Download
1 / 5
90 likes | 136 Views
This overview covers filtering, smoothing, and the MAP algorithm in Kalman Filter setting, highlighting efficient solution methods using multivariate Gaussian distributions. Learn how to compute marginal conditionals with the Kalman smoother!
E N D
Maximum A Posteriori (MAP) Estimation Pieter Abbeel UC Berkeley EECS TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAAAAAAAAAAA
Overview • Filtering: • Smoothing: • MAP: X0 X0 X0 Xt-1 Xt-1 Xt-1 Xt Xt Xt Xt+1 Xt+1 XT XT z0 z0 z0 zt-1 zt-1 zt-1 zt zt zt zt+1 zt+1 zT zT
MAP Naively solving by enumerating all possible combinations of x_0,…,x_Tis exponential in T ! • Generally:
MAP --- Complete Algorithm • O(T n2)
Kalman Filter (aka Linear Gaussian) setting • Summations integrals • But: can’t enumerate over all instantations • However, we can still find solution efficiently: • the joint conditional P(x0:T | z0:T) is a multivariate Gaussian • for a multivariate Gaussian the most likely instantiation equals the mean we just need to find the mean of P(x0:T | z0:T) • 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 • We already know how to do so: marginal conditionals can be computed by running the Kalman smoother.
