Think bayesian

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:

Chain rule

Sum rule

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)
Donate article here
Share the post