Questions tagged [importance-sampling]

Importance sampling is a variance reduction technique to approximate integrals/expectations which are not directly computable.

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Effective samples size for quantile estimates with weighted samples

I've been reading the paper Rethinking the Effective Sample Size. In it, the authors provide an estimator for effective sample size given weighted samples (eq. 5.1 in the paper) which specifically ...
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Simplying Bayes Theorem expression: SIS particle filter posteriori

In the book Beyond the Kalman Filter: Particle Filters for Tracking Applications on page 39 the weight update equation for the particle filter is derived. The derivations begins by introducing the ...
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Is there any proof of global convergence for 1D convex numeric optimization using cross entropy method?

Suppose we have the following 1D numeric optimization problem: $min_{x} f(x)$ given $0< x \le x_{MAX}$ where $f(x)$ is a convex function. And I want to apply the cross-entropy method to optimize ...
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Particle filter: Evaluating Optimal importance density

NOTE I posted this in the math stack exchange but I realized this may be the more appropriate place, old post here. I'm not sure if I should delete one of them so I just linked them in both? I am ...
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Estimate default by importance sampling (using R)

I want to use Importance sampling to estimate probability of default of an insurance company within the next $t$ years. The company starts with capital $C$ at $t=0$. Each year it gains $p > 0$ ...
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Stratified sampling to generate random numbers (eg. for Monte-Carlo applications)

I am using a Monte-Carlo method to compute a value of interest $y$ from some input parameters $x_{i}$, that I use to draw statistical sets from simple distribution laws. In my case, for a single Monte-...
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assessing the stability of importance (sampling) weights

I have read that when importance weights are used, the stability (variability) of the weights should be assessed (Levine and Casella, 2001) -- however, I wonder how this might be accomplished. For ...
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Can importance sampling be used as an actual sampling mechanism?

This question is a duplicate of How can we use importance sampling for drawing posterior distribution samples? , but that question seems to lack additional detail and goes unanswered (for more than 2 ...
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Evaluate an integral using importance sampling

Estimate $\int^{1}_{0}e^{x} dx$ using importance sampling. Should I use beta distribution as proposal distribution and uniform distribution as target ?
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Any papers or unpublished work on relationship of downsample/upweighting and importance sampling in the context of training imbalanced classes

I understood that the general Q of training highly imbalanced data has been asked and answered many times. I have skimmed through the other threads and felt fairly confident my specific question/point ...
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Evaluating Likelihood in Bootstrap Particle Filter

I am currently struggling with an attempt to apply a bootstrap particle filter to a linear, Gaussian state-space model $$s_t=A\,s_{t-1}+B\,\nu_t\qquad\text{( transition equation )}$$ $$\qquad z_t=C\,...
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Bootstrap Particle Filter (Gordon, Salmond, Smith, 2003) - Importance Weights

So, my endeavor to apply the is just for my own edificationI am currently struggling with an attempt to apply a bootstrap particle filter (Gordon, Salmond, Smith, 2003) to a linear, Gaussian state-...
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Why does weighted importance sampling work?

I am reading Sutton & Barto. I am trying to understand this passage from chapter 5: An important alternative is weighted importance sampling, which uses a weighted average, defined as $$ V (...
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38 views

Importance Sampling derivation

I'm learning about importance sampling from p139 of this book which has the following derviation: What I am confused about is the second step in the derivation, though the rest makes sense to me. I ...
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39 views

Generating samples from target distribution after importance sampling

So I'm trying to understand importance sampling. So far I have the algorithm implemented like this: ...
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Multiple Importance Sampling and Metropolis-Hastings on extended state space

Let $(E,\mathcal E,\lambda),(E',\mathcal E',\lambda')$ be measure spaces $k\in\mathbb N$ $p,q_1,\ldots,q_k:E\to(0,\infty)$ be probability densities on $(E,\mathcal E,\lambda)$ $w_1,\ldots,w_k:E\to[0,...
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Is there a reason why we should run the Metorpolis-Hastings algorithm with a target density approximating the density we're actually after?

