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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Using continuity correction for normal approximation or not?

Below is a question on a recent actuarial exam, Exam 3L of the CAS. I didn't know whether or not to use the continuity correction when using the normal approximation to do hypothesis testing ...
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Efficient MCMC using the normal approximation of the posterior

I can usually quickly get the normal approximation of the posterior distribution, but I sometimes struggle with setting up an efficient MCMC of the same model. Can I somehow use the results of the ...
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Analytical approximation of probability of one beta-distributed var being greater than another?

The answer to What is the probability P(X > Y) given X ~ Be(a1, b1), and Y ~ Be(a2, b2), and X and Y are independent? provides an analytical solution for this, but is there a less computationally ...
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Fast approximation to inverse Beta CDF

I am looking for a fast approximation to the inverse CDF of the Beta distribution. The approximation need not be precise, but more stress is on simplicity (I'm thinking Taylor expansion of the first 1 ...
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1answer
395 views

In a Monte Carlo approximation of a product of expectations, can the same samples be used for both expectations?

I am trying to approximate a product of expectations: $\operatorname{E}[f(x)]\operatorname{E}[g(x)]=\sum_x P(x) f(x) \sum_x P(x) g(x)$ with $N$ Monte Carlo samples $(x_1,x_2,...,x_N)$ from $P(x)$: $...
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172 views

Approximate distribution of normal squared

I am studying for a test, one section of which will cover the delta method. This problem came from that section: Let $X\sim N(\mu,n^{-1})$. Find an approximate distribution of $X^2$. (It also asks ...
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The distribution of a linear combinations of poisson random variables [duplicate]

I would like to know what the distribution is of linear combinations of Poisson random variables. I know that a linear combinations of Poisson random variables is not always a Poisson random variable,...
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112 views

The product distribution: how fast does dissimilarity increase as a function of number of samples?

If $\mathcal{D}$ is a distribution, let $\mathcal{D}^n$ denote the $n$-fold Cartesian product of $\mathcal{D}$. In other words, $\mathcal{D}^n$ is the distribution of $n$-tuples $(x_1,\dots,x_n)$ ...
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Generating non-zero binomial probabilities (n,k ) with small p and large n - k

I am trying to generate binomial probabilities (in R) as follows: ${N \choose{k}} p^{k} (1-p)^{(n-k)}$ My problem is given $p \approx 0.03$, and $N =400$, $k>270$, I get the probability equal to $...
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Normal approximation to the binomial distribution

I am having trouble getting to the bottom of this concept for two types of questions (hw is already passed, but I have a test this week and would like to do better). Hopefully someone can help me get ...
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Why bother with low rank approximations?

If you have a matrix with n rows and m columns, you can use SVD or other methods to calculate a low-rank approximation of the given matrix. However, the low rank approximation will still have n rows ...
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What is the normal approximation of the multinomial distribution?

If there are multiple possible approximations, I'm looking for the most basic one.
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How to go about selecting an algorithm for approximate Bayesian inference

I am wondering if there are any good rules of thumb for how to go about selecting an approximate inference algorithm for a problem/model (specifically when exact inference is intractable)? When you ...
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Approximate multidimensional sequences

I'm trying to find some method for approximating sequences. I found Keogh's SAX symbolic approximation but it works for real valued time series and I have a database of about 3000 sequences, each one ...
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1answer
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Error in normal approximation to a uniform sum distribution

One naive method for approximating a normal distribution is to add together perhaps $100$ IID random variables uniformly distributed on $[0,1]$, then recenter and rescale, relying on the Central Limit ...
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What is a “polynomially bounded” function, and why is this a requirement of the The Delta Method?

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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1answer
137 views

Approximation in a equality

Let $X_1,X_2$ be independent distributed with cdfs $F_1(x),F_2(x)$, so that $$\overline F_i(x) := 1 - F_i(x) = x^{-\alpha}L_i(x),\ \alpha \geq 0 \>,$$ where $L_i(x)$ is a slowly varying function, ...
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Good approximations for CDFs

As you probably know, the CDFs for many widely-used distributions are difficult, if not impossible, to express in closed form without the use of special functions The normal distribution uses $\...
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Combining Deterministic and Random Unbiased Estimators

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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Are machine learning techniques “approximation algorithms”?

