Representations for Probability distributions:

• Table of values

  • Discretization effects

• Particles

  • Use sampling with replacement to re-normalize

  • Use motion model to separate duplicates from sampling with replacement

• Closed form

  • Conjugate priors - http://www.cis.hut.fi/ahonkela/dippa/node23.html

    • Basically the initial distribution is of the same class as the transition model (eg both gaussian)

  • Normal (Gaussian) Distribution:

    • 

    • In multi-dimensions, mean becomes a vector and co-variance (measure of spread) becomes a matrix

    • Known as a Kalman filter when implemented with Matrix math

    • Assumes linear-gaussian transition and sensor models (linear function + gaussian noise)

    • Works well with "any system characterised by continuous state variables and noisy measurements" (R&N p557)

    • Use tagent as approximation to linear form (linearization) if a distribution is non-gaussian

  • Also: Beta, Dirichet distributions (the conjugate prior for multinomial distributions)

• Also covered decomposition of P(x,y) into P(x).P(y|x)

  • e.g. P(x, y, theta) split into P(x, y) and P(theta | x, y)

  • Much smaller tables, cannot represent correlations

Interesting Side Reading: The particle filter example of the robot in the hallway with rooms was done by University of Washington. Here's the page where they have that (and other examples) of work done in their lab with particle filters: http://www.cs.washington.edu/ai/Mobile_Robotics/mcl/ (lots of links to papers, info, and animations showing particle filters)