Questions tagged [importance-sampling]
Importance sampling is a variance reduction technique to approximate integrals/expectations which are not directly computable.
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How to obtain moment bound from importance sampling identity?
Let $m(t) =E[X^t].$ The moment bound states that for a > 0, $$P\{ X \geq a \}\leq m(t)a^{-t} \forall t > 0 .$$ How would you prove this result using importance sampling identity?
My answer:
...
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How calculate importance ratio for continuous states?
I am trying to understand importance sampling estimators, in particular for off-policy evaluation in reinforcement learning.
I am working with the definition:
The IS estimator provides an unbiased ...
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Minimizing cross entropy over a restricted domain?
Suppose $f(x;q)$ is the true distribution. The support of the random variable $X$ is $\Omega$. Suppose, I am interested in a particular subset of $\Xi \subset \Omega$. I would like to minimize the ...
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Radon-Nikodym derivative between $\eta$ and its empirical approximation obtained through importance resampling
Suppose I have a probability measure $\eta$ on a measurable space $(\mathsf{X}, \mathcal{X})$ and I have an empirical approximation of it
$$
\eta^N(dx) = \frac{1}{N}\sum_{i=1}^N \delta_{X^{(i)}}(dx)
$$...
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How to solve for an unkown probability distribution within a hierarchical model?
The Problem
Given probability distributions $P(\theta)$ and $P(X)$, and given an inverse function $Y=f^{-1}(X,\theta)$ that returns a unique $Y$. How can one estimate the unkown distribution $P(Y)$ in ...
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Importance sampling for a parameterized family of distributions using a wide distribution from the same family
I'm motivated here by a problem for robust Bayesian analysis. Let $l(Y|X)$ be the likelihood and let $\{p_\xi(X)\}$ be a parameterized family of prior distributions where $\xi$ denotes the ...
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Why do we want to minimise the variance of our importance weights in SIS with respect to the proposal distribution
Is there a clear and precise explanation of why minimising the variance of the weights in SIS with respect to a proposal ensures that the samples generated from the empirical distribution induced by ...
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Why is inverse - sampling inefficient?
One metric used to measure the efficiency of sampling in monte carlo sampling (given a dataset of size $n$) is the effective sample size $N_{eff}$.
*The efficiency of a sampling procedure depends ...
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Importance sampling weights for NN training
Say you're training an NN and have different groups of samples, say number of groups is ngroups.
Each group has a different number of samples, say ...
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In the β-TCVAE paper, can someone help with the derivation (S3) in Appendix C.1?
Paper: Isolating Sources of Disentanglement in VAEs
I follow as far as,
$$\mathbb{E}_{q(z)}[log[q(z)] = \mathbb{E}_{q(z, n)}[\ log\ \mathbb{E}_{n'\sim\ p(n)}[q(z|n')]\ ]$$ Subsequently, I don't ...
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How to calculate the variance of importance sampling estimate
I am given the following Hidden Markov Model:
$X_{k+1} = \alpha X_{k} + b W_{k+1}$
$Y_{k} = cX_{k} + dV_{k}$
Also, $V_{k}$ and $W_{k}$ are independent and iid following $N(0, 1)$
I am required to ...
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Is importance sampling and importance weighting the same thing?
I see that the formula for importance sampling and importance weighting are basically the same Expectation[x * weight]. So, are they the same thing?
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Estimating complicated conditional probabilities with less calculation/computation [closed]
How can we estimate P(A | B & C & D & E) with reasonable accuracy by only calculating something like P(A | B), P(A | C), P(A | D), P(A | E), P(A | B & C), P(A | B & E), P(A | C &...
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Difference in Normalizing constants for Annealed Importance Sampling and Sequential Monte Carlo
I have been looking into Annealed Importance Sampling (AIS, Neal, 2001) and Sequential Monte Carlo (SMC, Del Moral et al., 2006) methods lately. I was wondering where the difference in estimating the ...
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Importance Sampling Variance vs Importance sampling Size
Does the increase in importance sampling size guarantee the decrease in importance sampling variance?
Some context here: I'm trying to use importance sampling instead of equal probability sampling to ...
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Use bootstrap to calculate confidence interval for importance sampling results
For regular equal probability sampling, assume we have n samples: X1, X2, X3, ..., Xn, and we can calculate the point estimation for our estimator F. Then we can use bootstrapping to resample those n ...
