Podcast #128: We chat with Kent C Dodds about why he loves React and discuss what life was like in the dark days before Git. Listen now.

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.

Filter by
Sorted by
Tagged with
0
votes
1answer
47 views

Frequency of N Independent Events

I'm attempting to implement a numerical approximation to assess the frequency of outcomes given $N$ independent random variables. There is a slight twist however, each $N$ has a weight, Each variable ...
2
votes
1answer
149 views

Approximated relation between the estimated coefficient of a regression using and not using log transformed outcomes

Consider the two regression equations: \begin{align} Y &= \alpha+\beta X+\varepsilon \\ \log Y &= \gamma+\delta X+\eta \end{align} Is there a "simple" approximated relation between the ...
3
votes
1answer
164 views

Question about variational autoencoder gradient

The following is from Section 2.2 of the Auto-Encoding Variational Bayes paper, It says the gradient of the lower bound w.r.t $\phi$ is a bit problematic because the Monte Carlo estimator exhibits ...
0
votes
0answers
235 views

Double integral in the context of a bivariate normal distribution

This question is relevant to bivariate normal distribution. I figured out most of the steps but am currently stuck on finding the closed form of a double integral. Let $x$ and $y$ be of a bivariate ...
3
votes
0answers
48 views

For the two sample t-test statistics T, how to find the order with respect to the difference E(T| welch-t approximation) - E(T| exact distribution)?? [closed]

For the two sample t-test with unequal variances: $T=\frac{\bar{x}-\bar{y}}{s_x^2/n_1+s_y^2/n_2}$, we usually use welch approximation, thus $T\sim t(f)$, where $f=\frac{(\frac{\sigma_x^2}{n_1}+\frac{\...
3
votes
1answer
151 views

Chow-Liu trees and Kullback Leibler divergence

I'm reading David Barber's book on Bayesian Reasoning and Machine Learning. At Section 9.5.4 he covers Chow-Liu trees, and I am having difficulties understanding the flow of the equations after he ...
6
votes
0answers
37 views

What bounds can we place on approximation error for a moment-matching approximation with $N$ moments?

Suppose I have a distribution over the real line ($p$) and I'm approximating it by matching its first $N$ moments. What can I say about the approximation error as a function of $N$? Alternatively, ...
4
votes
0answers
753 views

Faster computation of high-dimensional multivariate normal probabilities

My goal is to find a faster way to calculate something like mvtnorm::pmvnorm(upper = rep(1,100)) that is, the tail probability of multivariate normal distribution ...
2
votes
0answers
75 views

What is the geometric mean of the first hitting time distribution of Wiener process?

I'm looking for an analytic formula. Approximate formulas are welcome, in which case I give more importance to simple and nice expressions rather than to precision of the approximation. I'm looking ...
0
votes
0answers
87 views

Distributions with uneven Min & Max boundaries that Approximates a Normal Distribution [on hold]

Suppose that you know the mean should equal 2.02, the minimum value should be 0.8, and the maximum value should be 3.6. What is the best method to create as close to an approximately normal ...
2
votes
1answer
307 views

Hessian for Laplace Approximation in Uncertainty Propagation

This is possibly a silly conceptual question, ... but anyway: Imagine I have a function: $f = F(\mathbf{x}) = F(x_1,x_2) = ax_1^2 + bx_2^3,$ where $x_1,x_2 \sim N(0,1)$ for example. For a naive ...
1
vote
0answers
123 views

Is there a way to approximately convert a standard deviation of log returns to a standard deviation of simple returns?

If I have a standard deviation of log returns, what do I need to know to convert it to a standard deviation of simple returns? I have got the following R functions which seem to give consistent ...
1
vote
0answers
23 views

A close-form solution to a low dimensional curve fitting

I have a very short vector of intensities (let's say 5 elements per vector). The intensities are a discrete sampling of a curve around its maxima (the maxima is not discrete though). What I'd like to ...
1
vote
0answers
39 views

Approximate computation of a linear function

I am not sure that this is the right stackexchange site for this, but here goes: Let $d_1,\ldots,d_n$ be known real numbers, some positive, some negative. Let $v_1,\ldots,v_n$ be non-negative numbers ...
3
votes
1answer
1k views

code Welch's t-test in C++ using approximations for student's t values

I need to code a Welch's t-test between two populations in C++ without using external libraries like, for example, boost. I know that given my two populations of size $N_1$ and $N_2$, I can calculate ...
0
votes
1answer
100 views

