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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Robustness of a model to learnt parameters

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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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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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Gaussian-Hermite quadrature

From Joint Models for Longitudinal and Time-to-Event Data by Dimitris Rizopoulos. \begin{equation} \begin{aligned} E&\{A(\theta, b_i) \mid T_i, \delta_i, y_i; \theta\} = \int A(\theta, b_i)p(b_i ...
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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 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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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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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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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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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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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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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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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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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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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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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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235 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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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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Sum of multivariate lognormals

Is it possible to approximate the sum of multivariate lognormals using Wilkinson approximation? Any reference?
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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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Approximate prediction interval in linear regression

Suppose we have a linear regression model of the following format : $$ y(x) = \beta_0 + \beta_1 x_1+ \beta_2x_2+\beta_3x_3+\epsilon$$ We know that the prediction interval associated with a level $\...
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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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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 $x$ ~${\rm log\,normal}(\mu, \sigma)$

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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1answer
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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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348 views

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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264 views

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 ...
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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 ...