Maximum A Posteriori (MAP) Estimation Pieter Abbeel UC Berkeley EECS

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Maximum A Posteriori (MAP) Estimation Pieter Abbeel UC Berkeley EECS. TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: A A A A A A A A A A A A A. Overview. Filtering: Smoothing: MAP:. X 0. X 0. X 0. X t-1. X t-1. X t-1. X t. X t. X t. X t+1.

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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:
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.