1、LDA数学定义
1）话题模型：传统的文本分类器，比如贝叶斯、kNN和SVM，只能将其分到一个确定的类别中。假设我给出3个分类“算法”“分词”“文学”让其判断，如果某个分类器将该文归入算法类，我觉得还凑合，如果分入分词，那我觉得这个分类器不够准确。
假设一个文艺小青年来看我的博客，他完全不懂算法和分词，自然也给不出具体的备选类别，有没有一种模型能够告诉这个白痴，这篇文章很可能(80%)是在讲算法，也可能(19%)是在讲分词，几乎不可能(1%)是在讲其它主题呢？
有，这样的模型就是话题模型。
2）LDA概念
潜在狄立克雷分配（Latent Dirichlet Allocation，LDA）主题模型是最简单的主题模型，它描述的是一篇文章是如何产生的。如图所示：
从左往右看，一个主题是由一些词语的分布定义的，比如蓝色主题是由2%几率的data，2%的number……构成的。一篇文章则是由一些主题构成的，比如右边的直方图。具体产生过程是，从主题集合中按概率分布选取一些主题，从该主题中按概率分布选取一些词语，这些词语构成了最终的文档（LDA模型中，词语的无序集合构成文档，也就是说词语的顺序没有关系）。
如果我们能将上述两个概率分布计算清楚，那么我们就得到了一个模型，该模型可以根据某篇文档推断出它的主题分布，即分类。由文档推断主题是文档生成过程的逆过程。
在《LDA数学八卦》一文中，对文档的生成过程有个很形象的描述：
3）概率模型
LDA是一种使用联合分布来计算在给定观测变量下隐藏变量的条件分布（后验分布）的概率模型，观测变量为词的集合，隐藏变量为主题。
联合分布
LDA的生成过程对应的观测变量和隐藏变量的联合分布如下：
式子有点长，耐心地从左往右看：
式子的基本符号约定——β表示主题，θ表示主题的概率，z表示特定文档或词语的主题，w为词语。进一步说——
β1:K为全体主题集合，其中βk是第k个主题的词的分布（如图1左部所示）。第d个文档中该主题所占的比例为θd，其中θd,k表示第k个主题在第d个文档中的比例（图1右部的直方图）。第d个文档的主题全体为zd，其中zd,n是第d个文档中第n个词的主题（如图1中有颜色的圆圈）。第d个文档中所有词记为wd，其中wd,n是第d个文档中第n个词，每个词都是固定的词汇表中的元素。
p(β）表示从主题集合中选取了一个特定主题，p(θd)表示该主题在特定文档中的概率，大括号的前半部分是该主题确定时该文档第n个词的主题，后半部分是该文档第n个词的主题与该词的联合分布。连乘符号描述了随机变量的依赖性，用概率图模型表述如下：
比如，先选取了主题，才能从主题里选词。具体说来，一个词受两个随机变量的影响（直接或间接），一个是确定了主题后文档中该主题的分布θd，另一种是第k个主题的词的分布βk（也就是图2中的第二个坛子）。
后验分布
沿用相同的符号，LDA后验分布计算公式如下：
分子是一个联合分布，给定语料库就可以轻松统计出来。但分母无法暴力计算，因为文档集合词库达到百万（假设有w个词语），每个词要计算每一种可能的观测的组合（假设有n种组合）的概率然后累加得到先验概率，所以需要一种近似算法。
基于采样的算法通过收集后验分布的样本，以样本的分布求得后验分布的近似。
θd的概率服从Dirichlet分布，zd,n的分布服从multinomial分布，两个分布共轭，所谓共轭，指的就是先验分布和后验分布的形式相同：
两个分布其实是向量的分布，向量通过这两个分布取样得到。采样方法通过收集这两个分布的样本，以样本的分布近似。
4）马氏链和Gibbs Sampling这是一种统计模拟的方法。
马氏链
所谓马氏链指的是当前状态只取决于上一个状态。马氏链有一个重要的性质：状态转移矩阵P的幂是收敛的，收敛后的转移矩阵称为马氏链的平稳分布。给定p(x)，假如能够构造一个P，转移n步平稳分布恰好是p(x)。那么任取一个初始状态，转移n步之后的状态都是符合分布的样本。
Gibbs Sampling
Gibbs Sampling是高维分布（也即类似于二维p(x,y)，三维p(x,y,z)的分布）的特化采样算法。
2、代码：
参考：https://github.com/hankcs/LDA4j
源于Gregor Heinrich的LdaGibbsSampler.java类。
/*
* (C) Copyright 2005, Gregor Heinrich (gregor :: arbylon : net) (This file is
* part of the org.knowceans experimental software packages.)
