Generative Model

This article is my notes on generative model for Lecture 5 and 6 of Machine Learning by Andrew Ng. What we do in logistic regression using generalized linear model is that, we approximate \(P(y|x)\) using given data. This kind of learning algorithms is discriminative, in which we predict \(y\) based on the input features \(x\). On the contrary, generative model is to model \(P(x|y)\), the probability of the features \(x\) given class \(y\). In other words, we want to study how the features structure looks like given a class \(y\). If we also learn what \(P(y)\) is, we can easily recover \(P(y|x)\), for example, in the binary classification problem,

$$\begin{equation} P(y=1|x) = \frac{P(x|y=1)P(y=1)}{P(x)}, \label{eqn:bayes} \end{equation}$$


where \(P(x) = P(x|y=0)P(y=0) + P(x|y=1)P(y=1)\).

In this article, we are going to see a simple example of generative model on Gaussian discriminant analysis and Naive Bayes.

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Generalized Linear Model (Examples)

This article is a companion article to my another post Generalized Linear Model. In this article, I will implement some of the learning algorithms in Generalized Linear Model. To be more specific, I will do some examples on linear regression and logistic regression. With some effort, google search gives me some very good example data sets to work with. The datasets collected by Larry Winner is one of the excellent sets, which will be used in the article.

The implementations here use Python. Required 3rd party libraries are:

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Generalized Linear Model

This article on Generalized Linear Model (GLM) is based on the first four lectures of Machine Learning by Andrew Ng. But the structure of the article is quite different from the lecture. I will talk about exponential family of distributions first. Then I will discuss the general idea of GLM. Finally, I will try to derive some well known learning algorithms from GLM.

Exponential Family

A family of distributions is an exponential family if it can be parametrized by vector $\eta$ in the form $$P(y; \eta) = b(y)\exp(\eta^{T} T(y)-a(\eta)),$$ where $T(y)$ and $b(y)$ are (vector-valued) functions in terms of $y$, and $a(\eta)$ is a function in terms of $\eta$.

\(\eta\) is called the natural parameter and \(T(y)\) is called the sufficient statistic.

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简单的python爬虫

一个网络爬虫大致可以分成三个部分:获取网页,提取信息以及保存信息。Python有很多爬虫框架,其中最出名要数Scrapy了。这也是我唯一用过的Python爬虫框架,用起来很省心。让我苦恼的是,Scrapy在我的Raspberry Pi Zero W安装起来很麻烦,而且我觉得我爬取的网页比较容易处理,没有必要用这么重量级的框架。抱着学习的心态,我开始自己造轮子了。

在造轮子之前我找到些轻量级的框架,Sukhoi是我比较喜欢的一个。该库作者iogf使用了自己的异步库、网络库来写这个框架。这让我很佩服他。我写的框架受到了Sukhoi很大的启发与影响。

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意志力

今天早上读完了鲍迈斯特的《意志力》。有趣的是,能看完这本书也是一种意志力的运用吧。

自我损耗造成的影响是双重的:一方面意志力减弱了,另外一方面渴望变强了。

“自我损耗”是作者以及书中提到的研究者在做实验的时候常用的手段。这也有警醒的意味:意志力减弱的时候,渴望还会变强!

面临一个让他们产生内心冲突(一方面非常想要,一方面真不该要)的新诱惑,如果他们已经挡住了之前的诱惑,特别是如果上个诱惑过去没多久新诱惑就来了,那么他们更容易屈服。

1.你的意志力是有限的,使用就会消耗。 2.你从同一账户提取意志力用于各种不同任务。

我们可以把意志力的运用分为四大类,控制思维,控制情绪,控制冲动,控制表现。

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Pelican Signals

Pelican的插件系统是使用blinkersignal实现的。Pelican所有可以用的signals可以在signals.py找到。本文的目的是记录这些signals是在Pelican运行中什么时候发出的。

(1) Pelican有一个叫做Pelican的类,含有程序的主体框架。当Pelican的一个实例pelican初始化完成之后(基本设置,加载插件),发出第一个signal。

signals.initialized.send(pelican)

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Lambda Calculus

This post is my note for What is seminar on Lambda Calculus.

Lambda calculus was created by Alonzo Church in the 1930s, and was used by him to solve Entscheidungsproblem in 1936, which is related to Hilbert's tenth problem. In the same year, Alan Turing independently solved Entscheidungsproblem using his invention Turing machine. Shortly after, Turing realized that these two models are actually equivalent as models of computation.

In this note, I will first give the formal definition of lambda expressions. Then with the help of Python, I am going to show how to do Boolean algebra and basic arithmetic using lambda calculus, which to some extend gives an illustration that Turing machine and lambda calculus are equivalent.

Definition

Lambda calculus consists of lambda expressions and operations on them. There are three basic elements in Lambda expression:

  1. variables: x, y, z, ...
  2. symbols in abstraction: λ and .
  3. parentheses for association: ()

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maupassant: a Pelican theme

我现在用的博客生成软件是Hexo。这个软件可以快速将Markdown格式的文章转成html格式,并且包含发布到github page的工具。Hexo是基于Nodejs,所以某天我就在想有没有一个基于Python的博客生成软件。为什么我会这样想?因为

人生苦短,我用Python。

当然会有基于Python的博客生成软件:Pelican是最为突出的一个。事实上,我很喜欢Pelican!其中一个亮点是它的主题都是基于Jinja,这种简单统一的模版格式让用户很容易设计自己喜欢的主题。出于练手的原因,我将我喜欢的一个Hexo主题maupassant移植到Pelican了。

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Cassini Ovals

This post is my note for What is seminar on Cassini Ovals.

Definition

An ellipse is a geometric object formed by locus of points which have fixed sum of distances to two fixes foci.

Ellipse

In the above figure, we can express the definition of ellipses in one simple equation:

$$|PF_1|+|PF_2|=c,$$


for some \(c>0\). What if we change the addition in the above equation to multiplication? This comes the definition of Cassini ovals.

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Rademacher functions

This post is my note for What is…? seminar on Rademacher function.

For a real number \(x\) in \([0, 1]\), let \((a_1(x), a_2(x), \cdots, a_n(x), \cdots)\) be its binary representation. That is to say, \[\begin{equation} x = \sum_{n=1}^\infty \frac{a_n(x)}{2^n}. \label{20170623-1} \end{equation}\]

For some \(x\), there might be two possible binary representation. Take \(\frac{1}{2}\) for example, it can be represented as \((1, 0, 0, \cdots)\) or \((0, 1, 1, \cdots)\). In this situation, we always prefer the former representation, in which all terms become \(0\) eventually.

Definition 1. Let \(n \ge 1\) be an integer. The \(n\)-th Rademacher function \(r_n: [0, 1] \to \{-1,1\}\) is defined to be \[r_n(x) = 1 - 2a_n(x).\]

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