Coursera Stochastic Processes 課程筆記, 共十篇:

• Week 0: 一些預備知識
• Week 1: Introduction & Renewal processes
• Week 2: Poisson Processes
• Week3: Markov Chains
• Week 4: Gaussian Processes
• Week 5: Stationarity and Linear filters
• Week 6: Ergodicity, differentiability, continuity
• Week 7: Stochastic integration & Itô formula
• Week 8: Lévy processes (本文)
• 整理隨機過程的連續性、微分、積分和Brownian Motion

Coursera Stochastic Processes 課程筆記, 共十篇:

• Week 0: 一些預備知識
• Week 1: Introduction & Renewal processes
• Week 2: Poisson Processes
• Week3: Markov Chains
• Week 4: Gaussian Processes
• Week 5: Stationarity and Linear filters
• Week 6: Ergodicity, differentiability, continuity
• Week 7: Stochastic integration & Itô formula (本文)
• Week 8: Lévy processes
• 整理隨機過程的連續性、微分、積分和Brownian Motion

Coursera Stochastic Processes 課程筆記, 共十篇:

• Week 0: 一些預備知識
• Week 1: Introduction & Renewal processes
• Week 2: Poisson Processes
• Week3: Markov Chains
• Week 4: Gaussian Processes
• Week 5: Stationarity and Linear filters
• Week 6: Ergodicity, differentiability, continuity (本文)
• Week 7: Stochastic integration & Itô formula
• Week 8: Lévy processes
• 整理隨機過程的連續性、微分、積分和Brownian Motion

Coursera Stochastic Processes 課程筆記, 共十篇:

• Week 0: 一些預備知識
• Week 1: Introduction & Renewal processes
• Week 2: Poisson Processes
• Week3: Markov Chains
• Week 4: Gaussian Processes
• Week 5: Stationarity and Linear filters (本文)
• Week 6: Ergodicity, differentiability, continuity
• Week 7: Stochastic integration & Itô formula
• Week 8: Lévy processes
• 整理隨機過程的連續性、微分、積分和Brownian Motion

Coursera Stochastic Processes 課程筆記, 共十篇:

• Week 0: 一些預備知識
• Week 1: Introduction & Renewal processes
• Week 2: Poisson Processes
• Week3: Markov Chains
• Week 4: Gaussian Processes (本文)
• Week 5: Stationarity and Linear filters
• Week 6: Ergodicity, differentiability, continuity
• Week 7: Stochastic integration & Itô formula
• Week 8: Lévy processes
• 整理隨機過程的連續性、微分、積分和Brownian Motion

Coursera Stochastic Processes 課程筆記, 共十篇:

• Week 0: 一些預備知識
• Week 1: Introduction & Renewal processes
• Week 2: Poisson Processes
• Week3: Markov Chains (本文)
• Week 4: Gaussian Processes
• Week 5: Stationarity and Linear filters
• Week 6: Ergodicity, differentiability, continuity
• Week 7: Stochastic integration & Itô formula
• Week 8: Lévy processes
• 整理隨機過程的連續性、微分、積分和Brownian Motion

Coursera Stochastic Processes 課程筆記, 共十篇:

• Week 0: 一些預備知識
• Week 1: Introduction & Renewal processes
• Week 2: Poisson Processes (本文)
• Week3: Markov Chains
• Week 4: Gaussian Processes
• Week 5: Stationarity and Linear filters
• Week 6: Ergodicity, differentiability, continuity
• Week 7: Stochastic integration & Itô formula
• Week 8: Lévy processes
• 整理隨機過程的連續性、微分、積分和Brownian Motion

Coursera Stochastic Processes 課程筆記, 共十篇:

• Week 0: 一些預備知識
• Week 1: Introduction & Renewal processes (本文)
• Week 2: Poisson Processes
• Week3: Markov Chains
• Week 4: Gaussian Processes
• Week 5: Stationarity and Linear filters
• Week 6: Ergodicity, differentiability, continuity
• Week 7: Stochastic integration & Itô formula
• Week 8: Lévy processes
• 整理隨機過程的連續性、微分、積分和Brownian Motion

Coursera Stochastic Processes 課程筆記, 共十篇:

• Week 0: 一些預備知識 (本文)
• Week 1: Introduction & Renewal processes
• Week 2: Poisson Processes
• Week3: Markov Chains
• Week 4: Gaussian Processes
• Week 5: Stationarity and Linear filters
• Week 6: Ergodicity, differentiability, continuity
• Week 7: Stochastic integration & Itô formula
• Week 8: Lévy processes
• 整理隨機過程的連續性、微分、積分和Brownian Motion

本篇回顧一些基礎的機率複習, 這些在之後課程裡有用到.
強烈建議閱讀以下文章:

• [測度論] Sigma Algebra 與 Measurable function 簡介
• [機率論] 淺談機率公理 與 基本性質
• A guide to the Lebesgue measure and integration
• Measure theory in probability

以下回顧開始:

PRML book sampling (chapter 11) 開頭把動機描述得很好, 也引用來當這篇文章的前言.
在用 machine learning 很多時候會遇到需要計算某個 function $f(x)$ 的期望值, 當 $x$ follow 某個 distribution $p(x)$ 的情況, i.e. 需計算

例如 EM algorithm 會需要計算 $\mathbb{E}_{p(z|x)}[f(x,z)]$, 參考 ref 的式 (23), (28)
又或者我們要做 Bayesian 的 prediction 時, 參考 ref 的式 (2)

這些情況大部分都無法有 analytical form. 不過如果我們能從給定的 distribution $p(x)$ 取 $L$ 個 sample 的話, 式 (1) 就能如下逼近