Questions tagged [sequential-monte-carlo]

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How to update particles' weights and what are some ensemble evaluation metrics for a multivariate particle filter?

I am trying to figure out how to perform a multivariate (standard/bootstrap) particle filter to assimilate observations into a hydrological model. Most of the literature (in my field at least) is ...
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14 views

Sanity check: Ensemble-based sequential state inference with different parameters (Ensemble Kalman Filter)

I have recently been invited to review a publication which employs the Ensemble Kalman Filter (EnKF) for the sequential inference of dynamic state variables $x_{1,...,t}=(x_1,...,x_t)$. In the study, ...
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21 views

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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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 ...
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27 views

Monte Carlo probability approximation vs Histogram

I am trying to learn the sequential Monte Carlo method (particle filter) in data assimilation. In this method, the aim is to approximate the CDF of the target variable having a random sample of the ...
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43 views

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 ...
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41 views

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 $Z$ - ...
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26 views

Use of the inversion method in sequential sampling to “invert” a random walk

Let $M\subseteq\mathbb R^3$ be Borel measurable, $\lambda$ be a $\sigma$-finite measure on $\mathcal B(M)$, $k\in\mathbb N$, $I:=\{0,\ldots,k\}$, $q$ be a probability density on $\left(E^I,{\mathcal E}...
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44 views

ABC SMC: How do weights scale proportionally with number of parameters

Having some problems with the ABC SMC algorithm. I'm trying to implement the methods taken from here: Simulation-based model selection for dynamical systems in systems and population biology How do ...
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42 views

Predictions after SMC

I have a statistical model given by $$ y_t\sim p(y_t|x_t, \theta)\\ x_t\sim p(x_t|x_{t-1},\theta)\\ \theta\sim p(\theta) $$ where $y$ is the only observed component. Using a sequential Monte Carlo ...
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Posterior as prior for correlated parameters [closed]

I want to use the posterior distribution of the model parameters $\theta$ given data in the time frame $[0,t]$ days, $P(\theta|y_{0:t})$; as a prior for the parameters in the time frame $[t+1, t+n]$ ...
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157 views

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 ...
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1answer
86 views

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 ...
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1answer
271 views

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 ...
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84 views

Sequential monte carlo : A simple example

I am attempting to understand how to implement the sequential monte carlo algorithm using this article. Here are the steps that the author proposes: Example problem: Say I have a self moving robot ...
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117 views

Particle filter - expectations

I've recently been implementing some particle filter algorithms and I've realized there is a small detail I might have been doing incorrectly. Unfortunately the descriptions of the algorithms in ...
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1answer
149 views

Sequential monte carlo, resampling

In particle filters when one is doing sequential importance sampling, the quantity of interest that is being approximated is usually a weighted sum: $$\hat x_t = \sum_{i=1}^M \Bigl [f(v^{(i)}_{t}) \...
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1answer
125 views

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 ...
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59 views

Variance of a mixture of Normals with same $\sigma^2_i$

Let $Y\sim \sum^N_{i=1}\omega_iN(m_i,h^2 V)$. The text I'm reading states that $Var(Y)=(1+h^2)V$, when $m_i=\theta_i$, where $\theta_i$ are draws taken from $P(\theta|D)$, and $V=Var(\theta|D)$ I ...
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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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71 views

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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177 views

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, ...
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1k 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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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 ...
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739 views

How do I calculate the weights in ABC-SMC

I have been reading through the Tutorial on ABC rejection and ABC SMC for parameter estimation and model selection by Tina Toni and Michael P. H. Stumpf. I can't work out how to calculate the weights ...
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1k views

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 ...