Ways to approximate multiple samples of same function in R

Example dataset (simplified): ...
0
votes
1answer
144 views

Interpretting Confidence Interval Questions

In a survey conducted by a mail order company a random sample of $200$ customers yielded $172$ who indicated that they were highly satisfied with the delivery time of their orders. Calculate an ...
1
vote
0answers
113 views

Computing the partition-function of an exponential family member

I am working on a Monte Carlo Expectation Propagation problem. In that context I got the following property: $ I = \sum\limits_i w^{(i)} \log p_\eta(x^{(i)}) $ where $\{w^{(i)}\}_i$ are weights, $...
1
vote
1answer
40 views

Ratio in a random subset without replacement

This function returns a random proportion. It appears normally distributed with mean $a/(a+b)$. What is its approximate variance? ...
7
votes
1answer
2k views

Why does the reconstruction error of truncated SVD equal the sum of squared singular values?

I saw this formula in a textbook: squared Frobenius norm of the original matrix $\mathbf X$ minus its truncated SVD $\mathbf X_k$ (which can be seen as the approximation error) equals the sum of ...
1
vote
1answer
35 views

Other types of mean error

Let $\tilde f$ be an approximation of the function $f(x) = \arccos(x)$. I'm using MATLAB to figure out how good this approximation is by calculating a mean error. My first idea was to use this formula ...
1
vote
0answers
28 views

Digits that are insignificant by virtue of representing extremely small quantities within a much larger quantity: what are they called?

In computer science, sometimes we can measure run-times with high accuracy. As a hypothetical scenario, suppose a computation takes a week to run and we can measure the run-time accurate to the ...
2
votes
1answer
256 views

68–95–99.7 rule for binomial distribution?

What I want to calculate is the probability that the size $k'$ of an intersection of two subgroups with sizes $n$ and $K$ that have been randomly selected from a group with size $N$ that is smaller ...
8
votes
4answers
5k views

normal approximation to the binomial distribution: why np>5?

Nearly every text book which discusses the normal approximation to the binomial distribution mentions the rule of thumb that the approximation can be used if $np\geq5$ and $n(1-p)\geq 5$. Some books ...
2
votes
1answer
454 views

Dirac Delta function Notation

I am trying to understand the delta function notation used to be express a monte carlo approximation of a probability distribution. The notation used in this (p10) is $\pi(x_{1:n}) = \frac{1}{N}\sum^...
2
votes
0answers
94 views

Training Restricted Boltzmann Machines according to the Likelihood Function

Is it possible to chose the parameters of a RBM to maximize the likelihood of the observed data? (I follow the notation of the deeplearning tutorial ). Denote the observable data by $x$, hidden data ...
6
votes
2answers
374 views

Bayes Factor approximation

A brute force method to approximate the Bayes Factor (the ratio of the denominators (normalizing constants) in the Bayes formula) is to do the following for the two models of interest: repeat ...
0
votes
0answers
213 views

mixture of Gaussians vs mixture of quadratic denominators (Cauchy)

It is known that mixture of Gaussians are dense in the set of all distribution functions. A 1-dimensional Gaussian has the following density: $$ \frac{1}{\sqrt{2\pi \sigma^2}} e^{-\frac{(\omega-\beta)^...
5
votes
0answers
192 views

Asymptotic distribution of a weighted sum of chi squared variables beyond CLT? [duplicate]

I have a sum $$ S = \sum_{i=1}^{n} d_i X_i^2, $$ where $X_i$ are independent standard normals, and $d_i > 0$ are fixed real numbers, for example $d_i = i$. The asymptotic distribution of this sum ...
3
votes
1answer
45 views

bad fit - nomenclature for breeds

Question: What is it called when one uses a basis, like the pure line instead of the sigmoid/logistic, in a manner that grossly departs from the "physics" of the problem? There should be a word for ...
8
votes
1answer
2k views

How to understand the geometric intuition of the inner workings of neural networks?

I've been studying the theory behind ANNs lately and I wanted to understand the 'magic' behind their capability of non-linear multi-class classification. This led me to this website which does a good ...
0
votes
1answer
253 views

glasso- Assumptions of Meinhausen-Buhlmann approximation?

First off, This is not an R question - it is a conceptual question, so there is no need to perform any code. Problem: I'm trying to invert a large dimensional covariance matrix of p features. For ...
0
votes
0answers
45 views

How can I estimate the approximation quality of the odds ratio based on a sample or how can I ignore infrequent items?

I am analyzing URLs found in tweets of the Twitter stream. I have two samples of tweets from that stream: a random sample of tweets all tweets of a subset of users I want to find URLs which are ...
1
vote
0answers
147 views

Are there any fast approximations to generalized linear mixed models?