Let $(E,\mathcal E,\lambda)$ be a measure space, $p:E\to[0,\infty)$ be $\mathcal E$-measurable with $$c:=\int p\:{\rm d}\lambda$$ and $$\mu:=\underbrace{\frac1cp}_{=:\:\tilde p}\lambda$$ denote the ...
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Test for significant difference between two probability distributions sampled from a model

Let's say I use importance sampling to sample conditional probability distributions for variable X (categorical variable with 3 levels) 50 times, for example, P(X|Y=1) = { (0.11, 0.21, 0.68), (0.09, ...
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Central limit theorem in a multidimensional sphere

In the book of David MacKay there's a chapter on Monte Carlo methods (http://www.inference.org.uk/mackay/erice.pdf) where, when discussing the importance sampling, it gives this example: Suppose a ...
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Importance sampling estimation of power function

Problem Suppose we are given $\text{Poisson}(\theta)$ model, and the null hypothesis is as follows: $$ H_0 : \theta = 0.1 \ \ \text{vs}. \ \ H_1 : \theta < 0.1 $$ Suppose we take sample of $n=...
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Literature on design of importance sampling distribution using MLE or point-estimates of highest modes

Suppose I have many distributions $p_i(\theta)$ I wish to take expectations over $$\mathbb{E}_{p_i}[\mathbf{f}_i(\theta)]$$ where the $\mathbf{f}_i$ are vector-valued. In my problem the $p_i$ share ...
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How to evaluate double Integral with importance sampling

I am trying to recreate the Bayesian Hierarchical Clustering algorithm using Python. The example in section two requires evaluating the following double integral (univariate case): \begin{align} p(...
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Variance of Monte Carlo integration with importance sampling

I am following these lecture slides on Monte Carlo integration with importance sampling. I am just implementing a very simple example: $\int_{0}^{1} e^{x}dx$. For the importance sampling version, I ...
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Importance sampling with partly deterministic samples

I am trying to estimate an expectation with a certain set of unlikely, but important (for the value of the expectation) events. In particular, let’s say I have a (normal) distribution p(x). Now, I ...
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importance sampling from posterior distribution in R

Today I read that Importance Sampling can be used to draw posterior distribution samples just like Rejection Sampling. However, my understanding of Importance Sampling is that its main purpose is to ...
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importance sampling for conditional probability

Stats experts, I am trying to estimate $G(x)$ for $x$ in a very small subset $A$ of a finite space $U$, i.e $E(G(x | x \in A))$. Determining if $x$ in $A$ has a cost. So it is expensive to uniformly ...
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227 views

in reinforcement learning off policy mc may not work

I noticed off-policy mc prediction(or control) will not work, as being descripted by boxed algorithm in page 110 of the book "reinforcement learning an introduction". The weight W should before C's(...
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63 views

Generalization performance in Bayesian errors-in-covariates model

I'm working on a model with this basic structure: The square nodes are data, and the round nodes are parameters and/or latent variables. The left plate represents the "training observations" we ...
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Would an “importance Gibbs” sampling method work?

I suspect this is a fairly unusual and exploratory question, so please bear with me. I am wondering if one could apply the idea of importance sampling to Gibbs sampling. Here's what I mean: in Gibbs ...
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28 views

importance sampling and exponential moving average

Lets say i have got a random variable $X$ with samples $x_t\sim X$ and density $p_X(x)$ and want to compute its mean via a moving average $ \mu_{t+1}=(1-c)\mu_t + c x_t$ Assume, I can not observe $X$...
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Flatten the target density in the Metropolis-Hastings algorithm

Let $(E,\mathcal E,\mu)$ be a measure space $F$ be a $\mathbb R$-Banach space $f\in\mathcal L^1(\mu;F)$ $f^\ast:E\to[0,\infty)$ be $\mathcal E$-measurable with $$b:=\int f^\ast\:{\rm d}\mu\in(0,\...
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importance sampling strategies

I am trying to approximate the expectation of the "complete-data likelihood" with respect to the distribution of some missing data, and I am having some trouble. This expectation can be written as $$ ...
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59 views

What's done with the expectations in this proof?

This is a proof of the per-decision importance sampling (theorem 1) from the appendix of: https://www.google.co.uk/url?sa=t&source=web&rct=j&url=http://scholarworks.umass.edu/cgi/...
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A doubt on the formula for updating the weights in Sequential Importance Sampling in a State-Space model

Let $x_{0:t}^{(i)}$ be the states from time $0$ to $t$ from sample $i$. Similarly for the observations $y_{1:t}$. The normalized weights are updated according to Where does the term $p(y_t|x_t^{(i)})...
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Conditional expectation on an estimator for defensive sampling

In Introducing Monte Carlo Methods, by Robert and Casella, we have How do we derive the second equality? Shouldn't it be $$E\left[\frac{f(X_i)}{g_{Y_i}(X_i)}|X_i\right]=\frac{f(X_i)}{g_1(X_i)}\rho+...
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SIR explanation in Robert and Casella Intro to Monte Carlo Methods - How to do this derivation?

