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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Nonlinear regression with derivative dependence

I am trying to perform a functional approximation on some experimental data. I have a set of measurements of a d-dimensional state $\mathrm x \in \mathbb{R}^d$, together with velocities $\dot{\mathrm ...
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A problem about making an approximation to the integral over parameters -- eq (3.70) of Bishop's Pattern Recognition and Machine Learning

The problem comes from the paragraph containing equation (3.70) at the bottom of page 162 of Bishop's "Pattern Recognition and Machine Learning" which talks about an approximation to the ...
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Best approximation for the size of a test

Let $X \sim \mathrm{Bernoulli}(\vartheta)$ for some unknown $\vartheta \in (0,1)$, and let $(X_1, …, X_n)$ be a moderately large IID sample for $X$. Let $\vartheta_0 \in (0,1)$. I want to test $H_0 \...
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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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How many components of a gaussian mixtures do I need to match moments up to the $r$-th order?

Suppose I have a ($k$-dimensional) random variable $X \sim D$ and I want to find a Gaussian Mixture $GM \sim \sum_{i=1}^C \pi_i \mathcal{N}(\mu_i, \Sigma_i)$ such that the moments of order $r'$, for $...
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Asymptotic Distribution of Likelihood Ratio under Nonlinear Hypothesis

Suppose we are testing $\mathbf h(\boldsymbol\theta) = \mathbf 0$ versus $\mathbf h(\boldsymbol\theta) \neq \mathbf 0$ for a vector of parameters $\boldsymbol\theta \in \boldsymbol\Theta\subset \...
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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 ...
2 votes
2 answers
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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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4 votes
1 answer
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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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Is there a closed form approximation for the composition of the Gamma CDF with the inverse Normal CDF?

Given $k$, $\theta$ fixed shape and scale parameters for some Gamma distribution which has a CDF $F$. Let $G^{-1}$ be the inverse CDF of the standard Normal distribution. Consider the composition $H(x)...
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Approximation of a polynomial via histogram

Note: I originally tried to pose this question generally, without discussing the specific type of stochastic process. I hope that this can still be an interesting question generally. Assume that we ...
3 votes
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How to solve negative binomial regression? [closed]

I want to estimate negative binomial regression for from scratch i.e. I want to write a script that will maximize maximum likelihood obtaining optimal parameters. To do so we can calculate derivatives ...
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1 answer
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Approximate Posterior Predictive Quantiles with Numerical Methods

I have a posterior function which is easy to approximate using numerical methods (the posterior has only 2 parameters, and is approximately Gaussian because of the large sample). However, I need to ...
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Unscented Transform - Combination of multiple Sets of Sigma Points?

Given an initial state distribution $x \sim N(m_x, S_x) \in R^{n_x}$ and transition function $y = f(x)$ one can use the unscented transform to approximate the distribution $p(y) \approx N(m_y, S_y)$ ...
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Distribution of sum of independent but not i.i.d. lognormal variables?

I am trying to find the distribution of the following variable Z: $X_i$ are each independent with Lognormal distribution ($\mu_i, \sigma^2_i$), $X_i \in L^2$ forall $\forall i$ Z = $\sum_i cX_i$ where ...
1 vote
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approximate fisher information for intractable likelihoods

Suppose I have a data set $X_1, \ldots, X_n$, and from that I compute a statistic $T(X_1, \ldots, X_n) := T$. I want to assess how reactive/sensitive this calculation is to changes in parameter values....
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How to approximate the expression to $\sum x_i$

How to approximate the expression on the left hand side to $\sum_{i=1}^Nx_i$ as $n\to \infty$ $$ \frac{\sum\limits_{i=1}^{N}x_i^2}{n-2\frac{\sum\limits_{i=1}^{N}x_i}{N}} \left(\sqrt{1+\frac{Nn\left(...
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If $X$ and $Y$ are uncorrelated random variables, then under what condition is $E[X \mid Y] \approx E[X]?$

Suppose $X$ and $Y$ are real random variables that are uncorrelated. Now, uncorrelated does not imply independence, so $E[X \mid Y] \ne E[X]$. However, can they be said to be approximately equal? If ...
1 vote
1 answer
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Normal approximation to Bernoulli variable

I'm looking for a normal approximation for a Bernoulli variable (so I can later sum multiple correlated approximated variables) The trivial approximation is taking the mean and variance of the ...
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Correlation of Financial Returns using Period-End vs. Period-Average Values

I have two time series of financial returns for assets $A$ and $B$ defined below for $n$ periods. The return $a_i$ is the percent growth in the asset price of $A$ using period-end values for $i-1$ and ...
1 vote
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How to prove that the two equations for the slope of the line of best fit are equivalent? [duplicate]

I was reading this great article on deriving the equation for the line of best fit (https://www.neelocean.com/simple-ols-estimators/), and got confused when I came across: Rearranging: $$\hat \beta = \...
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How do I approximate a multivariable polynomial equation using Neural Networks?

I've been trying to experiment and test the extents to which a neural network works. I was only able to make something with broad categorical variables function in an acceptable amount of time and in ...
1 vote
1 answer
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Can I use the delta method with a function that depends on n to approximate the distribution of a function of the sum of iid random variables?

Let $X_1, X_2,...$ be i.i.d. random variables with finite mean $\mu$ and finite variance $\sigma^2$. From the Central Limit Theorem, we know that $\sqrt{n}(\bar{X_n}-\mu)$ tends in distribution to $N(...
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Can a neural net approximate any conditional density asymptotically?

Assume that the conditional density of $ y \vert x $ is a Beta distribution for all values of x. Can a Beta distribution with parameters computed by a neural net, i.e. Beta($\hat{\alpha}$, $\hat{\beta}...
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1 vote
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Why does non-parametric approach break down when the joint distribution is estimated by a finite data sample?

I am currently reading the paper on Gradient Boosting Machines - J. H. Friedman, “Greedy function approximation: A gradient boosting machine,” Ann. Stat., vol. 29, no. 5, pp. 1189–1232, 2001, doi: 10....
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Predicting Repurchase Curves next value based on usual functional form

Some definitions first: Acquired customers: Customers placing an order for their first time. Cohort: Group of customers that have been acquired during the same time period. Repurchase: An order placed ...
2 votes
1 answer
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CLT theorem and Berry–Esseen bounds for this special case of sampling

Consider a finite set $S=\{s_1,s_2,..s_n\}$, where $a \leq s_i\leq b$ are integers. Each element in $S$ can be chosen to a subset $S'$ in probability $p$. We consider $n$ to be very large. My question:...
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What are the variations of Expectation Maximization?

To explain my question better, I will use this analogy: In the case of the Gradient-Descent method, we have multiple variations/expansions for the main algorithm, like stochastic gradient descent (SGD)...
2 votes
1 answer
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Derivation of confidence interval of incidence rate ratio

I am trying to understand the confidence interval equation for a Incidence Rate Ratio (IRR) given several places: $ 95\text{% CL(IRR)} = \exp(\log(\text{IRR}) \pm 1.96\times \text{SE(log(IRR))})$, ...
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Nonlinear Iterative Partial Least Squares algorithm can calculate accurately all Principal Components?

I wanted to demonstrate a small example in order to understand better the $\textbf{Nonlinear Iterative Partial Least}$ $\textbf{Squares algorithm}$. My goal is to calculate all the Principal ...
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