# 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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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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### 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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### 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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### 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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### 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 ...
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 ...
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 ...