University of Colorado Boulder ASEN 5070: Statistical Orbit Determination I Fall 2014 Professor Brandon A. Jones Lecture 14: Probability Wrap-Up and Statistical Least Squares

University of Colorado Boulder  Lecture Quiz Due by 5pm Friday (10/3)  Homework #5 Due 10/3 2

University of Colorado Boulder  Multivariate Gaussian Distribution  Central Limit Theorem  Bayes’ Theorem  Statistical Least Squares 3

University of Colorado Boulder 4 Multivariate Normal Distribution

University of Colorado Boulder  Multivariate: 5  Univariate:

University of Colorado Boulder 6

University of Colorado Boulder  It may be shown that: 7  Although the above assumes a bivariate normal distribution, the idea extends to higher dimensions with minor changes

University of Colorado Boulder  The conditional density function is also a normal distribution (anyone seeing a trend here?) 8

University of Colorado Boulder 9  The conditional PDF from the previous slide is a special case of the general conditional PDF

University of Colorado Boulder 10

University of Colorado Boulder 11 Central Limit Theorem

University of Colorado Boulder 12

University of Colorado Boulder 13

University of Colorado Boulder  When we have many observations, the CLT implies that we can treat the mean observation error as a Gaussian random variable  What about when we don’t have many observations? 14

University of Colorado Boulder 15 Bayes’ Theorem

University of Colorado Boulder 16

University of Colorado Boulder  Allows for updating a hypothesis’ probability when given additional information ◦ Known as Bayesian Inference  Modern estimation research is rooted in Bayesian Inference! 17

University of Colorado Boulder 18

University of Colorado Boulder 19

University of Colorado Boulder 20 Statistical Interpretation of Least Squares

University of Colorado Boulder 21

University of Colorado Boulder 22

University of Colorado Boulder 23

University of Colorado Boulder 24

University of Colorado Boulder 25

University of Colorado Boulder 26

University of Colorado Boulder 27

University of Colorado Boulder 28

University of Colorado Boulder 29 We may leverage a statistical interpretation of the observation errors to yield a description of the estimation errors