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Questions tagged [approximation]

Approximations to distributions, functions, or other mathematical objects. To approximate something means to find some representation of it which is simpler in some respect, but not exact.

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Continuity correction in a 2 proportion test, with different sample sizes

In a test of 2 proportions (binomial -> Normal), when the sample sizes are different, what does a continuity correction look like? Usually, in a 1 sample test, we would divide by $n$ (sample size) ...
An old man in the sea.'s user avatar
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Characterize conditions in which Taylor moment approximation is good

I am working with the Projected Gaussian, or Angular Gaussian distribution, which is given by $z = \frac{x}{||x||}$, where $x \sim \mathcal{N}(\mu, \Sigma)$. This is a distribution on the sphere in $\...
dherrera's user avatar
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Why is taking the mean RMSE sometimes so far off overall RMSE?

I'm working with a multi-threaded program, which splits a dataset into N chunks, and evaluates some regression model's performance, predicting a score for each item in each chunk. I'm using RMSD as ...
Seán Healy's user avatar
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22 views

Monte Carlo Approximation on integral of Gaussian pdf on Convex Domain

I have hard time on estimating the following integral on convex domain ($\mathcal D$) using Monte-Carlo approximation. $$a = \int_{\mathcal D} dx f(x;\mu,\Sigma) $$ where $x \in \mathbb R^d$ and $f$ ...
Interception's user avatar
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27 views

taylor approximation multivariate OLS coefficient

Say we have the following multivariate regression model: $ y = \beta_1 x_1 + \beta_2 x_2 + \varepsilon $ The OLS formula for the first coefficient looks like this $ \hat{\beta}_1 = \frac{Cov(\tilde{y}...
user9875321__'s user avatar
4 votes
2 answers
87 views

Universal approximation theorem for neural networks reference

On Wikipedia, a nice theorem is given: However, I can not find the stated theorem in the given references. So where is the stated theorem from?
tamtam_'s user avatar
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Getting extremely poor accuracy while doing function approximation using a neural networks in PyTorch [duplicate]

I have been given a task to approximate the function 5x^3 - 10x^2 - 5x - 9 using a neural network in pytorch. The training data is the set of integers in the range [-100,100] and I have to test the ...
Paarth's user avatar
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Approximation for a correlation matrix

I have a cross-correlation matrix of some parameter for each time period. E.g. expected economy growth for each months in the future, i.e. growth for Apr 2014, May 2014, ...., Dec 2018, and ...
guygsakjdfbnasdbff's user avatar
3 votes
1 answer
75 views

What paper did Hall suggest the queuing rule of thumb $s \geq \max ( 1, \rho + \sqrt{\rho})$?

According to this site: Hall (1991) cited an argument of his previous paper that operation research profession could and should be more scientific and less mathematical. In fact, Hall also suggested ...
Galen's user avatar
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3 votes
1 answer
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Montecarlo Confidence Interval of T distribution

Suppose: \begin{equation} x|\sigma^2 \sim \mathcal{N}(x; \mu, \sigma^2) \; \; st. \; \; \sigma^2 \sim \mathcal{X}^{-2}(\sigma^2; \psi, v) \end{equation} where $\mathcal{X}^{-2}$ is the inverse ...
Snowy Baboon's user avatar
1 vote
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Taylor approximation for function of a random variable [closed]

There is a function $f$ whose domain is the space of CDFs on $\mathbb{R}_+$ and whose range is $[0,1]$, e.g. $f$ maps a CDF on to a real number. Further, $f$ is continuous, increasing with respect to ...
user's user avatar
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3 votes
1 answer
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Moments and PDF of solution to random quadratic equation

Consider the following random quadratic equation, $$ x^2 + Z x + Y = 0, $$ where, $$ \begin{gathered} Z \sim \mathcal{N}(\mu_Z,\sigma_Z), \qquad Y \sim \mathcal{N}(\mu_Y,\sigma_Y). \end{gathered} $$ ...
Emmy B's user avatar
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Little's Law + Kingman's Formula --> Approximation of Expected Length of G/G/1?