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Using normal distribution to approximate t distribution in importance sampling
The question is Exercises 6 and 7 regarding importance sampling on page 273 of Bayesian Data Analysis 3 http://www.stat.columbia.edu/~gelman/book/BDA3.pdf.
Exercise 6 approximate a normal distribution ...
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Why do we need importance sampling?
Let's say we want to calculate the following expectation:
$$
\mathbb{E}_{z\sim p_z(z)}[f(z)]
$$
One issue, is that the samples from $p_z(z)$ could be not very informative:
We see here that $f(z)$ ...
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Train a model subject to max error
I would like to train a neural network by minimizing a loss over samples (as usual), but doing so in a way that the maximum error is bounded.
What options do I have? Some that come to mind are:
...
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Sampling according to a product of a known density and a probability function
Given a known density $p(x)$, I'd like to generate samples according to $q(x) \propto p(x) f(x)$, where $f(x)$ is some probability function, $\forall x f(x) \in [0, 1]$, e.g., a sigmoid function.
One ...
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On the optimal distribution for importance sampling
Let's say $X$ is a rv, $p(x)$ is its pmf.
I want to importance-sample $\mu := \mathbb E[f(X)]$,
for some bounded function $0<f<1$,
using another distribution $q(x)$.
Then what I should do is ...
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Combining importance sampling with enumeration for estimating expected value
I have a Monte Carlo simulation which, given an initial state, does some random stuff and outputs a scalar. Let this output be the random variable $Y$. The simulation takes place on an $K$x$K$ grid, ...
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Particle Filter Derivation based on Forward Algorithm
I have been studying the particle filter, sequential monte carlo methods, and sequential importance sampling.
I am interested in apply the particle filter equations to the standard forward algorithm:
$...
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Measure-Theoretic Importance Sampling: Do we need equivalence of measures?
Let $\pi$ and $\mu$ be the target and proposal measures on $(X, \mathcal{X})$ respectively, with $\pi \ll \mu$. Suppose $\lambda$ is the reference measure on $(X, \mathcal{X})$ and that $\pi\ll \...
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Question about Bayesian Melding?
I am going through Bayesian Melding paper by Poole and Raftery (2000). One of the ideas of the paper is demonstrated by Example 3.5, where there are three uniform distributions considered for $X\sim ...
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An alternative sampling without replacement
Consider a set $X := \{x_1, \ldots, x_n\}$ with corresponding weights $p_1, \ldots, p_n$. Suppose we would like to draw $m < n$ distinct (i.e. unique) elements in a way that the probability of ...
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Importance Sampling: using Target Distribution as Proposal Distribution to approximate normalizing constant
Importance Sampling is a method use to approximate expectations of a test function $\phi$ with respect to $p$ by instead sampling from a proposal distribution $q$
$$
\mathbb{E}_{p}[\phi(x)] = \int \...
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Sampling without replacement while avoiding an element
Let $p$ be a distribution over $N$ objects, pick an object $k$ from $N$, and define $p^*(x)$ to be 0 if $x = k$ and $p(x) / (1 - p(k))$ otherwise.
Suppose I want to sample $n$ items from $p^*$ without ...
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Estimate Ratio of Normalizing Constants from two datasets
Suppose I have a non-negative function $f:\mathbb{R}^N \to [0, +\infty)$ that defines two different (unnormalized) probability densities on two separate subsets $A, B \subset \mathbb{R}^N$ with $A \...
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Efficient sampling to render an expression made up of random variables
Let's say I have a few random variables, like
$
x_1 \sim N(0, 1)\\
x_2 \sim N(2, 1)\\
x_3 \sim U(0, 2)
$
I would now like to render the following distribution, an algebraic expression made up of these ...
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Do Particle Filters actually approximate the posterior distribution?
Im reading a tutorial paper about particle filters (Link) in which it is stated that as the number of samples tends to infinity the approximated posterior density given by
$p(x_k|z_{1:k}) \approx \...
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Improving samples from a distribution
I have a long-standing problem regarding generating samples from a desired distribution, $p(x)$. I know the analytic form of $p(x)$.
I have a mechanism that should draw samples from $p(x)$, and does ...
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Evaluation metrics for an RL model. How to select then?
I trained an RL model adapting the RL batch example (Jupyter Notebook) to the problem I was aiming to solve.
As for the training, everything went well but, even though the RL batch returned several ...
