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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18 views

Use stirling's formula to prove central limit theorem for binomial random variables [migrated]

First I must show $${n \choose k} p^k q^{n-k} \sim \frac{1}{\sqrt{2\pi n p q}}e^{-\frac{(k-np)^2}{2npq}}?$$ How do I do this? I know that Stirling's formula says that as $$n \rightarrow \infty.$$, $$...
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19 views

Definition of Derivative and the Delta [migrated]

I am doing some approximation for a function, during this task I came across the following equation and I was wondering if I can consider it as the derivative with respect to the variable t: . ...
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26 views

Approximation of chained functions

Say we want to approximate a set of $n$ continuous functions $f_n(g(x))=y$ where $x \in \mathbb{R}^d, y \in \mathbb{R}, g(x) \in \mathbb{R}^m$ by fitting them to $n$ different datasets $(X, Y)_n$ ...
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60 views

Expected value with dependent samples

It is well known that the expected value of a function can be approximated with i.i.d. samples: $$ E_X[f(X)] = \frac{1}{n}\sum_{i=1}^n f(x_i),\quad x_i\sim_{i.i.d.} X $$ What methods exist to ...
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Approximating the expected value of a random variable as a function of the prior mean of a parameter

I have a parameterised statistical model and I'm trying to calculate the expected value of a random variable. I know that the expected value is a function of the value of the unknown parameter. So I ...
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19 views

kl divergence of two different distributions of subsets

Suppose there is a set $S=\{1, 2, 3, ..., n\}$, then I need a distribution of its subsets with fixed size k, which can be denoted as $A=\{x_1, x_2, ..., x_k\}$ where $x_1$ to $x_k$ are from 1 to n. ...
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39 views

Approximate known non-linear function using linear regression

Consider the following model: $$ y_{i}=f\left(\boldsymbol{x}_{i};\theta\right)+\varepsilon_{i} $$ where $y_{i}$ is the dependent variable, $\boldsymbol{x}_{i}$ is a vector of explanatory variables, $...
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28 views

Maximum sample size for one-way ANOVA?

Lists of requirements for one-way ANOVA include the following: Samples should be mutually independent Samples should be from a population with a normal distribution Samples should have the same ...
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73 views

Estimating Quartiles with Moments

The Wikipedia article on Skewness indicates that the median of a distribution can be estimated from the mean, standard deviation, and skeweness with an error term that goes as $O(skewness^2)$. ...
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14 views

Binary Matrix Low Rank Matrix Factorization

Low Rank Matrix Factorization is a pretty popular problem in data mining. We need to find 2 matrices, $W, H$ such as $F = W \cdot H$. I know that this approximation is NPC problem, so we won't find ...
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87 views

Sampling distribution is not normal. How is that possible?

As central limit theorem suggests, sampling distribution is approaching normal on the large sample sizes regardless of the initial distribution of the variable. And it's always been true for me until ...
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Nystrom approximation with inexact/stochastic kernel evaluation

Suppose we have several data points $x_1,\ldots,x_m\in\mathbb R^n$ as well as a positive definite kernel $K(x,y):\mathbb R^n\times\mathbb R^n\to\mathbb R$ that can be written in the form $$K(x,y)=\...
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Robustness of a model to learnt parameters

There is a recent push to study how sensitive a model is to small changes in its input. This has also been studied from an adversarial point of view: e.g what is the smallest input perturbation that ...
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74 views

What is the difference between approximate bayesian computation vs approximate bayesian inference?

What are the main differences between approximate bayesian computation vs approximate bayesian inference? Are they essentially the same? Do they refer to the same of different family of models? My ...
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Approximate inverse of a Gaussian Process

I'm using a GP in order to learn the transition function of a continuous Markov Decision Process, i.e. P(s'|s,a). This works reasonably well, but I'm now also ...
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103 views

What is better in Monte Carlo integration: product of means or mean of products?

Let $X$ and $Y$ be two independent continuous random variables with pdfs $f_X$ and $f_Y$, respectively. Let $\varphi_1$ and $\varphi_2$ be two continuous functions from ${\mathbb R}$ to ${\mathbb R}$. ...
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How to transform $P[k_1\leq (x_i-\mu - \sigma\cdot Z)^2 \leq k_2]$ to $P[k_1\leq \frac{(x_i-\mu)^2}{\sigma^2}+e \leq k_2]$?

