1. Lecture Note 2 – Calculus and Probability Shuaiqiang Wang Department of CS & IS University of Jyväskylä http://users.jyu.fi/~swang/ shuaiqiang.wang@jyu.fi

2. Part 1: Calculus

3. Definition

4. Polynomial Function

5. Proof: Polynomial Function

6. Logarithm Function

7. Proof: Logarithm Function

8. Exponential Function Example:

9. Proof: Exponential Function

10. Exponential Function

11. Taylor Series

12. Partial Derivative and Gradient For example

13. Taylor Approximation Taylor Approximation Taylor Series

14. First-Order Taylor Approximation 1 dimension

15. Gradient Descent Optimization

16. Gradient Descent Algorithm

17. Part 2: Probability

18. Independent Events

19. Conditional Probability A person goes to sauna 6 times during the last 10 days, at most once per day. It snowed 8 days during the last 10 days. It snowed 4 days during the 6 sauna days. P(sauna | snow) = ? P(snow | sauna) = ? Example

20. Bayes’ Theorem Since Then

21. Bayes’ Theorem

22. Maximum Likelihood Estimation

24. Optimization

25. Any Question?