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.
486 questions
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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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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 parameterized statistical model and I am 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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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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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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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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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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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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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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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? I.e., what distributions can be estimated by multivariate ...
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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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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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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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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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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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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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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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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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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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Recovering Distribution from Percentiles
I have data on the 10th, 25th, 50th, 75th, and 90th percentiles of a probability distribution, together with the mean, and standard deviation. I am interested in recovering a continuous distribution ...
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Determining under what conditions an exponential function is linear
I'm working through an exercise to determine when an exponential function of the form:
y = ae^(bx)+c
is approximately or exactly linear (of the form ...
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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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"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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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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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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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
$$p(n,T) = \frac{1}{T}...
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Learn to mimic function adaptively
Assume I have a function $F: R^n \to R$ that is slow to evaluate, which I, therefore, would like to approximate with something faster by using machine learning. I have seen some work proceeding by ...
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Why is one of the two approximations in the bootstrap worse than the other?
My statistics text has the following diagram:
$$\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}}$$
...
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Double integral involving the normal CDF
I need to compute (or best approximate?) the following integral
$$\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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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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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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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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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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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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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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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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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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Proof of Approximate / Exact Bayesian Computation
The ABC algorithm is given as
Draw $\theta \sim \pi(\theta)$
Simulate data $X \sim \pi(x | \theta)$
Accept $\theta$ if $\rho(X, D) < \varepsilon$
where $\pi(\theta)$ is the prior, $\pi(x | \theta)$...
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CLT approximation - how large should sample be so probability is equal to 0.95? [duplicate]
We have a measurement which has mean $\mu$ and variance $\sigma^2$ = 25. Let $\bar{X}$ be average of $\textit{n}$ such independent measurements.
How large should $\textit{n}$ be in so that $P(|\bar{...
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Is there a universal approximation theorem for monotone functions?
The universal approximation theorem basically states that a feed-forward network with a single hidden layer containing a finite number of neurons can approximate continuous functions on compact ...
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Approximation of the critical value for $\alpha$ of $\Gamma(n-1,1)$
Say I have the critical region for a test statistic $T$ and some constant $c$, as follows,
$$
n(T - 1)^2 \ge c
$$
where $nT \sim \Gamma(n-1, 1)$ and the probability of rejection is $\alpha = P(n(T - ...
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Approximation of copulas
I'm studying copulas, finished the Introduction to Copulas by Nelsen. I'm interested in the latest/best known/etc approaches for approximating any Copula, or some families of copulas, so would be ...
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Approximating the first moment of $h(x)$ where $x$ ~${\rm log\,normal}(\mu, \sigma)$ [closed]
What is the best way to approximate $E(h(X))$, where $X$ ~ Lognomal($\mu, \sigma$)?
So far, I can think of Monte Carlo Methods and Gaussian Hermite quadrature as below:
\begin{align}
E(h(X)) &= ...
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Compute Mean of Normal Distribution where x% of Values are over y
I am looking for a way to determine the mean of a normal distribution (with given variance), where e.g. $z = 0,37 = 37\% $ of values should be above a certain value $a$ (e.g. 0,2)?
My first idea was ...
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How state aggregation in reinforcement learning works?
I am watching Prediction with linear approximation video course in the RL class by prof. Sutton. He presented state aggregation approach on a random walk problem. It seems that this approach just ...
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The Universal Approximation Theorem vs. The No Free Lunch Theorem: What's the caveat?
The universal approximation theorem:
A neural network with 3 layers and suitably chosen activation functions can any approximate continuous function on compact subsets of $R^n$.
The no free lunch ...
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Approximating the error of maximum likelihood estimation
I have a log likelihood function of a model and I want to find $\mu$ and $\sigma^2$ which maximize the log likelihood. Since the log lik function is quite complex, I decided to use Nelder-Mead ...
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