Recently there was a ML-like question over on cstheory stackexchange, and I posted an answer recommending Powell's method, gradient descent, genetic algorithms, or other "approximation algorithms". In ...
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1answer
859 views

Reference for generalized linear mixed models using Laplace approximation

I'm fitting a generalized linear mixed model in R using the Laplace approximation. I'm looking for a reference for the Laplace approximation used for that, or a reference regarding the comparison ...
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1answer
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One sided Chebyshev inequality for higher moment

Is there an analogue to the higher moment Chebyshev's inequalities in the one sided case? The Chebyshev-Cantelli inequality only seem to work for the variance, whereas Chebyshevs' inequality can ...
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How to compute the probability associated with absurdly large Z-scores?

Software packages for network motif detection can return enormously high Z-scores (the highest I've seen is 600,000+, but Z-scores of more than 100 are quite common). I plan to show that these Z-...
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How to calculate tridiagonal approximate covariance matrix, for fast decorrelation?

Given a data matrix $X$ of say 1000000 observations $\times$ 100 features, is there a fast way to build a tridiagonal approximation $A \approx cov(X)$ ? Then one could factor $A = L L^T$, $L$ all 0 ...
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Probability for finding a double-as-likely event

Repeating an experiment with $n$ possible outcomes $t$ times independently, where all but one outcomes have probability $\frac{1}{n+1}$ and the other outcome has the double probability $\frac{2}{n+1}$,...
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Approximate order statistics for normal random variables

Are there well known formulas for the order statistics of certain random distributions? Particularly the first and last order statistics of a normal random variable, but a more general answer would ...
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Are there any available implementations of density or conditional density tree learning?

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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3answers
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Evaluate definite interval of normal distribution

I know that an easy to handle formula for the CDF of a normal distribution is somewhat missing, due to the complicated error function in it. However, I wonder if there is a a nice formula for $N(c_{-}...
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Justifying normal approximation of experimental data

Background: In connection with the question here I came upon a more interesting question. I believe the question is large and distinct enough to have it's own thread. Of course I might be mistaken, ...
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How do extreme values scale with sample size?

Assume I have a random vector $X = \{x_1, x_2, ..., x_N\}$, composed of i.i.d. binomially distributed values. If it would simplify the problem substantially, we can approximate them as normally ...
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274 views

The range of non-identically distributed binomial variables

Lets say we have three independent variables: $$\eqalign{ X_{1}\sim &B(n,\frac{1}{2}+\beta) \cr X_{2}\sim &B(n,\frac{1}{2}) \cr X_{3}\sim &B(n,\frac{1}{2}-\beta). }$$ I'm looking for ...
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1answer
470 views

Central Limit Theorem Tails

I'm trying to compute some p-values for samples from a distribution of sums of ~1000 random variables. The exact distribution of these random variables isn't known, but I have empirical estimates that ...
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What are the factors that cause the posterior distributions to be intractable?

In Bayesian statistics, it is often mentioned that the posterior distribution is intractable and thus approximate inference must be applied. What are the factors that cause this intractability?
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Approximating $Pr[n \leq X \leq m]$ for a discrete distribution

What's the best way to approximate $Pr[n \leq X \leq m]$ for two given integers $m,n$ when you know the mean $\mu$, variance $\sigma^2$, skewness $\gamma_1$ and excess kurtosis $\gamma_2$ of a ...
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563 views

Bayes' Theorem and Agresti-Coull: Will it blend?

I'd like to use Bayes' Theorem on data obtained through a small random sample, and I want to use Agresti-Coull (or any other alternative technique) to know how big the uncertainty is. Here is Bayes' ...
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588 views

Issues using Hermite approximation to bivariate Gaussian in R

In "On Gaussian-like Densities of Order Greater than Two" (Willett, P. Thomas, J. B., 1987), section II, the author state: $\mathcal{N}(x,y,\rho)=\phi(x)\phi(y)\sum_{n=0}^{\infty}\rho^nH_n(x)H_n(y)$ ...
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769 views

Can we approximate this function by a polynomial?

For the more mathematically minded, we have $x \in \mathbb{R}^2$ and the function $h(x)$ defined as: $h(x)=\alpha_1x_1^2+\alpha_2x_2^2+\alpha_3x_1+\alpha_4x_2+\alpha_5x_1x_2+\alpha_6$ and the ...