# Think bayesian & Statistics review

## Main principles

1. Use prior konwledge
2. Chose answer that explains observations the most
3. Avoid extra assumptions

### example

A main is running, why?

1. He is in a hurry
2. He is doing exports (use principle 2 to exclude, does not waer a sports suit, contradicts the data)
3. He always runs (use principle 3 to exclude)
4. He saw a dragon (use principle 1 to exclude)

## Probability

for throw a dice, the probability of one side is 1/6

## Random variable

### Discrete

Probability Mass Function(PMF)

### Continuous

Probability Density Function(PDF)

### Independence

X and Y are independent if:

• P(x,y) -> Joint
• P(x) -> Marinals

## Conditional probability

Probability of X given that Y happened:

## Total probability

1. $B_1, B_2 \cdots$ 两两互斥，即 $B_i \cap B_j = \emptyset$ ，$i \neq j$, i,j=1，2，….，且$P(B_i)>0$,i=1,2,….;
2. $B_1 \cup B_2 \cdots = \Omega$ ，则称事件组 $B_1 \cup B_2 \cdots$ 是样本空间 $\Omega$ 的一个划分

## Bayes theorem

• $\theta$: parameters
• $X$: observations
• $P(\theta|X)$: Posterior
• $P(X)$: Evidence
• $P(X|\theta)$: Likelyhood
• $P(\theta)$: Prior

## Bayesian approach to statistics

### Frequentist

• Objective
• $\theta$ is fixed, X is random
• training
Maximum Likelyhood (they try to find the parameters theta that maximize the likelihood, the probability of their data given parameters)

### Bayesian

• Subjective
• X is random, $\theta$ is fixed
• Training(Bayes theorem)
what Bayesians will try to do is they would try to compute the posterior, the probability of the parameters given the data.
• Classification
• Training:
• Prediction:
• On-line learning (get posterior)

## How to build a model

Model is the “joint probability” of all variables model

### Example model
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