*/
/*
* LdaGibbsSampler is free software; you can redistribute it and/or modify it
* under the terms of the GNU General Public License as published by the Free
* Software Foundation; either version 2 of the License, or (at your option) any
* later version.
*/
/*
* LdaGibbsSampler is distributed in the hope that it will be useful, but
* WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or
* FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more
* details.
*/
/*
* You should have received a copy of the GNU General Public License along with
* this program; if not, write to the Free Software Foundation, Inc., 59 Temple
* Place, Suite 330, Boston, MA 02111-1307 USA
*/
/*
* Created on Mar 6, 2005
*/
package sk.ml;
import java.text.DecimalFormat;
import java.text.NumberFormat;
/**
* Gibbs sampler for estimating the best assignments of topics for words and
* documents in a corpus. The algorithm is introduced in Tom Griffiths' paper
* "Gibbs sampling in the generative model of Latent Dirichlet Allocation"
* (2002).<br>
* Gibbs sampler采样算法的实现
*
* @author heinrich
*/
public class LdaGibbsSampler {
/**
* document data (term lists)<br>
* 文档
*/
int[][] documents;
/**
* vocabulary size<br>
* 词表大小
*/
int V;
/**
* number of topics<br>
* 主题数目
*/
int K;
/**
* Dirichlet parameter (document--topic associations)<br>
* 文档——主题参数
*/
double alpha = 2.0;
/**
* Dirichlet parameter (topic--term associations)<br>
* 主题——词语参数
*/
double beta = 0.5;
/**
* topic assignments for each word.<br>
* 每个词语的主题 z[i][j] := 文档i的第j个词语的主题编号
*/
int z[][];
/**
* cwt[i][j] number of instances of word i (term?) assigned to topic j.<br>
* 计数器，nw[i][j] := 词语i归入主题j的次数
*/
int[][] nw;
/**
* na[i][j] number of words in document i assigned to topic j.<br>
* 计数器，nd[i][j] := 文档[i]中归入主题j的词语的个数
*/
int[][] nd;
/**
* nwsum[j] total number of words assigned to topic j.<br>
* 计数器，nwsum[j] := 归入主题j词语的个数
*/
int[] nwsum;
/**
* nasum[i] total number of words in document i.<br>
* 计数器,ndsum[i] := 文档i中全部词语的数量
*/
int[] ndsum;
/**
* cumulative statistics of theta<br>
* theta的累积量
*/
double[][] thetasum;
/**
* cumulative statistics of phi<br>
* phi的累积量
*/
double[][] phisum;
/**
* size of statistics<br>
* 样本容量
*/
int numstats;
/**
* sampling lag (?)<br>
* 多久更新一次统计量
*/
private static int THIN_INTERVAL = 20;
/**
* burn-in period<br>
* 收敛前的迭代次数
*/
private static int BURN_IN = 100;
/**
* max iterations<br>
* 最大迭代次数
*/
private static int ITERATIONS = 1000;
/**
* sample lag (if -1 only one sample taken)<br>
* 最后的模型个数（取收敛后的n个迭代的参数做平均可以使得模型质量更高）
*/
private static int SAMPLE_LAG = 10;
private static int dispcol = 0;
/**
* Initialise the Gibbs sampler with data.<br>
* 用数据初始化采样器
*
* @param documents 文档
* @param V vocabulary size 词表大小
*/
public LdaGibbsSampler(int[][] documents, int V) {
this.documents = documents;
this.V = V;
}
/**
* Initialisation: Must start with an assignment of observations to topics ?
* Many alternatives are possible, I chose to perform random assignments
* with equal probabilities<br>
* 随机初始化状态
*
* @param K number of topics K个主题
*/
public void initialState(int K) {
int M = documents.length;
// initialise count variables. 初始化计数器
nw = new int[V][K];
nd = new int[M][K];
nwsum = new int[K];
ndsum = new int[M];
// The z_i are are initialised to values in [1,K] to determine the
// initial state of the Markov chain.
z = new int[M][]; // z_i := 1到K之间的值，表示马氏链的初始状态
for (int m = 0; m < M; m++) {
int N = documents[m].length;
z[m] = new int[N];
for (int n = 0; n < N; n++) {
int topic = (int) (Math.random() * K);
z[m][n] = topic;
// number of instances of word i assigned to topic j
nw[documents[m][n]][topic]++;
// number of words in document i assigned to topic j.
nd[m][topic]++;
// total number of words assigned to topic j.
nwsum[topic]++;
}
// total number of words in document i
ndsum[m] = N;
}
}
public void gibbs(int K) {
gibbs(K, 2.0, 0.5);
}
/**
* Main method: Select initial state ? Repeat a large number of times: 1.
