浅谈隐式马尔可夫模型 - A Brief Note of Hidden Markov Model (HMM)-优快云博客

本文链接：https://blog.youkuaiyun.com/philthinker/article/details/78630999

本文介绍了隐式马尔可夫模型（HMM），包括问题定义、贝叶斯决策理论、马尔可夫模型以及HMM的三个主要问题：评估、解码和训练。详细讨论了前向算法、后向算法、维特比算法以及baum-welch重估计方法，旨在帮助读者深入理解HMM在概率序列识别中的应用。

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文章目录

Introduction
HMM Problems and Solutions
Reference

Introduction

Problem Formulation

Now we talk about Hidden Markov Model. Well, what is HMM used for? Consider the following problem:

*Given an unknown observation: $O$ , recognize it as one of $N$ classes with minimum probability of error. *

So how to define the error and error probability?
Conditional Error: Given $O$ , the risk associated with deciding that it is a class $i$ event:
$R(S_{i} | O) = \sum_{j=1}^{N}e_{ij}P(S_{j} | O)$ where $P(S_{j} | O)$ is the probability of that $O$ is a class $S_{j}$ event and $e_{ij}$ is the cost of classifying a class $j$ event as a class $i$ event. $e_{ij}>0, e_{ii}=0$ .
Expected Error:
$\mathcal{E}=\int R(S(O) | O)p(O)dO$ where $S (O)$ is the decision made on $O$ based on a policy. Then the question can be considered as:

How should $S (O)$ be made to achieve minimum error probability? Or $P (S (O) ∣ O)$ is maximized?

Bayes Decision Theory

If we institute the policy: $S(O) = S_{i} = \arg\max_{S_{j}}P(S_{j} | O)$ then $R(S(O)|O)=\min_{S_{j}}R(S_{j} | O)$ . It is the so-called Maximum A Posteriori (MAP) decision. But how do we know $P(S_{j}|O), i=1,2,\dots,M$ for any $O$ ?

Markov Model

States : $S=\{S_{0}, S_{1},S_{2},\dots, S_{N}\}$
Transition probabilities : $P(q_{t}=S_{i} |q_{t-1}=S_{j})$
Markov Assumption:
$P(q_{t}=S_{i} |q_{t-1}=S_{j}, q_{t-1}=S_{k},\dots)=P(q_{t}=S_{i} |q_{t-1}=S_{j})=a_{ji}, \quad a_{ji}\geq 0, \sum_{i=1}^{N}a_{ji}=1,\forall j$