# 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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### Binomial to Poisson Approximation

So, a little context. The image you see is from the GCE A-LEVEL syllabus where they have defined the conditions for approximating binomial to poisson. But why did they have mention that the ...
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### Bivariate normal CFD approximation using characteristic function

The normal distribution CFD can be approximated using $$F_X (x)=P[X≤x]=\frac{1}{2}-\frac{1}{π} \int^{\infty}_{0}\operatorname{Re}\left[\frac{e^{-iux}\phi_X (u)}{iu}\right]du$$ where the characteristic ...
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### How much error will there be if one models the multivariate hypergeometric distribution using a multinomial distribution?

See the title. I stumbled upon this answer which explains the approximation in more detail. One can approximate the multivariate hypergeometric distribution by using the multinomial distribution. ...
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### Why approximation and estimation errors (of generalisation error decomposition) are called as such?

Approximation error: The degree to which the underlying distribution (D) can be well approximated by the hypothesis class (H) Estimation error: Provided H, the degree to which the underlying ...
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### Can GMM approximate any given probability density function?

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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### How do you define a good approximation for a probability distributions?

We know a series of probability distribution approximations that are considered good as long as some condition holds. A few examples are: Binomial can be approximated by Normal if $np(1-p) > 10$ ...
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### How to prove the mean of a sample approximates well to the mean of the population

Suppose I have a population whose distribution is definitely not normal but both the population and sample size will be large. Is there any way I can prove/ show that the mean of a large enough sample ...
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### Approximate distribution of test statistic for weighted sample mean

Let $$R_{i}(t) \sim \mathcal{N}(\mu_i, \sigma_i^2),$$ denote the one period return distribution for asset $i$, from which we observe the iid samples $\{R_i(t)\}_{t=1}^{n_i}$. The MLE sample mean and ...
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### Modifying Variational Inference to be robust to outliers?

Normally, for variational inference, you have some evidence data $Z$, you have some true distribution $P(X|Z)$, and you have a simpler parameterized distribution $Q(X|\theta)$, and you're trying to ...
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### Best way to approximate head point having only face keypoints

I'm using the BlazeFace model from TensorFlow which only has this few keypoints: I need those keypoints plus a head keypoint, like this one: My question is, which would be the best way to ...
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### LSA and cosine similarity approximation with large matrix

This question is the non-r related version of this one I have a tdm with 16k x 350k dimension, for which I am trying to get the document-document similarity (cosine type). With RSpectra I have found a ...
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### Distribution of the approximation error in Gaussian Process Regression (finite data setting)

I am learning about Gaussian Process Regression. I would like to have some references or results regarding the distribution of the error between a given function, and the posterior obtained in ...
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### Universal approximation of Gaussians

Can gaussian kernels reproduce non continuous L2 integrable functions? ( Do non continuous L2 integrable functions lie in the RKHS constructed by a Gaussian Kernel?) Edit: I think my question is being ...
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### Simplifying the Kullback-Leibler divergence for a sum of distributions

I want to find an approximation of a mixture of probability distributions that minimises the Kullback-Leibler divergence (KLD). I need to verify my result, as it seems suspect. We have a joint ...
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### Universal Approximation Capabilities of Mixture of Weibulls

Can a mixture of $N$ Weibull distributions approximate any continuous density with non-negative support, if $N$ is sufficiently large? (If so, a reference to the proof would be greatly appreciated). (...
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### Is there a way to correct for degrees of freedom when using a generalized linear model with a Poisson distribution featuring random effects?

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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### Conservative coverage probability when an biased estimator is used for the variance

Suppose that $X_n\sim N(\mu, \sigma^2_n)$. Thus, to construct a 95% CI for $\mu$, we can use $X_n\pm 1.96 \sigma_n$. The coverage probability, $P(\mu\in [X_n\pm 1.96 \sigma_n])$, is equal to 95%. ...
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### Confidence interval for multiple samples of ratios of counts (in R)

Data and objective I have count data from two groups, A and B, from across multiple samples. I want to estimate the average ratio of A to B across all samples, along with a confidence interval. Issues ...
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### Mean squared value error only useful for policy evaluation?

The mean squared value error is defined as: $\overline{VE}(\mathbf{w}) \equiv \sum_{s \in \mathcal{S}} \mu (s) \left[v_{\pi}(s)- \hat{v}(s,\mathbf{w})\right]^2$ Just by looking at this metric, it ...
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### What is known, in principle, about the possibility of approximating the random discrepancy between a statistical estimate and its parameter?

The difference between the value of a statistical estimate and its parameter's value is almost never exactly $0$. For example, $r - \rho$, for a unique sample $r$, is likely to be some non-zero ...
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### A nondeterministic covariance-stationary process approximated by an ARMA process

We know that the Wold Decomposition Theorem says that any purely nondeterministic covariance-stationary process, $x = [x_t : t \in \mathbb{Z}]$, can be written as a linear combination of lagged values ...
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### Fast likelihood evaluation for Gaussian distribution with diagonal plus low rank covariance

Let's assume the likelihood $$y \sim\mathcal N_p(0, \Sigma + \Lambda\Lambda^\top)$$ where $\Sigma$ is diagonal and $\Lambda$ is a $p \times d$ matrix with $d \ll p$. What is the fastest way to ...
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### Number of points a one hidden layer neural-network can interpolate

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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### Connection between mean field inference and mean field theory (physics)

In variational inference, the mean-field family of probability distributions is the set of distributions that factors over its terms (i.e. each component is independent of all others). This allows us ...
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