrapolate concepts from real or categorical variables to sets things become confusing ...
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2 votes
1 answer
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Averaging a list of variances

In a setting where I have n clients each owns a subset of the dataset, each computes the mean and variance on his data and send them to the server. The server would average all means and variances. ...
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2 votes
1 answer
56 views

Is this statistical group summation unambiguously reversible?

Let $X$ be a finite multisubset of $\mathbb{N}^2$. Let's introduce the following notation: $A$ is a set of all first elements of pairs from $X$ and $B$ is a set of all second elements of pairs from $X$...
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2 votes
0 answers
24 views

Algorithm to mapping given probabilities to empirical probabilities

Consider following problem statement: You have given $n$ actions. You can perform any of them. Each action gives you success with some probability. The challenge is to perform given finite number of ...
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2 answers
122 views

Find the average number of steps to sort an array by randomly selecting two elements

I have sequence of an unique numbers from 1 to 10 in randomly order (for example: list = [7, 5, 3, 4, 2, 6, 10, 1, 9, 8]). I can choose two random number and if the list from left number larger then ...
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1 vote
1 answer
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SGD statistical guarantee

I have a question regard online learning with SGD. Is there a way to give a statistical guarantee that the value obtained after $n$ samples deviates at most $\epsilon$ from the real value? Thank you ...
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0 answers
33 views

How to handle distribution of values with same attributes into different classes

I'm a student studying a data mining course and have come across a problem. I need to explain the problem with the help of an example scenario as I do not know how to explain the problem in any other ...
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1 vote
1 answer
50 views

Chebyshev inequality comparison

Given $n$ distinct values $(x_i)_{i=1}^{n}$ with mean $\mu$ and standard deviation $s$, for all $i$, we have $|x_i−\mu| ≤ s \sqrt{n − 1}$. How does this inequality compare with Chebyshev inequality as ...
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1 vote
1 answer
520 views

Efficiently selecting a random subset of size $m$ from a set of size $n$

This is a cross post of my question here on math.se. I have a list of $n$ items and would like to randomly select an $m$ set from it efficiently (in terms of time complexity). Also, I want all ...
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2 votes
0 answers
51 views

In a machine learning system, why use differentially private SGD if our input data is already perturbed by a DP mechanism?

I'm trying to implement my own version of a deep neural network with differential privacy to preserve the privacy of the parties involved in the training dataset. I'm using the method by Abadi et al. ...
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1 answer
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Gaussian distribution with condition?

What does this expression mean? Normal distribution with condition I am reading a research paper and found the following expression (Eq.28 in the paper below). It means a Gaussian distribution, but ...
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1 vote
2 answers
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Algorithm for Estimating Number of Unique Monthly Visitors

Is there a way to estimate the number of unique monthly visitors to a site based on a limited sample of one week of data? I have information about when a given user visited the site. This isn't as ...
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8 votes
1 answer
1k views

What is the optimal algorithm for playing the hangman word game?

Suppose we are playing the game hangman. My opponent and I both have access to the dictionary during the game. My opponent picks a word from the dictionary with knowledge of the algorithm which I will ...
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1 vote
1 answer
39 views

Minimizing Trimmed Distances

I'm trying to understand what is the minimum of the following function, $$ f(\mu) = \frac{1}{n}\sum_{i=1}^n F\left(\frac{\pi(i)}{n}\right) (x_i - \mu )^2 $$ where $F$ is a step function that assign $...
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Does this problem have a formal name?

I have come across the following problem but am unable to understand the solution for it. Hence I would like to know if it has a formal name then, I can search for it and read about it in more detail. ...
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0 answers
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How to find clusters of a set of points in n-dimensional space that are furthest from unwanted points

I have a list of 25 points and their coordinates in a 512-dimensional space. I have 8 target points and 17 points I need to avoid (the 17 points to avoid also have differences in priority of how ...
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1 vote
1 answer
237 views

Amount of expected loop iterations when searching an array by random index

Lets say we have an array A of size n. It has 1 as its first index and n as its last index. It contains a value x, with x occurring k times in A where 1<=k<=n If we have a search algorithm like ...
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0 answers
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Kullback-Liebler Divergence

For $P(x)=N(\mu,\sigma^2)$ and $Q(x)=N(0,1)$ I am supposed to calculate $KL(P(x)||Q(x))$, here is what I did \begin{align*} KL(P(x)||Q(x)) & = \int P(x) \cdot \log\left(\frac{P(x)}{Q(x)}\...
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3 votes
1 answer
166 views

Measures of performance in machine learning

I'm new to machine learning and struggle to interpret the results I get from different measures of performance. If for several prediction models I have e.g. accuracy, precision, recall, F1, FPR, and ...
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1 vote
0 answers
98 views

Suggestion for a good statistics book for computer scientists, in preparation of machine learning

We (a group of CS postdocs and Ph.D. students) are starting a shared reading of a machine learning book ("An Introduction to Statistical Learning", James, Witten, Hastie, Tibshirani). Before diving ...
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2 votes
0 answers
52 views

The No-Free-Lunch Theorem and K-NN consistency

In computational learning, The NFL theorem states that there is no universal learner. For every learning algorithm , there is a distribution that causes the learner to output a hypotesis with a large ...
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3 votes
0 answers
31 views

What is the purpose of standardization in machine learning?

I'm just getting started with learning about K-nearest neighbor and am having a hard time understanding why standardization is required. Reading through, I came across a section saying When ...
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3 votes
1 answer
61 views

Estimation of the number of solutions by Counting

This is a question from a quantum computation textbook. Consider a classical algorithm for counting the number of solutions to a problem. The algorithm samples uniformly and independently $k$ times ...
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0 answers
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What kind of standard deviation must be used in optimization algorithms?

I would like to ask about the standard deviation of objective function value. There are two types of standard deviations: Population standard deviation Sample standard deviation In metaheuristic ...
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1 answer
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Assumption of a generation of the dataset by a probability distribution

Consider the following paragraph from the deeplearningbook The training and test data are generated by a probability distribution over datasets called the data-generating process. We typically ...
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1 answer
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How can we get small test error reducing only train error?

My question is about mathematical part of machine learning algorithms, especially about using it in neural networks. We train network reducing train error and I was thinking about how then test error ...
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3 votes
3 answers
121 views

How good (or bad) is my makeshift PRNG?

Say I have designed a makeshift PRNG for my personal amusement, now I would like to see how good it is. How do I benchmark its "randomness"? Ideally, I want to know a statistics test, such that if I ...
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1 vote
0 answers
36 views

How to go about designing a quiz that changes difficulty based on user performance?

I thought about using a machine learning algorithm to learn the performance patterns of the quiz taker and model the next quiz based on how he/she performed. So each question will have a difficulty ...
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2 votes
2 answers
79 views

identify abrupt frame changes in a video

The frames of live streams or videos, in general, do not flow smoothly. A sports game would have multiple cameras recording the game and hence, there will be camera switches in the stream. I am ...
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1 vote
2 answers
348 views

probability of collision

my data's range is from 1 to 9 and I have two subsets of integers from this range. the hash function takes each of this subsets and calculate product of these three integers and maps this set to the ...
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