# 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 ...
395 views

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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.
183 views

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