1. Physics Fluctuomatics (Tohoku University) 1 Physical Fluctuomatics 2nd Probability and its fundamental properties Kazuyuki Tanaka Graduate School of Information Sciences, Tohoku University kazu@smapip.is.tohoku.ac.jp http://www.smapip.is.tohoku.ac.jp/~kazu/

2. Physics Fluctuomatics (Tohoku University) 2 Probability a.Event and Probability b.Joint Probability and Conditional Probability c.Bayes Formula, Prior Probability and Posterior Probability d.Discrete Random Variable and Probability Distribution e.Continuous Random Variable and Probability Density Function f.Average, Variance and Covariance g.Uniform Distribution h.Gauss Distribution This Talk Next Talk

3. Physics Fluctuomatics (Tohoku University) 3 Event, Sample Space and Event Experiment: Experiments in probability theory means that outcomes are not predictable in advance. However, while the outcome will not be known in advance, the set of all possible outcomes is known Sample Point: Each possible outcome in the experiments. Sample Space ： The set of all the possible sample points in the experiments Event ： Subset of the sample space Elementary Event ： Event consisting of one sample point Empty Event ： Event consisting of no sample point

4. Physics Fluctuomatics (Tohoku University) 4 Various Events Whole Events Ω:Events consisting of all sample points of the sample space. Complementary Event of Event A: A c =Ω ╲ A Defference of Events A and B: A ╲ B Union of Events A and B: A ∪ B Intersection of Events A and B : A ∩ B Events A and B are exclusive of each other: A∩B=Ф Events A, B and C are exclusive of each other: [A∩B=Ф] Λ [B∩C=Ф] Λ [C∩A=Ф]

5. Physics Fluctuomatics (Tohoku University) 5 Empirically Definition of Probability Statistical Definition: Let us suppose that an event A occur r times when the same experiment are repeated R times. If the ratio r/R tends to a constant value p as the number of times of the experiments R go to infinity, we define the value p as probability of event A. Definition by Laplace: Let us suppose that the total number of all the sample points is N and they can occur equally Likely. Probability of an event A with N sample points is defined by p=n/N.

6. Physics Fluctuomatics (Tohoku University) 6 Definition of Probability Axion 1: Axion 2: Axion 3: For every events A, B that are exclusive of each other, it is always valid that Definition of Kolmogorov: Probability Pr{A} for every event A in the specified sample space Ω satisfies the following three axioms:

7. Physics Fluctuomatics (Tohoku University) 7 Joint Probability and Conditional Probability A B Conditional Probability of Event A when Event B has happened. Probability of Event A Joint Probability of Events A and B

8. Physics Fluctuomatics (Tohoku University) 8 Joint Probability and Independency of Events A B In this case, the conditional probability can be expressed as Events A and B are independent of each other A B

9. Physics Fluctuomatics (Tohoku University) 9 Marginal Probability Let us suppose that the sample space  is expressed by Ω=A 1 ∪ A 2 ∪ … ∪ A M where every pair of events A i and A j is exclusive of each other. Marginal Probability of Event B for Joint Probability Pr{A i,B} Marginalize AiAiAiAi B AB Simplified Notation Summation over all the possible events in which every pair of events are exclusive of each other.

10. Physics Fluctuomatics (Tohoku University) 10 Four Dimensional Point Probability and Marginal Probability Marginal Probability of Event B AB C D Marginalize

11. Physics Fluctuomatics (Tohoku University) 11 Derivation of Bayes Formulas

12. Physics Fluctuomatics (Tohoku University) 12 Derivation of Bayes Formulas

13. Physics Fluctuomatics (Tohoku University) 13 Derivation of Bayes Formulas

14. Physics Fluctuomatics (Tohoku University) 14 Derivation of Bayes Formulas

15. Physics Fluctuomatics (Tohoku University) 15 Derivation of Bayes Formulas

16. Physics Fluctuomatics (Tohoku University) 16 Derivation of Bayes Formulas

17. Physics Fluctuomatics (Tohoku University) 17 Derivation of Bayes Formulas A B

18. Physics Fluctuomatics (Tohoku University) 18 Bayes Formula Posterior Probability Prior Probability A B It is often referred to as Bayes Rule. Bayesian Network

19. Physics Fluctuomatics (Tohoku University) 19 Summary a.Event and Probability b.Joint Probability and Conditional Probability c.Bayes Formulas, Prior Probability and Posterior Probability d.Discrete Random Variable and Probability Distribution e.Continuous Random Variable and Probability Density Function f.Average, Variance and Covariance g.Uniform Distribution h.Gauss Distribution The present talk Next talk