1. 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

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

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

4. University of Colorado Boulder 4 Multivariate Normal Distribution

5. University of Colorado Boulder  Multivariate: 5  Univariate:

6. University of Colorado Boulder 6

7. 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

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

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

10. University of Colorado Boulder 10

11. University of Colorado Boulder 11 Central Limit Theorem

12. University of Colorado Boulder 12

13. University of Colorado Boulder 13

14. 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

15. University of Colorado Boulder 15 Bayes’ Theorem

16. University of Colorado Boulder 16

17. 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

18. University of Colorado Boulder 18

19. University of Colorado Boulder 19

20. University of Colorado Boulder 20 Statistical Interpretation of Least Squares

21. University of Colorado Boulder 21

22. University of Colorado Boulder 22

23. University of Colorado Boulder 23

24. University of Colorado Boulder 24

25. University of Colorado Boulder 25

26. University of Colorado Boulder 26

27. University of Colorado Boulder 27

28. University of Colorado Boulder 28

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