哈尔滨工业大学深圳 模式识别 2017 考试重要知识点

λ(αi | ωj ) be the loss incurred for taking action αi when the state of nature is ωj. action αi assign the sample into any class-Conditional risk ∑===cj j j j i i x P x R 1)|()|()|(ωωαλα for i = 1,…,aSelect the action αi for which R(αi | x) is minimumR is minimum and R in this case is called the Bayes risk = best reasonable result that can be achieved!λij :loss incurred for deciding ωi when the true state of nature is ωjg i (x) = - R(αi | x)max. discriminant corresponds to min. riskg i (x) = P(ωi | x)max. discrimination corresponds to max. posteriorg i (x) ≡ p(x | ωi ) P(ωi ) g i (x) = ln p(x | ωi ) + ln P(ωi )问题由估计似然概率变为估计正态分布的参数问题极大似然估计和贝叶斯估计结果接近相同，但方法概念不同Please present the basic ideas of the maximum likelihood estimation method and Bayesian estimation method. When do these two methods have similar results ?请描述最大似然估计方法和贝叶斯估计方法的基本概念。

什么情况下两个方法有类似的结果？I．Maximum-likelihood view the parameters as quantities whose values are fixed but unknown. The best estimate of their value is defined to be the one that maximizes the probability of obtaining the samples actually observed.II．Bayesian methods view the parameters as random variables having some known prior distribution. Observation of the samples converts this to a posterior density, thereby revising our opinion about the true values of the parameters.III．Under the condition that the number of the training samples approaches to the infinity, the estimation of the mean obtained using Bayesian estimation method is almost identical to that obtained using the maximum likelihood estimation method.最小风险决策通常有一个更低的分类准确度相比于最小错误率贝叶斯决策。

然而，最小风险决策能够避免可能的高风险和损失。

贝叶斯参数估计方法。

Vectorize the samples.Calculation of the mean of all training samples.Calculation of the covariance matrixCalculation of eigenvectors and eigenvalue of the covariance matrix. Build the feature space.Feature extraction of all samples. Calculation the feature value of every sample. Calculation of the test sample feature value.Calculation of the samples of training samples like the above step.Find the nearest training sample as the result.Exercises1. How to use the prior and likehood to calculate the posterior ? What is the formula ? 怎么用先验概率和似然函数计算后验概率？公式是什么？P(ωj | x) = p(x | ωj ) . P(ωj ) / p(x)∑=1)(j P ω, ∑=1)|(x P j ω2. What’s the difference in the ideas of the minimum error Bayesian decision and minimum risk Bayesian decision? What’s the condition that makes the minimum error Bayesian decision identical to the minimum risk Bayesian decision?最小误差贝叶斯决策和最小风险贝叶斯决策的概念的差别是什么？什么情况下最小误差贝叶斯决策和最小风险贝叶斯决策是一致的（相同的）？答：在两类问题中，若有12222111λλλλ-=-，即所谓对称损失函数的情况，则这时最小风险的贝叶斯决策和最小误差的贝叶斯决策方法显然是一致的。

the minimum error Bayesian decision: to minimize the classification error of the Bayesian decision.the minimum risk Bayesian decision: to minimize the risk of the Bayesian decision.if R(α1 | x) < R(α2 | x) action α1: “decide ω1” is takenR(α1 | x) = λ11P(ω1 | x) + λ12P(ω2 | x)R(α2 | x) = λ21P(ω1 | x) + λ22P(ω2 | x)3. A person takes a lab test of nuclear radiation and the result is positive. The test returns a correct positive result in 99% of the cases in which the nuclear radiation is actually present, and a correct negative result in 95% of the cases in which the nuclear radiation is not present. Furthermore, 3% of the entire population are radioaetively eontaminated. Is this person eontaminated?一人在某实验室做了一次核辐射检测，结果是阳性的。

当核辐射真正存在时，检测结果21返回正确的阳性概率是99%；当核辐射不存在时，结果返回正确的阴性的概率是95%。

而且，所有被测人群中有3%的人确实被辐射污染了。

那么这个人被辐射污染了吗？ 答： 被辐射污染概率1()0.03P ω=未被辐射污染概率2()0.97P ω=X 表示阳性，X 表示阴性，则有如下结论：1(|)0.99P X ω=，2(|)0.95P X ω=。

则11121(|)()0.990.03(|)0.380.990.03(10.95)0.97(|)()i ii P X P P X P X P ωωωωω=⨯==≈⨯+-⨯∑ 21(|)1(|)0.62P X P X ωω=-=根据贝叶斯决策规则有：21(|)(|)P X P X ωω>所以这个人未被辐射污染。

4. Please present the basic ideas of the maximum likehood estimation method and Bayesianestimation method. When do these two methods have similar results ?请描述最大似然估计方法和贝叶斯估计方法的基本概念。

什么情况下两个方法有类似的结果？答：I. 设有一个样本集χ，要求我们找出估计量ˆθ，用来估计χ所属总体分布的某个真实参数θ使得带来的贝叶斯风险最小，这就是贝叶斯估计的概念。

(另一种说法：把待估计的参数看成是符合某种先验概率分布的随机变量；对样本进行观测的过程，就是把先验概率密度转化为后验概率密度，这样就利用样本的信息修正了对参数的初始估计值)II. 最大似然估计法的思想很简单：在已经得到试验结果的情况下，我们应该寻找使这个结果出现的可能性最大的那个θ作为真θ的估计。

III.在训练样本数目接近无穷时，使用贝叶斯估计方法获得的平均值估计几乎和使用最大似然估计的方法获得的平均值一样题外话：Prior + samplesI．Maximum-likelihood view the parameters as quantities whose vales are fixed but unknown. The best estimate of their value is defined to be the one that maximizes the probability of obtaining the samples actually observed.II．Bayesian methods view the parameters as random variables having some known prior distribution. Observation of the samples converts this to a posterior density, thereby revising our opinion about the true values of the parameters.III．Under the condition that the number of the training samples approaches to the infinity, the estimation of the mean obtained using Bayesian estimation method is almost identical to that obtained using the maximum likehood estimation method.5.Please present the nature of principal component analysis.请描述主成分分析法的本质答：主成分分析也称主分量分析，旨在利用降维的思想，把多指标转化为少数几个综合指标。