* Select an element 2. Update conditional on other elements. If
* appropriate, output summary for each run.<br>
* 采样
*
* @param K number of topics 主题数
* @param alpha symmetric prior parameter on document--topic associations 对称文档——主题先验概率？
* @param beta symmetric prior parameter on topic--term associations 对称主题——词语先验概率？
*/
public void gibbs(int K, double alpha, double beta) {
this.K = K;
this.alpha = alpha;
this.beta = beta;
// init sampler statistics 分配内存
if (SAMPLE_LAG > 0) {
thetasum = new double[documents.length][K];
phisum = new double[K][V];
numstats = 0;
}
// initial state of the Markov chain:
initialState(K);
System.out.println("Sampling " + ITERATIONS
+ " iterations with burn-in of " + BURN_IN + " (B/S="
+ THIN_INTERVAL + ").");
for (int i = 0; i < ITERATIONS; i++) {
// for all z_i
for (int m = 0; m < z.length; m++) {
for (int n = 0; n < z[m].length; n++) {
// (z_i = z[m][n])
// sample from p(z_i|z_-i, w)
int topic = sampleFullConditional(m, n);
z[m][n] = topic;
}
}
if ((i < BURN_IN) && (i % THIN_INTERVAL == 0)) {
System.out.print("B");
dispcol++;
}
// display progress
if ((i > BURN_IN) && (i % THIN_INTERVAL == 0)) {
System.out.print("S");
dispcol++;
}
// get statistics after burn-in
if ((i > BURN_IN) && (SAMPLE_LAG > 0) && (i % SAMPLE_LAG == 0)) {
updateParams();
System.out.print("|");
if (i % THIN_INTERVAL != 0)
dispcol++;
}
if (dispcol >= 100) {
System.out.println();
dispcol = 0;
}
}
System.out.println();
}
/**
* Sample a topic z_i from the full conditional distribution: p(z_i = j |
* z_-i, w) = (n_-i,j(w_i) + beta)/(n_-i,j(.) + W * beta) * (n_-i,j(d_i) +
* alpha)/(n_-i,.(d_i) + K * alpha) <br>
* 根据上述公式计算文档m中第n个词语的主题的完全条件分布，输出最可能的主题
*
* @param m document
* @param n word
*/
private int sampleFullConditional(int m, int n) {
// remove z_i from the count variables 先将这个词从计数器中抹掉
int topic = z[m][n];
nw[documents[m][n]][topic]--;
nd[m][topic]--;
nwsum[topic]--;
ndsum[m]--;
// do multinomial sampling via cumulative method: 通过多项式方法采样多项式分布
double[] p = new double[K];
for (int k = 0; k < K; k++) {
p[k] = (nw[documents[m][n]][k] + beta) / (nwsum[k] + V * beta)
* (nd[m][k] + alpha) / (ndsum[m] + K * alpha);
}
// cumulate multinomial parameters 累加多项式分布的参数
for (int k = 1; k < p.length; k++) {
p[k] += p[k - 1];
}
// scaled sample because of unnormalised p[] 正则化
double u = Math.random() * p[K - 1];
for (topic = 0; topic < p.length; topic++) {
if (u < p[topic])
break;
}
// add newly estimated z_i to count variables 将重新估计的该词语加入计数器
nw[documents[m][n]][topic]++;
nd[m][topic]++;
nwsum[topic]++;
ndsum[m]++;
return topic;
}
/**
* Add to the statistics the values of theta and phi for the current state.<br>
* 更新参数
*/
private void updateParams() {
for (int m = 0; m < documents.length; m++) {
for (int k = 0; k < K; k++) {
thetasum[m][k] += (nd[m][k] + alpha) / (ndsum[m] + K * alpha);
}
}
for (int k = 0; k < K; k++) {
for (int w = 0; w < V; w++) {
phisum[k][w] += (nw[w][k] + beta) / (nwsum[k] + V * beta);
}
}
numstats++;
}
/**
* Retrieve estimated document--topic associations. If sample lag > 0 then
* the mean value of all sampled statistics for theta[][] is taken.<br>
* 获取文档——主题矩阵
*
