Lecture Note 2 – Calculus and Probability Shuaiqiang Wang Department of CS & IS University of Jyväskylä

Part 1: Calculus

Definition

Polynomial Function

Proof: Polynomial Function

Logarithm Function

Proof: Logarithm Function

Exponential Function Example:

Proof: Exponential Function

Exponential Function

Taylor Series

Partial Derivative and Gradient For example

Taylor Approximation Taylor Approximation Taylor Series

First-Order Taylor Approximation 1 dimension

Gradient Descent Optimization

Gradient Descent Algorithm

Part 2: Probability

Independent Events

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

Bayes’ Theorem Since Then

Bayes’ Theorem

Maximum Likelihood Estimation

Optimization

Any Question?