Taste estimation As an example consider an experiment conducted to determine the optimal concentration of salt in popcorn. The concentration of salt in sample $i$ is denoted by ${x_i}$. The subject ...
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101 views

Multivariate Gaussian FItting

When trying to approximate a distribution of random vectors D by using multivariate gaussian what properties must we ensure that D has ie; what distributions can be estimated by Multivariate gaussian ...
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Computing KL Divergence for distributions over sets

I have a distribution over a set of (hundreds of) discrete terms, and I'd like to describe the difference between I see a couple of options, and none seems really attractive: Take the KL divergence ...
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66 views

what is the probability of detecting departure from H0?

The desired percentage of SiO$_2$ in a certain type of aluminous cement is 5.5. To test whether the true average percentage is 5.5 for a particular production facility, 16 independently obtained ...
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22 views

Poisson distribution question

An airline has found that the number of people booked on flights who do not arrive at the airport follows a Poisson distribution at the rate of 2% per flight.For a flight with 146 seats ,150 are sold ...
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43 views

How can I apply the Poisson ($\mu$) distribution to two series of random draws?

I have the following question: A box contains 1000 balls, of which 2 are black and the rest are white. If two series of 1000 draws are made at random from this box, what approximately, is the chance ...
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124 views

How to get better approximation than Central Limit Theorem

This is continuation of my problem Calculate variance of sum random variables Suppose random variable $X$ takes 3 values $1, 2, 3$ with probability $\frac{1}{2}$, $\frac{1}{3}$ and $\frac{1}{6}$. ...
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53 views

How to approximate the distribution of the sum of multiple multinomial random variables?

Say we have $T$ independent Multinomial random variables $X_1,X_2\dots X_T$, with $p(X_t=m)=p_{t,m},m\in\{1,2,...M\}$. What would be the distribution of $X_1+X_2+...+X_T$? If there is no closed-form, ...
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Approximation for the sampling error of the number of positive cases in a Bernoulli trial

Reading the book "Energy for Future Presidents" I found a way of approximating the binomial proportion sampling error which I never saw before, and I would like to know if my derivation is correct. ...
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242 views

Inverse-normal CDF approximation in Excel, Python or R

I read that the implementations of Inverse-normal cumulative distribution function (CDF) /quantile / ppf in R, Python (scipy) and Excel give similar results. However, I can't find the very formulae ...
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Order of continuity of an ANN approximation dependent on the activation functions used?

If I have understood this correctly, a result from Hornik et al.'s Universal Approximation of an Unknown Mapping and Its Derivatives Using Multilayer Feedforward Networks essentially states that, if ...
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Definition of Mean Squared Value Error with respect to action-value functions in Reinforcement Learning algorithms

I am referring to page 199 of Sutton and Barto book on Reinforcement Learning available here: book There the Mean Squared Value Error for an vector-parameterized function approximation $\hat{v}(s,\...
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25 views

Approximate a density function from sampled data

Let $(E,\mathcal E,\mu)$ be a measure space $E_0\in\mathcal E$ with $\mu(E_0)\in(0,\infty)$ and $\mathcal E_0\subseteq\left.\mathcal E\right|_{E_0}$ be finite and disjoint with $$E_0=\biguplus\...
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Asymptotic approximation of log-probability using first four moments

Consider a random variable $X \sim p_{n,\theta}$ where the first four moments are given by known functions: $$\begin{matrix} \ \ \ \ \ \ \mathbb{E}(X) \equiv \mu(n,\theta) & & & \ \ \ \ \ ...
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53 views

Analytical solution to the multivariate CDF given multivariate pdf

Is there any way of approximating or analytically solving the below CDF (let's say even for $n\to\infty$)? I am trying to find the below probability: \begin{align} &P\left[X_{2}-X_{1} \leq 0,...
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128 views

Can we minimize counting cost function for perceptron algorithm?