* @return theta multinomial mixture of document topics (M x K)
*/
public double[][] getTheta() {
double[][] theta = new double[documents.length][K];
if (SAMPLE_LAG > 0) {
for (int m = 0; m < documents.length; m++) {
for (int k = 0; k < K; k++) {
theta[m][k] = thetasum[m][k] / numstats;
}
}
} else {
for (int m = 0; m < documents.length; m++) {
for (int k = 0; k < K; k++) {
theta[m][k] = (nd[m][k] + alpha) / (ndsum[m] + K * alpha);
}
}
}
return theta;
}
/**
* Retrieve estimated topic--word associations. If sample lag > 0 then the
* mean value of all sampled statistics for phi[][] is taken.<br>
* 获取主题——词语矩阵
*
* @return phi multinomial mixture of topic words (K x V)
*/
public double[][] getPhi() {
double[][] phi = new double[K][V];
if (SAMPLE_LAG > 0) {
for (int k = 0; k < K; k++) {
for (int w = 0; w < V; w++) {
phi[k][w] = phisum[k][w] / numstats;
}
}
} else {
for (int k = 0; k < K; k++) {
for (int w = 0; w < V; w++) {
phi[k][w] = (nw[w][k] + beta) / (nwsum[k] + V * beta);
}
}
}
return phi;
}
/**
* Print table of multinomial data
*
* @param data vector of evidence
* @param fmax max frequency in display
* @return the scaled histogram bin values
*/
public static void hist(double[] data, int fmax) {
double[] hist = new double[data.length];
// scale maximum
double hmax = 0;
for (int i = 0; i < data.length; i++) {
hmax = Math.max(data[i], hmax);
}
double shrink = fmax / hmax;
for (int i = 0; i < data.length; i++) {
hist[i] = shrink * data[i];
}
NumberFormat nf = new DecimalFormat("00");
String scale = "";
for (int i = 1; i < fmax / 10 + 1; i++) {
scale += " . " + i % 10;
}
System.out.println("x" + nf.format(hmax / fmax) + "\t0" + scale);
for (int i = 0; i < hist.length; i++) {
System.out.print(i + "\t|");
for (int j = 0; j < Math.round(hist[i]); j++) {
if ((j + 1) % 10 == 0)
System.out.print("]");
else
System.out.print("|");
}
System.out.println();
}
}
/**
* Configure the gibbs sampler<br>
* 配置采样器
*
* @param iterations number of total iterations
* @param burnIn number of burn-in iterations
* @param thinInterval update statistics interval
* @param sampleLag sample interval (-1 for just one sample at the end)
*/
public void configure(int iterations, int burnIn, int thinInterval,
int sampleLag) {
ITERATIONS = iterations;
BURN_IN = burnIn;
THIN_INTERVAL = thinInterval;
SAMPLE_LAG = sampleLag;
}
/**
* Inference a new document by a pre-trained phi matrix
*
* @param phi pre-trained phi matrix
* @param doc document
* @return a p array
*/
public static double[] inference(double alpha, double beta, double[][] phi, int[] doc) {
int K = phi.length;
int V = phi[0].length;
// init
// initialise count variables. 初始化计数器
int[][] nw = new int[V][K];
int[] nd = new int[K];
int[] nwsum = new int[K];
int ndsum = 0;
// The z_i are are initialised to values in [1,K] to determine the
// initial state of the Markov chain.
int N = doc.length;
int[] z = new int[N]; // z_i := 1到K之间的值，表示马氏链的初始状态
for (int n = 0; n < N; n++) {
int topic = (int) (Math.random() * K);
z[n] = topic;
// number of instances of word i assigned to topic j
nw[doc[n]][topic]++;
// number of words in document i assigned to topic j.
nd[topic]++;
// total number of words assigned to topic j.