Little's Law gives us $$\mathbb{E}[L] = \lambda \mathbb{E}[W]$$ where $L$ is the number of customers in the queue + being served $\lambda$ is the arrival rate $\mu$ is the service rate $W$ is the ...
Galen's user avatar
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3 votes
2 answers
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Diffrence in logs vs. a % changes in econometrics: why is the dif log approvimation almost always used when the exact quantity is easily available?

I have observed that in econometrics work people almost always use the difference in logs rather than the actual percentage change. This makes no sense to me. I understand that the difference in logs ...
andrewH's user avatar
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Is there a heavy traffic approximation for percent under benchmark?

Suppose I have an waitlist of patients waiting to be served. Each patient has a benchmark number of days that the service should be completed by as a goal (not a hard constraint of the modelling). ...
Galen's user avatar
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3 votes
2 answers
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Approximating the distribution of the product of iid beta variates

Background I am interested in the distribution of $$\theta_0=1-\prod_{i=1}^n(1-\theta_i)$$ where the $\theta_{i>0}$ are iid beta random variates: $$\theta_{i>0}\sim\text{Beta}(\alpha,\beta)$$ In ...
jblood94's user avatar
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3 votes
1 answer
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Approximating the standard normal density with the logistic density: How to numerically optimize $\infty$-norm?

Let's say that we want to use the logistic distribution as an approximation to the standard normal density. As the location parameter of the logistic distribution is $0$, the scale parameter $s$ is ...
COOLSerdash's user avatar
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4 votes
1 answer
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Formulas or approximations for $\mathbb{E}\left( \frac{X}{\|X\|} \right)$, $X\sim N(\mu, Id)$?

This is a cross-posting of this math SE question. I want to compute or approximate the following expected value with some analytic expression: $\mathbb{E}\left( \frac{X}{||X||} \right)$ , where $X \in ...
dherrera's user avatar
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Approximating a bivariate distribution with another distribution, which method to use?

Let $X \sim F(;\theta)$ and $Y \sim G(;\eta)$ be two independent continuous random variables. The greek symbols represent the parameters of those distributions. I can easily sample from these ...
Coolio's user avatar
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2 votes
1 answer
98 views

Finding Sample Range of Fisher's z-distribution via Approximating Hypergeometric $\,_2F_1\left(\frac{1}{2},\frac{x+1}{2};\frac{3}{2};-z^2\right)$

Recently, I have encountered Hypergeometric function $\,_2 F_1\left(\frac{1}{2},\frac{x+1}{2};\frac{3}{2};-z^2\right)$ in the context of order statistics. In particular, I am trying to evaluate an ...
anatolvitold's user avatar
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1 answer
62 views

Extrapolating a Discrete Distribution to a Continuous one

Say I have a list of the letter grades of a class (meaning some number of As, Bs, Cs, Ds and Fs). Is there any way for me to take this discrete distribution and extrapolate it to the most likely ...
Ghull's user avatar
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0 answers
37 views

Methods to approximate Area under Precision-Recall Curve

average_precision_score from sklearn uses formula: ap = sum( (recall[k+1] - recall[k]) * precision[k+1] ) But trapezoidal rule implies: ...
Ars ML's user avatar
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2 votes
0 answers
21 views

Differential Privacy guarantee that takes into account the approximate density (e.g., the pseudo randomness) used in practice?

In theory the differential privacy guarantee comes from adding randomness to an algorithm so whatever is output is a sample from a target distribution (e.g., the Laplacian, Gaussian, Exponential ...
travelingbones's user avatar
2 votes
1 answer
69 views

Certain approximation in the setting of three expectation values does not make sense to me

I'm currently going through some lecture notes in the field of Bayes optimization and I'm currently looking at a expression looking like this: $$\mathbb{E}_{x^*} \left[\mathbb{E}_y\left[\left\{\mathbb{...
SphericalApproximator's user avatar
1 vote
0 answers
36 views

Universal approximation theorem in the $\mathcal{C}^1$-norm [closed]

Let's say we are given a one dimensional FFNN $f_N: \mathbb{R} \rightarrow \mathbb{R}$ with hidden dimension $N$ $$f_N(x)=W_1\phi(W_2x+b_2)+b_1$$ with $W_2 \in \mathbb{R}^{N },b_2 \in \mathbb{R}^{N},...
NicAG's user avatar
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0 answers
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How to visualize the proximity between two stochastic processes?

