1. Convergence of Sequential Monte Carlo Methods
Dan Crisan, Arnaud Doucet

2. Problem Statement X: signal, Y: observation process
X satisfies and evolves according to the following equation,
Y satisfies

3. Bayes’ recursion
Prediction
Updating

4. A Sequential Monte Carlo Methods
Empirical measure
Transition kernel
Importance distribution
: abs. continuous with respect to
: strictly positive Radon Nykodym derivative
Then is also continuous w.r.t. and

5. Algorithm Step 1:Sequential importance sampling sample:
evaluate normalized importance weights
and let

6. Step 2: Selection step Step 3: MCMC step
multiply/discard particles with high/low importance weights to obtain N particles
let assoc.empirical measure
Step 3: MCMC step
sample ,where K is a Markov kernel of invariant distribution and let

7. Convergence Study denote convergence to 0 of average mean square error
under quite general conditions
Then prove (almost sure) convergence of toward under more restrictive conditions

8. Bounds for mean square errors
Assumptions
1.-A Importance distribution and weights
is assumed abs.continuous with respect to
for all
is a bounded function in argument
define

9. There exists a constant s. t. for all
there exists with s.t.
There exists s. t.
and a constant s.t.

10. 2.-A Resampling/Selection scheme

11. First Assumption ensures that
Importance function is chosen so that the corresponding importance weights are bounded above.
Sampling kernel and importance weights depend “ continuously” on the measure variable.
Second assumption ensures that
Selection scheme does not introduce too strong a “discrepancy”.

12. Lemma 1 Lemma 2 Let us assume that for any then after step 1, for any
then for any

13. Lemma 3 Lemma 4 Let us assume that for any then after step 2, for any
then for any

14. Theorem 1
For all , there exists independent of s.t. for any