Why is it an exact simulation from $f$, and not only an approximation? I get $\begin{split} P(X^*\in A) & = \sum_i^n P(X^*\in A , X^* = X_i)=\sum_i^n P(X^*\in A | X^* = X_i)P(X^* = X_i) \\ & ...
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Looking for a recursive formula for asymptotic variance of importance sampling estimator (self-normalized)

Looking for a recursive formula to approximate variance of importance sampling estimator $Var_q\big[\delta_{IS}\big]\approx\sum_{i=1}^n\tilde w(X_i)^2\big[h(X_i)-\delta_{IS}\big]^2$. This is an ...
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654 views

Optimal proposal for self-normalized importance sampling

Consider a function $f: \mathcal X \to \mathbb R$ and a probability distribution $p$ with the support on $\mathcal X$ which we can evaluate up to a normalizing constant, i.e. we can only evaluate $\...
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Why does Off-Policy Monte Carlo Control only learn from the “Tails of Episodes”?

I was reading through section 5.7 of the second edition of Sutton and Barto's "Reinforcement Learning: An Introduction" when I came across this passage: where the "method" that the author is ...
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Derivation of squared coefficient of variation (Importance Sampling)

Text: Computational Statistics 2E by Givens and Hoetings Section: 6.3.2.3 Weight Degeneracy, Rejuvenation, and Effective Sample Size I am having trouble following another result in the text. Below ...
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Stuck on part of proof that a random variable $\bf X$ drawn with the SIR algorithm has distribution that converges to $f$ as $m \rightarrow \infty$

Text: Computational Statistics by Givens and Hoeting Section 6.3.1: Sampling Importance Resampling Algorithm The authors provide a proof that random variable $\bf X$ drawn with the SIR algorithm has ...
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Hyper-parameters which minimize the variance of transformed multi-variate Guassian variable

Let $k < p$ be positive integers and $g: \mathbb R^k \rightarrow \mathbb R^p$ be a smooth Lipschitz continuous function. Let $y_1,\ldots, y_N \in \mathbb R^p$ and $a = (a_1,\ldots,a_N) \in \mathbb ...
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how to sample data for regression that is the most informative?

Background I have a unknown function $$f(x_1, x_2)$$ But I have access to evaluate this function finite $L$ times, $$y_j = f(x_1^j, x_2^j), j=1,\ldots,L $$ Then I have a model $\hat{f}$ which I ...
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887 views

Understanding Sequential Importance Sampling and Particle Filtering

I am struggling with SIS for particle filtering in the following aspect: In particle filtering (as per this book), the objective is to estimate the full posterior $p( x_{0:k} \mid y_{1:k} )$ rather ...
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Low-variance estimate for the mean of the sotfmax transformation of a variable

Consider a set of infintiely-differentiable convex functions real-valued functions $f_i: \mathcal X \rightarrow \mathbb R$, where $i$ varies from $1$ to $m$, and suppose we know all the moments of $...
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how to prove that estimator converges to posterior distribution

Let $X_1, X_2, \dots X_n \sim f(x|\theta)$, $\theta$ has prior $\pi$. Generate $\theta_1, ... \theta_m$ from $\pi$, calculate $q_i = L(\theta_i|\textbf{x}) / \sum_j L(\theta_j|\textbf{x})$ where $L(\...
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Off-policy importance sampling for TD(0)

Consider the off-policy value update $V(s) \leftarrow V(s) + \alpha\frac{\pi(a\mid s)}{b(a\mid s)}[r_t+\gamma V(s') - V(s)]$ Where $\pi$ is the target policy (from which greedy actions are determined)...
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Monte Carlo - Importance sampling using normal distribution as sampling distribution

Suppose that I want to approximate an integral over finite range, say for example 0 to 10 using the Monte Carlo method. Can I choose a normal distribution as the sampling distribution even though the ...
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About the variance of a weighted sum

What can be said about the variance of the following quantity $$ \frac{1}{n}\sum_{i=1}^n \left(\frac{f(x_i)}{\sum_{j=1}^nf(x_j)}-b\right)g(x_i) ? $$ Here, $b \in [0, 1]$, and the $x_i$s are i.i.d ...
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ABC with non-uniform prior

I had asked some similar questions in the past, but I never got either the answers or the discussion I was hopping for. So I will rephrase the problem to see if I can understand it myself. I'm trying ...