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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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Let $\mathbf{a}_1,\ldots,\mathbf{a}_n \in \mathbb R^d$ and $b_1,\ldots,b_n \in \mathbb R$ be fixed. For $\mathbf{x} \sim \mathcal U([0,1]^d)$ and $j \in \{1,\ldots,n\}$, consider the real variable ...
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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, ...
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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 ...
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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 $$K(x,y)=\...
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In the book Asymptotic theory of statistics and probability by Anirban DasGupta (2008, Springer Science & Business Media) in page 109 Example 8.13 I found the following approximation $$\Phi^{-1}\...
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For bivariate zero-mean normal distribution $P(x_1,x_2)$, the quadrant probability is defined as $P(x_1>0,x_2>0)$ or $P(x_1<0,x_2<0).$ $P(x_1>0,x_2>0) = \frac{1}{4}+\frac{sin^{-1}(\...
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I am working on joint and conditional density trees for approximating clique potentials in Bayesian Belief Networks. A brief introduction to topic is available from this paper in case you'd like to ...
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I am currently studying on Bayesian models, and still new to probability theory. I learned that Gaussian Mixture Model is used to represent the distribution of given population as a weighted sum of ...
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We have a bivariate random variable $(X,Y)$ for which sampling is challenging. If we were to know how to sample from the conditionals $(X|Y)$ and $(Y|X)$, we could get samples from the joint using ...
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I am trying to approximate a set of quantiles from the estimated mean, variance, skewness and kurtosis of a random variable with unknown distribution. I tried to apply the Cornish-Fisher expansion of ...
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Suppose that data X have a Normal distribution with some mean $\mu$ and some variance $\sigma^2$. However, you don't get to see X. Instead, you see $Y = g(X)$ where $g$ is a known function. Assume ...
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When an observation $x$ is generated by $P(x|\theta)$ for a parameter $\theta$ the Bayesian optimal estimator for the value of $\theta$ is $\hat\theta_{BEST}=\mathbb{E}[\theta|x]=\frac{1}{P(x)}\int d\...
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I am trying to compute an expectation $E[f(X;\theta,n)]$ where $\theta$ and $n$ are known parameters. I have an easy-to-compute deterministic function $\tilde{f}(\theta,n)$ that provides an ...
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Consider the following random quadratic equation, $$ x^2 + Z x + Y = 0, $$ where, $$ \begin{gathered} Z \sim \mathcal{N}(\mu_Z,\sigma_Z), \qquad Y \sim \mathcal{N}(\mu_Y,\sigma_Y). \end{gathered} $$ ...
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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 ...
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We am trying to understand the number of points that a neural network of a particular size can interpolate. I think this may be isomorphic to its degree of freedom? We are not interested in whether ...
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Suppose you have a smooth function $f^*:D_1 \times D_2\rightarrow\mathbb{R}$ that you observe with error as $f$ such that $$f(x,y)=f^*(x,y)+\epsilon$$ where $\epsilon$ has zero expectation (you can ...
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I'm developing a logistic regression used for prediction. I have pre-selected, based on prev. literature, 15 candidate predictors (fitting my ~200 events). Now, I want a reduced/more parsimonious ...
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I am exploring ways to reduce the noise of a covariance matrix estimator when the number of variables is greater than the number of observations, i.e. $n > t$. First, I tried using a low rank ...
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There is a ton of literature (see, for example, a highly cited paper by Huang et al. (2006)) on neural networks with random weights (NNRWs), i.e. neural networks whose weights are random except for ...
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Let $T$ be a compact, connected, proper subset of $\mathbb{R}^3:\quad T \subset \mathbb{R}^3$. Further let $\left\{ \boldsymbol{\mu}_i \right\}_{i=1}^n$ be a given finite set of $n$ point in $T$: $$ \...
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I am working with a blackbox prediction model which takes known inputs and outputs a single mean response. I know this model's residuals to be heteroskedastic, but also can assume the error term of ...
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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 ...
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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 ...
