Questions tagged [sequential-monte-carlo]
30 questions
11
votes
1 answer
Rao-Blackwellization of sequential Monte Carlo filters
In the seminal paper "Rao-Blackwellised Particle Filtering for Dynamic Bayesian Networks" by A. Doucet et. al. a sequential monte carlo filter (particle filter) is proposed, which makes use of a linear substructure $x^L_k$ in a markov process $x_k =…
Jakob
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4
votes
2 answers
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 than the marginal posterior $ p( x_k \mid y_{1:k} )…
bonanza
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3
votes
1 answer
SMC Samplers - Optimal Backward Kernel Explanation
In Sequential Monte Carlo Samplers of Del Moral (2006) we see that the optimal backward kernel is
$$
L_{n-1}^{\text{opt}} (x_{n-1} \mid x_n) = \frac{\eta_{n-1}(x_{n-1}) K_n(x_n \mid x_{n-1})}{\eta_n(x_n)}
$$
I am very confused by the notation used…
Physics_Student
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3
votes
1 answer
Help understanding the proof that Monte Carlo methods that the expectation of $R$ samples is the expectation of the function
The above proof I got from UofT's CS412 lecture slides. So I have a few questions regarding this notation that I don't understand
is $x \sim p(\{x^{(r)}\}^R_{r=1})$ supposed to represent $R$ number of samples taken from $p(x)$? Why is it $x \sim$…
user8714896
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3
votes
1 answer
Why is it necessary to perform resampling step in particle filtering (or sequential monte carlo)?
I read the Wikipedia page on particle filter, it says that during 'prediction-updating', the samples from the distribution are weighted by a likelihood that represents the probability of that particle being sampled from the probability density…
zoozoo
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3
votes
1 answer
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) \\
& = \sum_i^n P(X_i\in A)\frac{f(X_i)}{ng(X_i)} \\
& =…
An old man in the sea.
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3
votes
1 answer
Importance weight of conditioned particle in conditional SMC
In a generic particle filter, I understand the importance weights for each particle are calculated as
$w_t^s \propto w_{t-1}^s \frac{p(y_t \mid z_t^s) p(z_t^s \mid z_{t-1}^s)}{q(z_t^s \mid z_{t-1}^s, y_t)}$.
I am confused about the weight of the…
Gonzalo Benegas
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3
votes
0 answers
Soft Question: What background do I need to understand Feynmann Kac Formulae by Pierre Del Moral?
I am attempting to understand Sequential Monte Carlo(SMC) deeply, but with little theoretical background on probability theory and stochastic processes. Usually, the 'statistics' perspective of markov chains and stochastics is well understood by me.…
tintinthong
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2
votes
0 answers
Particle Filter for structural credit risk model
Kwon (2012)* proposes a structural credit risk model where the asset value process and the noise are estimated based on the observed equity prices:
$S$ - equity prices
$V$ - value of the assets
$Z$ - noise which contaminates the equity…
Sandu Ursu
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2
votes
1 answer
sequential Monte Carlo sampler, why the extended space and backward kernel?
Hello cross validated,
I am currently studying sequential Monte Carlo samplers.
My current understanding is as follows:
We are interested in the marginal distribution of some sequence of joint distributions, increasing in dimension. We are…
user
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2
votes
1 answer
Kernel for MCMC moves in sequential monte carlo
I'm trying to understand how to employ MCMC moves in a sequential Monte Carlo procedure for estimating static parameters as in the setting described by Chopin.
He proposes, for example, the usage of a Metropolis-Hastings kernel admitting the…
noosesan
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2
votes
1 answer
Are the Sequential Monte Carlo algorithm invariant to the step at which we resample?
In a usual textual description (according to SMC in Practice book ) of a SMC algorithm for State-Space models, we usually expand the particles according to the distribution from the transition equation, or from a proposal/importance distribution,…
An old man in the sea.
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2
votes
2 answers
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 $p(y_t|x_t^{(i)})$ come from? $\pi$ is the proposal/importance…
An old man in the sea.
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1
vote
0 answers
How does Particle Filters work?
I'm trying to figure out how particle filter works.
Assume that I have selected propability function called $a \sim Gauss(\mu, \sigma)$. We call it proposial (Gaussian) Distribution.
Then we have another function who corresonds to the real world…
Nazi Bhattacharya
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1
vote
0 answers
Calculating the weights in ABC SMC (2 parameters and more)
Im trying to implement ABC SMC for ODE model which has 2 parameters to estimate. I stopped in the step when calculating the weights as it appear in this answer. My question is should I calculate the weights for each parameter separately and then use…
Sarah
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