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Importance sampling - computing the mean of unnormalised importance weights
I am completing an assignment for self-study, and am experiencing some confusion over some elementary algebra concerning importance sampling.
The context is as follows:
Given a random distribution $p(...
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Multi-walker MCMC
I just have a brief question regarding Markov Chain Monte Carlo with multiple walkers. I'm currently using this technique to calculate integrals and I'm not 100% on how to combine the statistics of ...
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Consistency of likelihood importance sampling estimator
In a lecture recently our lecturer described a method for approximating the expectation of a function over a posterior distribution using likelihood importance sampling. That is:
$$ \mathbb{E}_{p(x|D)}...
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Using importance sampling for prior sensitivity analysis in Bayesian modeling
I read a section on Bayesian sensitivity analysis in the following book by Carlin and Louis (2009), 'Bayesian Methods for Data Analysis' (3rd ed.), CRC Press.
The context is a sensitivity analysis of ...
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Compute the inverse of a conditional quantile regression output
Brunello et al (2009) show that extended compulsory schooling leads to increased wages respectivly to the individual gender. Their empirical model first uses quantile regression to show the impact of ...
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Sample from a distribution and plot in python
I am trying to understand Particle Filter and Importance Sampling Principle from a UniFreiburg Course and this USNA document on particle filters.
Simultaneously, I am also trying to write a document ...
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Importance sampling and Metropolis MC
I am evaluating numerically integral
$$I(\theta) = \int_{-\infty}^{+\infty} dx_1 dx_2 dx_3 dx_4 \int_0^{+\infty} dy_1 dy_2 dy_3 dy_4 \prod_{k=0}^4\left[w_n(x_k)w_e(y_k)\right]F(x_1, x_2, x_3, x_4, y_1,...
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Ways to sample from a distribution that are more efficient than random
I am trying to sample from a known distribution (somewhat complicated in that a transformed random variable has random noise from a scale mixture of normals added to it and is then back-transformed - ...
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Sampling from a continuous 2 dimensional probability distribution function for importance sampling
I just want to clarify a few points with regarding to sampling from a continuous 2-dimensional probability density function. If I want to sample from this pdf, I could sample from a 1D pdf, $P(x)$, ...
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Particle Filtering: Derivation that mean of weights is the marginal likelihood
I see everywhere the following (for the Bootstrap Filter)
$$
p(y_t \mid y_{1:t-1}) \approx \frac{1}{N} \sum_{i=1}^N W(x_{0:t}^i)
$$
where $W(x_{0:t}^i)$ are the normalized weights defined as
$$W(x_{0:...
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Importance sampling - approximation of an integral in R [closed]
So I am given this integral $$\mathrm{I}=\int_{38}^{\infty} \mathrm{f}(\mathrm{x}) \mathrm{d} \mathrm{x}=\int_{38}^{\infty} \mathrm{e}^{-\mathrm{x}} \mathrm{x}^{2} \mathrm{dx}$$
and i am also given ...
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Importance Sampling - By Hand Calculation and Example
To understand Importance sampling, I did a little experiment as described here. Since I'm getting high deviations when proposal distribution is not uniform, I would like to know if my steps are ...
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Can sampling be difficult even with access to the normalized version of the distribution?
There is a vast wealth of literature on (approximate) sampling from computationally difficult distributions. Generally, the techniques I have seen assume that we only have queries to a proportional ...
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The connection between the expectation in expectation maximization and the importance sampling?
The log-likelihood of the EM algorithm can be expressed as
\begin{align}
\ell(\theta, x) &= \log p(x|\theta) \\
&= \log \sum_z p(x, z|\theta) \\
&= \log \sum_z \frac{q(z|x)}{q(z|x)}p(x,z|...
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Combining importance sampling with optimization - does this yield an unbiased estimate?
I'm wondering if it is OK to combine importance sampling with optimization to choose the parameters for the substitute distribution.
I have a non-negative random variable $X$ on $\mathbb{R}^d$ with ...
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is off-policy Monte Carlo control really off-policy?
I'm reading the "Reinforcement learning: An introduction" by Sutton and Burto (http://incompleteideas.net/book/bookdraft2017nov5.pdf)
The off-policy MC control algorithm puzzles me, please if anyone ...
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Weighted importance sampling (WIS) and Importance sampling (IS)
I am currently reading papers about off-policy evaluation (or counterfactual evaluation) of reinforcement learning policies, including ones about the doubly robust estimator.
As in this paper https://...
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