I am studying a certain convergence of time series. And I would like to "make sure" that the convergence to a certain process actually happens. So, consider the following simplification. ...
PSE's user avatar
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Approximate distribution of random variable similar to studentized mean R.V?

It is well known that the distribution of the studentized mean, i.e., $T_0 = \frac{n^{1/2} (\bar{x}- \mu)}{\left(n^{-1} \sum \limits_{i=1}^n (x_i^2 - \bar{x}^2)\right)^{1 / 2}} $, can be approximated (...
Jaimin Shah's user avatar
1 vote
0 answers
67 views

Could one use mixtures of Gaussians to turn MCMC posterior samples into a new prior?

Theoretically in Bayesian inference one could use one experiment's posterior as another experiment's prior, such that knowledge of the parameters accumulates from $p(\theta) \rightarrow p(\theta|\...
Durden's user avatar
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67 views

Which metric to compare two probability density?

I need to compare two distribution $p$ and $q$. But I don't have access to the distribution $p$, I want to approximate it by distribution $q$ that I construct iteratively by choosing design point. ...
YP BARRY's user avatar
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0 answers
87 views

Approximation on Inverse Mills ratio for the normal R.V

I've come across several approximations for Mills ratio, but I haven't found any good ones for the Inverse Mills ratio. Is there any known closed-form approximation for the Inverse Mills ratio (link) ...
Jaimin Shah's user avatar
2 votes
1 answer
100 views

An approximate confidence interval for the $\alpha$ parameter of a Pareto Type II distribution when $\lambda$ is known

The Pareto Type II distribution, also known as the Lomax distribution, has the following density, $$f(x|\alpha,\lambda)=\frac{\alpha\lambda^{\alpha}}{(\lambda+x)^{\alpha+1}}, \qquad x>0,\ \alpha>...
29703461's user avatar
1 vote
0 answers
37 views

Approximate X given 5 function values and y values

Given 5 Lines(table) X values and corresponding Y1,Y2,...Y5. How can I calculate the approximate X value given the corresponding Y's? How can I tweak the formula if I want to weight to bias the ...
Jonathan Mercado's user avatar
9 votes
1 answer
664 views

XGBoost: universal approximator?

There are various "universal approximation theorems" for neural networks, perhaps the most famous of which is the 1989 variant by George Cybenko. Setting aside technical conditions, the ...
Dave's user avatar
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3 votes
0 answers
79 views

Understanding the ridge leverage scores sampling from an arXiv paper

I give a try to read the arXiv paper Distributed Adaptive Sampling for Kernel Matrix Approximation, Calandriello et al. 2017. I got a code implementation where they compute ridge leverage scores ...
emonhossain's user avatar
2 votes
0 answers
49 views

What are effective methods to maximize an unknown noisy function?

I have a function that takes a few hundred parameters and which returns a score I want to optimize for - It's a piece of software attempting to play a game against another player. The parameters ...
Borborbor's user avatar
1 vote
0 answers
100 views

Can the error be modeled in the approximation of expectation

I have a function $s(\omega)$ that is a sum of a function with random numbers $a_m$ and looks something like the following. $$ s(\omega) = \sum_{m = 1}^{M} f(a_m, \omega) $$ where all the $a_m$ are ...
CfourPiO's user avatar
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0 answers
61 views

approximating the density of the studentized range distribution

Is anyone aware of an approximation to the density function for the studentized range distribution https://en.wikipedia.org/wiki/Studentized_range_distribution ? I've found a fast approximation for ...
Brian Powers's user avatar
1 vote
0 answers
26 views

