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									  Recursive Smoothing with Particle Filters

                  Sumeetpal S. Singh
     Cambridge University Engineering Department
      joint work with P. Del Moral and A. Doucet


Cambridge Statistics Initiative One-day Meeting


                 25 September 2009
Hidden Markov Model




      X −valued (e.g. Rp ) hidden Markov process

                    Xk | (Xk−1 = xk−1 ) ∼ fθ ( ·| xk−1 )

      Y−valued (e.g. Rq ) observed process

                       Yk | (Xk = xk ) ∼ gθ ( ·| xk )
Recursive Smoothing




   Aim is to compute expectations of the type
                               n
                     Sn ≡ E         sk (Xk−1 , Xk ) y0:n
                              k=1

   recursively in time
Particle Filters




   A class of simulation algorithms to approximate {pθ (xn |y1:n )}n≥1 :
                                         N
                                                 (i)
                   pθ ( dxn | y1:n ) =         Wn δX (i) (dxn )
                                                       n
                                         i=1

            (i)        N    (i)
   where Wn > 0,       i=1 Wn     =1
Examples




      Computing the score vector,        log pθ (y0:n )
      The filter sensitivity   pθ (xn |y1:n )
      To compute the E-step in the EM
Two solutions




      One with computational complexity O(N) but asymptotic
      variance of         √       N
                             N Sn − Sn

      grows quadratically with time n
      The other with computational complexity O(N 2 ) but
      asymptotic variance grows linearly with time n

								
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