Greedy Importance Sampling Dale Shuurmans Generalized importance sampling

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Greedy Importance Sampling Dale Shuurmans

Greedy Importance Sampling Dale Shuurmans

Generalized importance sampling q We are interested in computing expectation q Draw independent points

Generalized importance sampling q We are interested in computing expectation q Draw independent points from a simpler “proposal” distribution q Weight these points by to obtain a “fair” representation of

Block Importance Sampling q Importance sampling is effective when Q approximates P over most

Block Importance Sampling q Importance sampling is effective when Q approximates P over most of the domain. q It fails when Q misses high probability regions of P and systematically yields samples with small weights. q To overcome this problem it is critical to obtain data points from importance regions of P. q Explicitly searching for significant regions in the target distribution P. q Sampling blocks of points instead of just individual points. q Let B be a partition of X into finite blocks B where

“Block” importance sampling q Draw independently from q For , recover block q Create

“Block” importance sampling q Draw independently from q For , recover block q Create a large sample out of the blocks q Weight each by q For a random variable by q For discrete spaces , estimate

“Sliding window” importance sampling q allow countably infinite blocks that each have a discrete

“Sliding window” importance sampling q allow countably infinite blocks that each have a discrete total order q unbiased estimator

“Sliding window” importance sampling q Draw independently from q For , recover block and

“Sliding window” importance sampling q Draw independently from q For , recover block and let § Get ‘s successors m-1 steps from § Get predecessors m-1 steps from § Weight by climbing up by climbing down q Create final sample from successor points q For a random variable , estimate

Greedy importance sampling : 1 -dimensional case q Draw q For , let independently

Greedy importance sampling : 1 -dimensional case q Draw q For , let independently from § compute successors by taking m-1 size steps in the direction of increase § compute predecessors by taking m-1 size steps in the direction of decrease. § If an improper ascent or descent occurs, truncate paths § Weight q Create final sample from successor points q For a random variable , estimate

Greedy importance sampling q optimal proposal distribution for importance sampling which minimize variance q

Greedy importance sampling q optimal proposal distribution for importance sampling which minimize variance q follow direction of increasing , taking fixed size steps q nontrivial issue : maintain disjoint search paths q At a collision, the largest ascent point must be allocated to a single path.

Experimental Result Direct mean 0. 1884 bias 0. 001 stdev 0. 071 Greed Imprtn

Experimental Result Direct mean 0. 1884 bias 0. 001 stdev 0. 071 Greed Imprtn Metropolis 0. 1937 0. 1810 8. 3609 0. 0075 0. 0052 8. 1747 0. 1374 0. 1762 22. 1212