Questions tagged [numerical-integration]
A class of algorithms to approximate definite integrals.
175 questions
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Using Monte Carlo algorithm to evaluate marginal density
Let $X, Y, Z$ be three random variables with joint p.d.f. $p(x,y,z)$.
Suppose that $z$ affects $y$ and $y$ affects $x$ independent of $z$, i.e. $p(x|y)= p(x|y,z)$,
and that $z$ has a known marginal p....
user398751
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Integrate an unknown, integrable function numerically with Metropolis-Hastings or other MCMC
I'm looking into evaluating an integral - $\int_{\Omega}f(x)\mathrm{d}x$ - for a colocation method and I don't want to set up a grid and integrate over some mesh, because it would be otherwise mesh-...
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How to numerically/using simulations evaluate the efficiency of an estimator when the estimator does not have a closed form?
I have a joint estimator of two parameters $x$ and $y$, i.e., $\hat{z}=h(\hat{x},\hat{y})$. The observed data is $u$ and $v$, and we have joint PDF of $x$ and $y$ given $u$ and $v$
$f(x,y|u,v)$. I ...
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Compute a truncated binomial expectation accurately
I need to compute the following expectation over a binomial random variable
$$E_{S \sim \rm{Binom}(\frac{k}{M}; N)} \left[\left(\frac{k-1}{k}\right)^S 1_{\{S \leq k\}}\right] = \sum_{s=0}^k \binom{N}{...
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Efficient Methods for Approximating High-Dimensional Integrals with Gaussian-Like Factors
I'm seeking a computationally efficient method to approximately evaluate high-dimensional integrals of the form:
$$\int f(\textbf{x}) \prod_i g_i(x_i) \, d\textbf{x}$$
where $f(\mathbf{x}) = (\mathbf{...
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Subtraction of Monte Carlo integrals - Catastrophic cancellation
I am attempting to estimate a quantity $Q$ which is given by the difference between two functions of Monte Carlo integrals over some set of points $\{x_i\}_{i=1}^N$, call the estimator $\hat{Q}$:
$$ \...
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How to compute the expected value of a function of a random variable given its log-density function? [closed]
Given a log-density function $\mathcal{L}f_{X}(x)$ of an 1d continuous random variable $X \in \mathcal{L}^{\infty}$ and an 1d polynomial function $h: \mathcal{I}(X) \to \mathbb{R}$, the expected value ...
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Can Fisher Information Matrix be calculated numerically through finite differentiation?
In GLMs (generalized linear models), the negative of the Fisher information matrix takes the form of a cross product between covariates $\mathbf{X}$ and a diagonal matrix:
$$
\mathbf{X}^T \mathbf{D} \...
- 607
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$E(XY)$ for a truncated bivariate normal
If $(X, Y)$ follows a bivariate Gaussian distribution with mean ${\bf \mu}$ and covariance ${\bf \Sigma}$ with truncation bounds $(a_x, b_x, a_y, b_y)$, can we compute $E(XY)$ in closed form? If not, ...
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Why use integrals when building exponential composite functions?
I recently read this paper, which describes a generalisation for statistical exponential decay models used in ecology. Essentially, the parameter $k$ of the exponential decay function $f(x) = ce^{-kx}$...
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(Bayesian) Quadrature when the density can be factorised with respect to a directed graph
I am very new to (Bayesian) quadrature and am trying to understand if or how we can use additional information to approximate our definite integral.
Specifically, I am interested in the expected value....
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Numerical moments of a multivariate Poisson Log-normal posterior
I have a log-density of the form:
$$P(\mathbf{x}) \propto \exp\left( - \mathbf{b}^{\top} e^{ \mathbf{x} } - \frac{1}{2}\mathbf{x}^{\top}A\mathbf{x} \right)$$
where $A$ is a symmetric positive definite ...
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Numerical quadrature for Pareto distribution
I would like to numerically evaluate an integral of the following type, when evaluating $f(x)$ at any given point is numerically costly:
$$
\int_{x_m}^\infty x^{-\alpha}f(x) \, dx, \quad \alpha >1, ...
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The meaning of probability density functions' product followed by an integration
Scipy's KDE object allows integration of a function multiplied by another KDE object. I assume that this is meant to be used for the estimation of distance between two distributions. As far as I ...
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Numerical Optimization of Marginal Likelihood that Explodes
I have a model with a marginal likelihood of the following form:
$$\mathcal{L}(\theta_1, \theta_2, \theta_3|\{x_{i,j}\}_{i=1, j=1}^{N, M_i})=\prod_{i=1}^{N}\int_{0}^{1} f(p_i;\theta_1) \prod_{j=1}^{...
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