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For particle filtering weight update, often the transitional prior is used for the proposal density, in which case the weight update simplifies to just the likelyhood function. However, in one Ensemble Kalman Filter Particle Filter ("An Improved Particle Filter Algorithm Based on Ensemble Kalman Filter and Markov Chain Monte Carlo Method", Haiyun Bi et. al, 2015), a proposal density is created from updated Kalman filters for each analysis particle Xa (see above). So I'm confused how the weight update proceeds with this type of formula (Above Equations 17 through 20). In Equation 20, the p(zk | xa(i,k) term is just the likelyhood function, so it is deterministic (I assume?) just plugging in the measurement to a guassian with mean h(xa(i,k) and cov Rk. But how about the other terms ? Are these generated with drawn samples, i.e. p(xa(i,k) | xa(i,k-1) = f(xa(i,k-1)) + n with cov(n) = Qk-1 ???? So these are non-deterministic ? Same goes for denominator term. Do we generate a noisy sample, or do we plug in the xa(i,k) into the gaussian and get a deterministic number like the likelyhood function. Thanks for the help!

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