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Maximum A Posteriori (MAP) Estimation
Pieter Abbeel UC Berkeley EECS
n Filtering: n Smoothing: n MAP:
Overview
Xt-1 Xt X0 zt-1 zt z0 Xt-1 Xt Xt+1 XT X0 zt-1 zt zt+1 zT z0 Xt-1 Xt Xt+1 XT X0 zt-1 zt zt+1 zT z0
Maximum A Posteriori (MAP) Estimation Pieter Abbeel UC Berkeley - - PowerPoint PPT Presentation
Maximum A Posteriori (MAP) Estimation Pieter Abbeel UC Berkeley - - PowerPoint PPT Presentation
Maximum A Posteriori (MAP) Estimation Pieter Abbeel UC Berkeley EECS Overview X 0 X t-1 X t n Filtering: z 0 z t-1 z t n Smoothing: X 0 X t-1 X t X t+1 X T z 0 z t-1 z t z t+1 z T n MAP: X 0 X t-1 X t X t+1 X T z 0 z t-1 z t z t+1 z T
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n Generally:
MAP
Naively solving by enumerating all possible combinations
- f x_0,…,x_T is
exponential in T !
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MAP --- Complete Algorithm
n O(T n2)
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n
Summations à integrals
n
But: can’t enumerate over all instantiations
n
However, we can still find solution efficiently:
n the joint conditional P(x0:T | z0:T) is a multivariate Gaussian n for a multivariate Gaussian the most likely instantiation equals the
mean à we just need to find the mean of P(x0:T | z0:T)
n the marginal conditionals P(xt | z0:T) are Gaussians with mean equal to the
mean of xt under the joint conditional, so it suffices to find all marginal conditionals
n We already know how to do so: marginal conditionals can be computed
by running the Kalman smoother.
n
Alternatively: solve convex optimization problem