Restrictions on sample cumulants/moments for truncated Edgeworth expansion

I'm trying to approximate an unknown distribution by a truncated Edgeworth series, with cumulants/central moments estimated from a large sample. I notice though that I am getting negative tail ...
Frido's user avatar
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1 vote
0 answers
122 views

Normal approximation to the posterior distribution

Suppose we have a posterior sample of parameter $\theta$ obtained by fitting some Bayesian model to $n$ data points. In black is the empirical posterior density and in red is a normal approximation to ...
user7064's user avatar
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0 answers
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Overflow when computing binomial distribution for large n [duplicate]

How do you compute a binomial probability distribution for large $n$? If I try the following, I get an integer overflow in any programming language: ...
at01's user avatar
  • 111
3 votes
2 answers
146 views

How should I deduce the variance and expectation of the log of a variable?

I read this paper "voom: precision weights unlock linear model analysis tools for RNA-seq read counts", in the methods, the "Delta rule for log-cpm" section: The RNA-seq data ...
Dan Li's user avatar
  • 179
3 votes
1 answer
57 views

Robustness of Posterior distribution wrt likelihood function

Suppose we have $$ X_1, \ldots, X_n \mid \theta \, \mathop{\sim}^{iid} \, L(\cdot \mid \theta), \quad \theta \sim \pi $$ By Bayes' theorem, the corresponding posterior distribution is $$ \pi_n(\mathrm ...
mariob6's user avatar
  • 550
2 votes
2 answers
295 views

How can I use the Central Limit Theorem to calculate the distribution of $\bar{X}$?

The central limit theorem says that $$ \frac{\bar{X}-\mu}{\frac{\sigma}{\sqrt{n}}} \stackrel{\mathcal{D}}{\rightarrow} N(0,1) $$ What is the distribution of $\bar{X}$? I've seen it given as $\sum X \...
user1141170's user avatar
0 votes
1 answer
62 views

Extract the functional mapping between input and output from a machine learning model

A lot of ML models, such as neural networks, are Universal Function Approximators. But when evaluating ML models, we use usually metrics, such as MSE or accuracy, to assess the performance of a ML ...
Frank Gallagher's user avatar
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0 answers
28 views

Probability bound on Maxima under random sampling

I have a set $S$ = {$e_1,e_2,..e_{400}$} of 400 elements and a non-linear function $f:2^{(S)}\to[0,1]$ that takes a subset of $S$ and returns a real number in $[0,1]$. I want to compute the subset for ...
CSStudent's user avatar
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1 vote
0 answers
53 views

Do Neural Networks tend to have Zero Mean Errors in each Output?

My NN (a few linear layers with ReLUs + batch normalization, no activation in the last layer) learns to approximate vector-valued labels $y_z$ from data $z\sim\rho_z$ in a supervised way, i.e. net$(z)=...
joinijo's user avatar
  • 111
1 vote
1 answer
162 views

Algorithm for approximating linear-interpolated curve

Goal Given a curve defined by a set of (x, y) coordinates with linear interpolation, we want to find the best approximation using a smaller set of points (w/ linear interpolation) that fall along a ...
Kungfunk's user avatar
1 vote
1 answer
172 views

Comparing Gibbs sampler and variational inference

I am learning about variational inference and Gibbs simpler. I am in the process of deriving variational inference on my own. In this process, I need to make a comparison with Gibbs sampler. I am ...
sam's user avatar
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0 answers
89 views

Can $e^{E(\log(x))}$ be calculated approximating $E(\log x) $ using a second order Taylor expansion around the mean?

While searching around, I found this question Expected value of a natural logarithm dealing with the expected value of the natural log. The top answer references a paper that approximates the expected ...
Jama's user avatar
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0 answers
59 views

Aproximate maximum of two multivariate Gaussians with multivariate Gaussian

Given two multivariate Gaussians $G_1(\mathbf{x}), G_2(\mathbf{x})$ (not PDFs!) with the same center at the coordinate origin and different covariance matrix $\mathbf{F}_1, \mathbf{F}_2$, where $\...
logocar3's user avatar

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