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SUMMARY:Unbiased approximations of products of expectations - Anthony Lee 
 (University of Warwick)
DTSTART:20170704T084500Z
DTEND:20170704T093000Z
UID:TALK73139@talks.cam.ac.uk
CONTACT:INI IT
DESCRIPTION:I will describe recent work with Simone Tiberi (Zurich) and Gi
 acomo Zanella (Bocconi)\, on the&nbsp\;unbiased&nbsp\;approximation&nbsp\;
 of a product of n&nbsp\;expectations. Such products arise\, e.g.\, as valu
 es of the likelihood function in latent variable models\, and&nbsp\;unbias
 ed&nbsp\;approximations&nbsp\;can be used in a pseudo-marginal Markov chai
 n to facilitate inference. A straightforward\, standard approach consists 
 of&nbsp\;approximating&nbsp\;each term using an independent average of M i
 .i.d. random variables and taking the product of these&nbsp\;approximation
 s. While simple\, this typically requires M to be O(n) so that the total n
 umber of random variables required is N = Mn = O(n^2) in order to control 
 the relative variance of the&nbsp\;approximation. Using all N random varia
 bles to&nbsp\;approximate&nbsp\;each&nbsp\;expectation&nbsp\;is less waste
 ful when producing them is costly\, but produces a biased&nbsp\;approximat
 ion. We propose an alternative to these two&nbsp\;approximations&nbsp\;tha
 t uses most of the N samples to&nbsp\;approximate&nbsp\;each&nbsp\;expecta
 tion&nbsp\;in such a way that the estimate of the product of&nbsp\;expecta
 tions&nbsp\;is&nbsp\;unbiased. We analyze the variance of this&nbsp\;appro
 ximation&nbsp\;and show that it can result in N = O(n) being sufficient fo
 r the relative variance to be controlled as n increases. In situations whe
 re the cost of simulations dominates overall computational time\, and fixi
 ng the relative variance\, the proposed&nbsp\;approximation&nbsp\;is almos
 t n times faster than the standard approach to compute.
LOCATION:Seminar Room 1\, Newton Institute
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