# 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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### How well can an AR(p) process model any given stationary time series?

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?
223 views

### natural log approximation

I've got an equation that contains x^p - 1 x is any positive number, such as 2, and p is a small positive number close to 0 such as 0.001 For some reason (that ...
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### How to deal with increasing action space in TD learning using linear function approximation

I am working on an application of reinforcement learning in an environment in which the number of possible actions increasing throughout an episode. The first step has only 8 possible actions, which ...
638 views

### Lower Bound on the Total Variation Distance between two Binomials

Let $X= B(n,1/2)$, $Y=B(n,1/2 + \delta)$, for a small $\delta >0$ be two Binomial Distributions. Question 1. I am looking for a lower bound on the Total Variation Distance the two Binomials ...
72 views

### What are some of the common techniques for density estimation?

I'm trying to estimate the probability density function of a real random variable given its iid realizations. What are some of the standard techniques to do this? One method I have heard of is the ...
121 views

### How to calculate or approximate the integral $\int_{-\infty}^{\infty}\sigma(x)(1-\sigma(x))\mathcal{N}(x|0,1)dx$

$\sigma(x)$ is the sigmoid function, that is, $\sigma(x)=\frac{1}{1+e^{-x}}$. And $x$ is from a Normal distribution of 0 mean and 1 variance. Now I'd like to calculate the expectation of the ...
58 views

### Is approximate normality limited to the exponential family?

A GLM looks like $$g(\mu) = X\beta,\ \ \mu = EY_i$$ where $Y_i$ is an exponential family. It is commonly assumed for a decent sample size that $\beta$ is approximately normal with mean $\hat{\beta}$ ...
48 views

### Is vcov the information matrix /n from the theory?

If I fit a model m <- glm/gam/gamm/lme/whatever(y ~ x + z, family = some exponential family) and extract coef(m) vcov(m) ...
37 views

### Simultaneously finding least squares predictors of every feature based on all others

Given set of $d$ datapoints and $f$ features in a $X_{f,d}$ data matrix, I'm trying to fit a least squares predictor of every feature based on every other feature. In other words for every feature $i$...
238 views

### Kahn Pseudo Normal distribution

I read about a so-called "Kahn Pseudo Normal" distribution in a forum post. It says when $U$ is uniformly distributed over $[0,1]$, then $\log(1/U-1)$ is approximately normal distributed, that is ...
177 views

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### Finding the optimal number of hidden nodes and training epochs for function approximation [duplicate]

In an attempt to find the optimal number of hidden nodes and number of training epochs (to obtain optimal performance while not over-fitting) in a single-hidden layer neural network, I generated the ...
2k views

### What is the CDF of the sum of weighted Bernoulli random variables?

Let's say we have a random variable $Y$ defined as the sum of $N$ Bernoulli variables $X_i$, each with a different, success probability $p_i$ and a different (fixed) weight $w_i$. The weights are ...
1k views

### Function approximation using multilayer perceptron (neural network)

I've been asked to solve a problem for a project. I'm working on Python or R. I need to approximate a function with multiplayer perceptron (neural network). The function is: $y= 2\text{cos}(x)+4$ on ...
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### Data approximation with unequal intervals

For some sampled function of real variable, defined on [a, b], how can [a, b] be split into subintervals of unequal width, so that for each interval one can say, "On this interval, function is ...
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### Difference between contrastive divergence and importance sampling

The terms Contrastive Divergence and Importance Sampling are sometimes used interchangeably. I understand that both are used to approximate partition functions (normalization terms for probabilities)...
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### approximate a nonparametric CDF in R

I have two vectors of same length. The first vector is a collection of realization from an unknown random variable. The second vector is the distribution computed at each particular realization. A ...
37 views

### Can the data from two resources be mixed?

There are several sources of temperature-dependent viscosity data, which goes as following: |:--------------------------------| | 273.15 | 320.5 | 330.25 | |---------------------------------| | 1....
406 views

### Quantile approximation using Cornish-Fisher expansion

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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### Regression without known covariates: approximating the likelihood

I would like to perform (linear) regression for an outcome $y$ but do not have access to the covariates $x$. Instead, each record $i$ belongs to one of $K$ groups and I know the distribution of $x_i$ ...
179 views

### Gaussian Exponential Sum Approximation

Let $x_j,\ j\in\{1,\cdots,n\}$ be independent standard Gaussian random variables and $a_j,\ j\in\{1,\cdots,n\}$ be constants. What would a function $f$ and a standard Gaussian random variable $x_0$ ...
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### Moment/cumulant series expansion of probability distribution?

Let $\mu_n = \langle x^n \rangle$ and $\kappa_n$ be the moments and cumulants of a probability density function $p(x)$. Given a finite number of the moments, $\mu_1, \dots, \mu_N$, or equivalently ...
I'm attempting to implement a numerical approximation to assess the frequency of outcomes given $N$ independent random variables. There is a slight twist however, each $N$ has a weight, Each variable ...