Are there any recommended methods or approximations that would help speed up the estimation of fixed effects components of a generalized linear mixed effects model? Specifically, my dataset includes ...
5
votes
1answer
556 views

Bound for weighted sum of Poisson random variables

Suppose I have some independent Poisson-distributed random variables $X_1 \ldots X_N$ with parameters $\lambda_1 \ldots \lambda_N$. These can be thought of as processes where each arrival/event ...
1
vote
1answer
821 views

Confidence interval from R's prop.test() differs from hand calculation and result from SAS

I'm wondering if anyone has insight into how prop.test() in R calculates its confidence intervals. Although it doesn't state it explicitly in its documentation, my ...
2
votes
1answer
1k views

Finding a “skew normal distribution” for given data

I am given a set of $n$ pairs $(x_i, y_i)$, where the $x$-coordinates can be interpreted as the measured values of a random variable $X$ and the $y$-coordinates can be interpreted as some "scaled" ...
1
vote
1answer
33 views

Statistical problem related to product lifespan

I design electronics, and while I have some basic education in the principles of statistics, I don't believe I'm qualified to determine whether this problem is tractable as it was posed to me. So I ...
3
votes
2answers
374 views

If Machine learning is about making accurate predictions, isn't it a catch-22 situation with the no free lunch theorem?

If Statistics is about making approximate summaries while Machine Learning is about making predictions as accurate as possible, doesn't it create a paradox due to the no free-lunch theorem?
1
vote
0answers
80 views

Is there any paper about the distribution of difference of log-normal variable?

I am working on the problem relating to the difference of log-normal distribution. I have found several papers about this topic, however, none of them gives me the answer I want. More specifically, ...
11
votes
1answer
186 views

Approximating $\log( E(X))$

I was casually reading an article (in economics) which had the following approximation for $\log(E(X))$: $\log(E(X)) \approx E(\log(X))+0.5 \mathrm{var}(\log(X))$, which the author says is exact if ...
6
votes
1answer
304 views

Approximating P(A,B,C) using P(A,B), P(A,C), P(B,C), and P(A), P(B), P(C)

For some events $A$, $B$, $C$ I know the occurrence probabilities $P(A),\: P(B),\: P(C)$ I also know the pairwise co-occurance probabilities $P(A,B),\: P(A,C),\: P(B,C)$ I want to approximate the ...
0
votes
1answer
165 views

Approximate probability distribution for a given expression of posterior probability

I am dealing with a discretized 2D space and I have an expression to calculate posterior probability of an event to happen at a grid point in the space. Given a measurement, the posterior probability ...
1
vote
1answer
80 views

Can I approximate the variance of a ratio to the ratio of two variances?

I've been slowly working my way through problems in Casella and Burger's Statistical Inference and I stumbled upon this problem: I haven't been able to get past (a). I've tried using the ...
12
votes
1answer
673 views

Assessments of “Approximately Normal” for t-tests

I am testing equality of means using Welch's t-test. The underlying distribution is far from normal (more skewed than the example in a related discussion here). I can obtain more data but would like ...
5
votes
3answers
166 views

Algorithm for approximating a density by a mixture density

Given a density $f(x)$ (e.g. the log-normal distribution or log-$t_{\nu=3}$ distribution), I was wondering what algorithm are known/typically used to find a mixture of distributions $g_r(x)$ from ...
0
votes
0answers
43 views

Express $E(x^{\alpha})$ in terms of $E(e^{-\zeta x})$? to a 1st or second order?

I have a random variable, $X$, and am able to find $\mathbf{E}(e^{-\zeta X})$ for many $\zeta$ (through the Laplace transform solving an ODE as this actually evolves over time) Is there any way I can ...
2
votes
1answer
199 views

Distribution over the product of three, or $n$, independent Beta random variables

This is a re-post of a question on the Mathematica stack exchange, as per the advice of another user (see here). I am pursuing a computational solution there, but thought it might be worth looking for ...
1
vote
0answers
103 views

Theoretical properties of Gaussian Process Emulator

I am studying Guassian Prcess Emulator (GPE) to approximate computationally expensive computer models. Basically, we suppose the computer model, or simulator, is denoted by $f(x)$, where $x$ is the ...
1
vote
0answers
35 views

The probability that several normally distributed random variables have a particular order

Let's say we have n random, independent variables $X_1,\ldots,X_n$ with normal distributions. Is there a reasonable way how to compute the probability that they have a particular order, for example $$...