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I am trying to approximate a multivariate function $y = f(x_1, ...x_n)$, which I have reason to believe will be well approximated by a classification and regression tree. Some of the variables are ...
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Consider a probability distribution $P(x)$, a set observed samples $S = \{x_1,\cdots, x_n\}$ where $x_i \stackrel{iid}{\sim} P(x)$ for $i \leq n$, and a symmetric function $h(x,y)$. How can one ...
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I was curious whether it is possible to obtain approximation error bounds on continuous densities from approximation error bounds of random variables. To make my question more precise: We consider ...
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For me, the first thing that comes to mind when I come across the term continuity correction is that when $X\sim\mathrm{Bin}(n,p)$, one approximates $\Pr(X\le x)=\Pr(X<x)$ by $\Pr(Y<x+1/2)$ ...
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According to various sources, the variance of ML estimators can be obtained from the Hessian matrix of the likelihood function. If $H$ is the Hessian of the negative log-likelihood function, then $H^{-...
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I am reading a paper "A note on the Delta Method" by Gary Oehlert, JASA, 1992. I am trying to estimate the variance of a function of a random variable, but first I want to understand the limitations ...
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I have been using Taylor expansion to get some feelings and many approximating results in trying to find innovative ideas for my research. And I have seen a lot of approximate equals or asymptotically ...
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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 ...
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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 ...
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I am running a generalized linear mixed effect model with a Poisson distribution to analyse count data. The model has a random effect that takes into account multiple observation obtained by the same ...
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Suppose I have a data set $X_1, \ldots, X_n$, and from that I compute a statistic $T(X_1, \ldots, X_n) := T$. I want to assess how reactive/sensitive this calculation is to changes in parameter values....
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I've been trying to experiment and test the extents to which a neural network works. I was only able to make something with broad categorical variables function in an acceptable amount of time and in ...
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I'd like to draw samples from some "target" probability density function $f(x)$. However, I don't have a way to do that -- instead I just have access to $N$ samples, each drawn from one of $...
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I've encountered a sentence: In information theory, Kullback–Leibler divergence is regarded as a measure of the information lost when probability distribution Q is used to approximate a true ...
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Let $m\geq 1$ be an integer and $F\in \mathbb{R}[x_1, \dots, x_m]$ be a polynomial. I want to approximate $F$ on the unit hypercube $[0, 1]^m$ by a (possibly multilayer) feedforward neural network. ...
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Consider a random uniform distribution of $N$ points in $2D$ space bounded by $[0, 1]$ in both dimensions. Example: If I want to estimate the mean area of their Voronoi cells, I have to obtain the ...
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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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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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In Bayesian statistics, you have a likelihood and a prior, $f(x_1,\ldots,x_n \mid \theta)$ and $\pi(\theta)$ respectively, and you use these to obtain the posterior $\pi(\theta \mid x_1, \ldots, x_n) \...
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I am trying to grapple with the following problem. I have an application that develops empirical distributions. In essence, I end up with a histogram of equally spaced $x$ values, with both a $max$...
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I'm trying to understand the proof of an expression for the asymptotic bias in local polynomial regression of degree $p\ge0$. Specifically, I'm distraught with equation $(3.59)$ on page 102 of this ...
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Are there any theorems which tell us how well AR(p) models are able to approximate any stationary finite time series? If so, what are the relevant results?
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If I fit a model m <- glm/gam/gamm/lme/whatever(y ~ x + z, family = some exponential family) and extract coef(m) vcov(m) ...
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The most common form of linear regression estimates the best values of $\vec{\beta}$ and $\sigma^2$ assuming that data is sampled from a model $y = \vec{\beta} \cdot \vec{x} + \vec{\epsilon}$ where $\...
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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 ...
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I learnt from section 10.3 of statistical inference that both Wald test statistic $\frac{W_n-\theta_0}{S_n}\approx\frac{W_n-\theta_0}{\sqrt{\hat I_n(W_n)}}$ and score test statistic $\frac{S(\theta_0)}...
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