# Questions tagged [importance-sampling]

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

178 questions
Filter by
Sorted by
Tagged with
34 views

### 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: ...
6 views

### 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 ...
46 views

### 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 ...
19 views

### 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 $$\eta^N(dx) = \frac{1}{N}\sum_{i=1}^N \delta_{X^{(i)}}(dx)$$...
1 vote
38 views

### 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 ...
1 vote
13 views

### 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 ...
18 views

### 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 ...
1 vote
27 views

### 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 ...
20 views

### 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 ...
1 vote
77 views

### 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 ...
1 vote
46 views

### 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 ...
28 views

### 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?
1 vote
33 views

### 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 &...
1 vote
51 views

### 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 ...
299 views

### 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 ...
45 views

### 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 ...
346 views

### 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 ...
2k views

### 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 $f(z)$ ...
122 views

### 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: ...
1 vote
43 views

### 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 ...
1 vote
250 views

### 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 ...
49 views

### 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, ...
84 views

46 views