nwsum[topic]++;
}
// total number of words in document i
ndsum = N;
for (int i = 0; i < ITERATIONS; i++) {
for (int n = 0; n < z.length; n++) {
// (z_i = z[m][n])
// sample from p(z_i|z_-i, w)
// remove z_i from the count variables 先将这个词从计数器中抹掉
int topic = z[n];
nw[doc[n]][topic]--;
nd[topic]--;
nwsum[topic]--;
ndsum--;
// do multinomial sampling via cumulative method: 通过多项式方法采样多项式分布
double[] p = new double[K];
for (int k = 0; k < K; k++) {
p[k] = phi[k][doc[n]]
* (nd[k] + alpha) / (ndsum + K * alpha);
}
// cumulate multinomial parameters 累加多项式分布的参数
for (int k = 1; k < p.length; k++) {
p[k] += p[k - 1];
}
// scaled sample because of unnormalised p[] 正则化
double u = Math.random() * p[K - 1];
for (topic = 0; topic < p.length; topic++) {
if (u < p[topic])
break;
}
if (topic == K) {
throw new RuntimeException("the param K or topic is set too small");
}
// add newly estimated z_i to count variables 将重新估计的该词语加入计数器
nw[doc[n]][topic]++;
nd[topic]++;
nwsum[topic]++;
ndsum++;
z[n] = topic;
}
}
double[] theta = new double[K];
for (int k = 0; k < K; k++) {
theta[k] = (nd[k] + alpha) / (ndsum + K * alpha);
}
return theta;
}
public static double[] inference(double[][] phi, int[] doc) {
return inference(2.0, 0.5, phi, doc);
}
/**
* Driver with example data.<br>
* 测试入口
*
* @param args
*/
public static void main(String[] args) {
// words in documents
int[][] documents = {
{1, 4, 3, 2, 3, 1, 4, 3, 2, 3, 1, 4, 3, 2, 3, 6},
{2, 2, 4, 2, 4, 2, 2, 2, 2, 4, 2, 2},
{1, 6, 5, 6, 0, 1, 6, 5, 6, 0, 1, 6, 5, 6, 0, 0},
{5, 6, 6, 2, 3, 3, 6, 5, 6, 2, 2, 6, 5, 6, 6, 6, 0},
{2, 2, 4, 4, 4, 4, 1, 5, 5, 5, 5, 5, 5, 1, 1, 1, 1, 0},
{5, 4, 2, 3, 4, 5, 6, 6, 5, 4, 3, 2}}; // 文档的词语id集合
// vocabulary
int V = 7; // 词表大小
int M = documents.length;
// # topics
int K = 2; // 主题数目
// good values alpha = 2, beta = .5
double alpha = 2;
double beta = .5;
System.out.println("Latent Dirichlet Allocation using Gibbs Sampling.");
LdaGibbsSampler lda = new LdaGibbsSampler(documents, V);
lda.configure(10000, 2000, 100, 10);
lda.gibbs(K, alpha, beta);
double[][] theta = lda.getTheta();
double[][] phi = lda.getPhi();
System.out.println();
System.out.println();
System.out.println("Document--Topic Associations, Theta[d][k] (alpha="
+ alpha + ")");
System.out.print("d\\k\t");
for (int m = 0; m < theta[0].length; m++) {
System.out.print(" " + m % 10 + " ");
}
System.out.println();
for (int m = 0; m < theta.length; m++) {
System.out.print(m + "\t");
for (int k = 0; k < theta[m].length; k++) {
// System.out.print(theta[m][k] + " ");
System.out.print(shadeDouble(theta[m][k], 1) + " ");
}
System.out.println();
}
System.out.println();
System.out.println("Topic--Term Associations, Phi[k][w] (beta=" + beta
+ ")");
System.out.print("k\\w\t");
for (int w = 0; w < phi[0].length; w++) {
System.out.print(" " + w % 10 + " ");
}
System.out.println();
for (int k = 0; k < phi.length; k++) {
System.out.print(k + "\t");
for (int w = 0; w < phi[k].length; w++) {
// System.out.print(phi[k][w] + " ");
System.out.print(shadeDouble(phi[k][w], 1) + " ");
}
System.out.println();
}
// Let's inference a new document
int[] aNewDocument = {2, 2, 4, 2, 4, 2, 2, 2, 2, 4, 2, 2};
double[] newTheta = inference(alpha, beta, phi, aNewDocument);
for (int k = 0; k < newTheta.length; k++) {
// System.out.print(theta[m][k] + " ");
System.out.print(shadeDouble(newTheta[k], 1) + " ");
}
System.out.println();
}
static String[] shades = {" ", ". ", ": ", ":. ", ":: ",
"::. ", "::: ", ":::. ", ":::: ", "::::.", ":::::"};
static NumberFormat lnf = new DecimalFormat("00E0");
/**
* create a string representation whose gray value appears as an indicator
* of magnitude, cf. Hinton diagrams in statistics.
*
* @param d value
* @param max maximum value
* @return
*/
public static String shadeDouble(double d, double max) {
int a = (int) Math.floor(d * 10 / max + 0.5);
if (a > 10 || a < 0) {
String x = lnf.format(d);
a = 5 - x.length();
for (int i = 0; i < a; i++) {
x += " ";
}
return "<" + x + ">";
}
return "[" + shades[a] + "]";
}
}
要理解这段代码，还是要深入其定义的数学形式，统计学方面相关的分布知识。
参考：http://www.hankcs.com/nlp/lda-java-introduction-and-implementation.html
对于机器学习的很多知识点，目前先汇聚起来，后续需要应用时一个个专题研究。