# Shannon–Hartley theorem

1. 其带宽为 $W$.
2. 其噪声为功率谱密度为 $N_0$ 的高斯白噪声
3. 信号的平均功率为 $P$.

# The Noisy-Channel Coding Theorem

## Definitions & Notations

• 用 $X, Y, Z$ 来表示随机变量，$\mathcal{A}_{X}$ 表示变量的取值集合，$X^N$ 则表示将 $N$ 个独立的 $X$ 组成的整体作为一个随机变量
• 信息熵： $H(X) = -\sum\limits_{p_i}\log{p_i}$
• 条件信息熵： $H(X|Y) = \sum\limits_{y \in \mathcal{A}_Y} P(y) H(X|Y=y) = - \sum\limits_{xy \in \mathcal{A}_X\mathcal{A}_Y} P(x, y) \log{P(x|y)}$，满足:

$H(X, Y) = H(X) + H(Y|X) = H(Y) + H(X|Y)$

• 互信息： $I(X; Y) \equiv H(X) - H(X|Y) = H(Y) - H(Y|X)$，度量了两个随机变量相互蕴含了多少关于对方的信息
• 信道容量： $C \equiv \underset{\mathcal{P}_X}{\mathrm{max}}\ I(X;Y)$，表示输出变量 $Y$ 最多能够蕴含多少关于输入 $X$ 的信息

# Style mod for GoodReader with vimperator

## The Pain Point

Like I’ve mentioned in other posts, most of my reading is done with digital materials. And the application that I use most often for that purpose is named GoodReader(IOS platform). There certainly are better alternatives, nevertheless, I’m too lazy(poor) to alter.

The old fashioned UI design of the app seems a little bit complicated and confusing, however the front-end page for file transfer via a WLAN is too simple to be satisfying.

• no design not all
• progressing info is too inconspicuous

# Hammersley Clifford Theorem

## Preface

PRML 中 8.3.2 小节简单描述了 Markov Random Fields 的分解特性，其中最核心的部分就是 Hammersley Clifford Theorem, 然而它并没有证明这个定理，只是在末尾的时候提到了这个结论，导致我在阅读中间部分的时候一头雾水。好在我 google 到了一个优雅的证明，顺便翻译在此。

# Add TOC indexes to PDF

## PDF books without Indexes are like Planets with Individuals

I prefer e-books to paper books, actually I may have read only 1-2 paper books in recent 3 years, while e-books of a much larger amount. The pros of e-books are really obvious:

1. portability
2. context can be copied
3. easy to manage
4. easy to navigate

I don’t want to argue about the superiorities here. I just wanna to convey that the 4th feature is a crucial one, and it is mainly achieved via indexes, bookmarks, reference links, etc. There’s no doubt that among those approaches, indexes play the most significant role. So I say that e-books without indexes are like planets with individuals, i.e. huge gaps among separated pieces of text.

# SVD++ Implementation in GraphX

apache/spark.

## SVD++ Intro

### SVD in CF

$$U = \begin{bmatrix} u_{11} & \cdots & u_{1k} \\ \vdots & \ddots & \vdots \\ u_{m1} & \cdots & u_{mk} \end{bmatrix} \begin{bmatrix} i_{11} & \cdots & i_{1n} \\ \vdots & \ddots & \vdots \\ i_{k1} & \cdots & i_{kn} \end{bmatrix}$$.