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Kalman filters are optimal estimators when dynamics and sensory measurements are linear and noise is Gaussian.

Explicit or implicit representations of world dynamics are necessary for optimal controllers since they have to anticipate state changes before the arrival of the necessary sensor data.

An estimator is a deterministic function which maps from measurements to estimates.

Given probability density functions (PDF) $P(X)$ and $P(X\mid M)$ for a latent variable $X$ and an observable $M$, an optimal estimator for $X$ wrt. the loss function $F$ is given by $$ f_{opt} = \mathrm{arg min}_f \int P(x) \int P(x\mid m) L(x,f(m))\;dx\;dm $$

The maxiumum a posteriori estimator (MAP) arises from an error function which penalizes all errors equally.

Optimality of an estimator is relative to

  • loss function,
  • measurement probability,
  • prior,
  • (depending on the setting) a family of functions.

A representation of probabilities is not necessary for optimal estimation.