In perceptron algorithm (the following analysis might apply to other classification algorithms), a smooth approximation of perceptron cost function $$\sum_i^n{\max(0, -y_i\mathbf{w}^T\mathbf{x}_i)}$$ ...
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76 views

Expectation Value of a Product of Many IID variables

First of all, I apologize for not being rigorous, but I am not a statistitian by background. Imagine you have $N$ i.i.d. positive random variables $X_1...X_N$ and you are trying to compute a ...
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30 views

“At least” approximation calculation

I have a vector of different probabilities to get 1, for example probs = [0.1, 0.5, 0.2, 0.9, 0.25, 0.55] I have to calculate the probability of having at least ...
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Query complexity of ANN based on Hamming distance

Ilya R provides the following query complexity of ANN for Hamming distance based on coordinate sampling https://www.ilyaraz.org/static/class_2018/files/20181023.pdf When he says ...
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Back-Transformation for Ln(X+1) of zero rich data

I have seen and read several similar questions, but mine pertains specifically to zero rich data. I will be back transforming my data based on a first order Taylor series approximation. As outlined ...
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37 views

Approximation of non linear function with multiple linear functions

How can a non-linear function be approximated by an appropriate amount of linear functions? In the picture below, it would be quite easy to draw 10-15 linear functions to describe all data points ...
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approximation the erf function for finding Z [duplicate]

I am looking for the following probability where X follows normal distrubition; P(X$\leq$x)= ($\frac{1}{2}$(1+erf($\frac{x}{\sqrt2}$)) I have a constraint in my mathematical model like P(X≤x) $\leq$...
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42 views

An approximation to the cdf of the normal from a pdf?

In this paper (p. 36), authors wrote $$p(n,T) = \Phi \Big(\frac{n}{T},\mu,\sigma \Big) - \Phi \Big (\frac{n-1}{T},\mu,\sigma \Big)\; (3) $$ Bellow we will use the approximation $$p(n,T) =...
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Learn to mimic function adaptively

Assume I have a function $F: R^n \to R$ that is slow to evaluate, which I, therefore, would like to approximate with something faster by using machine learning. I have seen some work proceeding by ...
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46 views

Why is one of the two approximations in the bootstrap worse than the other?

My statistics text has the following diagram: $$\mathbb{V}_F(T_n) \overbrace{\approx}^{ \text{not so small} } \mathbb{V}_{\hat{F}_n}(T_n) \overbrace{\approx}^{ \text{small} } v_{\text{boot}}$$ ...
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239 views

Double integral involving the normal CDF

I need to compute (or best approximate?) the following integral $$\int_0^\infty \int_0^\infty (1 + \alpha u)^{-1}(1 + v)^{-1} \Phi\left(\frac{\beta}{\sqrt{\gamma + uv}}\right) \text{d}u \text{d}v,\...
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46 views

Approximate the data to a single curve

The question might be simple, but I am not able to find the answer. Hence I am asking here. I did search google but didn't get an answer. I have a continuous stream of data coming from an API in the ...
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57 views

Finding mode of posterior using Newton method in R

I am attempting to approximate the posterior $\tilde{\pi_{G}}(z|\theta,Y)$ which is the Gaussian approximation to the full conditional of $z$, and in order to do this I need to find the mode $z^{*} \...
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350 views

Variance of Normal Order Statistics

Suppose we have $X_1, \cdots, X_n \overset{\textrm{i.i.d.}}{\sim} \mathcal{N}(0, 1)$ with $n > 50$, and let $X_{(1)}, \cdots, X_{(n)}$ be the associated order statistics. Are there any references ...
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31 views

at what probability will the probability we start considering the data?

For example I have this problem, Do Americans tend to vote for the taller of the two candidates in a presidential election? In 30 presidential elections since 1856, 18 of the winners were taller than ...
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73 views

Sum of product as product of sums

Assuming I have two random non-independent vectors $A,B$ which are within [-1,1]. I want to approximate their sum of product by product of sums (everything is a dot product), i.e. $\sum_{i=1}^NA_iB_i ...
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30 views

Simple arithmetic approximations to categorical analyses

Suppose I have a two by two table: $$ \begin{array}{c|ccc} & Y & \neg Y & \\ \hline X & a & b& &\\ \neg X & c & d& &\\ \end{array} $$ and I am interested ...
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52 views

Estimation of function using Spline Interpolation

My problem is the following: Estimate the function from given data (below) and show that the estimated function has the following properties: (i) $f(0)=0$ (ii) $f(x)>0, x>0$ and $f(x)<0, x<...
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529 views

Can a Bernoulli distribution be approximated by a Normal distribution?

$$\sum_{i=1}^n bernoulli(p) = binomial(n,p) \approx \mathcal N(np, np(1-p)) = \sum_{i=1}^n \mathcal N(p, p(1-p))$$ Can I conclude that $\mathcal N(p, p(1-p))$ could represent an approximation of $...

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