Bayesian input variable selection using posterior probabilities and expected utilities

贝叶斯信息准则 rmse
贝叶斯信息准则（Bayesian Information Criterion, BIC）是一种用于模型选择的统计量，常用于评估模型的拟合程度和复杂度。

BIC通过平衡模型的拟合优度和参数的数量，提供了一种可靠的方式来选择最佳的模型。

在使用BIC进行模型选择时，我们通常会比较不同模型的BIC值。

BIC的计算公式为BIC = n * ln(RMSE) + k * ln(n)，其中n是样本量，RMSE是模型的均方根误差，k是模型的参数个数。

BIC值越小，说明模型的拟合优度越好。

使用BIC可以避免过拟合问题。

过拟合是指模型过于复杂，过度拟合了训练数据，但在新数据上的预测效果却很差。

BIC考虑了模型的复杂度，并对参数个数给予了惩罚，因此可以有效地避免过拟合的发生。

BIC在实际应用中具有广泛的用途。

例如，在回归分析中，我们可以使用BIC来选择最佳的回归模型。

在聚类分析中，BIC可以帮助我们确定最佳的聚类数目。

在时间序列分析中，BIC可以用来选择最合适的模型来预测未来的值。

贝叶斯信息准则是一种重要的模型选择工具，可以帮助我们评估模型的拟合程度和复杂度。

通过使用BIC，我们可以选择最佳的模型，并避免过拟合问题的发生。

无论是在科学研究还是实际应用中，BIC
都发挥着重要的作用。

贝叶斯空间计量模型贝叶斯空间计量模型⼀、采⽤贝叶斯空间计量模型的原因残差项可能存在异⽅差，⽽ML 估计⽅法的前提是同⽅差，因此，当残差项存在异⽅差时，采⽤ML ⽅法估计出的参数结果不具备稳健性。

⼆、贝叶斯空间计量模型的估计⽅法（⼀）待估参数对于空间计量模型（以空间⾃回归模型为例）ερ+=Wy y假设残差项是异⽅差的，即),,(),0(~212n v v v diag V V N =σε上述模型需要估计的参数有：n v v v 21σρ共计n+2个参数，存在⾃由度问题，难以进⾏参数检验。

为此根据⼤数定律，增加了新的假设：v i 服从⾃由度为r 的卡⽅分布。

如此以来，待估参数将减少为3个。

（⼆）参数估计⽅法采⽤MCMC（Markov Chain Monte Carlo）参数估计思想，具体的抽样⽅法选择吉布斯抽样⽅法（Gibbs sampling approach）在随意给定待估参数⼀个初始值之后，开始⽣成参数的新数值，并根据新数值⽣成其他参数的新数值，如此往复，对每⼀个待估参数，将得到⼀组⽣成的数值，根据该组数值，计算其均值，即为待估参数的贝叶斯估计值。

三、贝叶斯空间计量模型的类型空间⾃回归模型far_g()空间滞后模型（空间回归⾃回归混合模型）sar_g()空间误差模型sem_g()⼴义空间模型（空间⾃相关模型）sac_g()四、贝叶斯空间模型与普通空间模型的选择标准⾸先按照参数显著性，以及极⼤似然值，确定普通空间计量模型的具体类型，之后对于该确定的类型，再判断是否需要进⼀步采⽤贝叶斯估计⽅法。

标准⼀：对普通空间计量模型的残差项做图，观察参数项是否是正态分布，若⾮正态分布，则考虑使⽤贝叶斯⽅法估计。

技巧：r=30的贝叶斯估计等价于普通空间计量模型估计，此时可以做出v的分布图，观察其是否基本等于1，若否，则应采⽤贝叶斯估计⽅法。

标准⼆：若按标准⼀发现存在异⽅差，采⽤贝叶斯估计后，如果参数结果与普通空间计量⽅法存在较⼤差异，则说明采⽤贝叶斯估计是必要的。

贝叶斯分类的优缺点
贝叶斯分类（Bayesian classification）是一种基于贝叶斯定理的分类方法，该方法通过计算给定特征的条件下，目标变量的概率来进行分类预测。

贝叶斯分类的优点和缺点如下：
优点：
1. 简单有效：贝叶斯分类器是一种非常简单的分类方法，易于理解和实现。

它只需要估计类别的先验概率和给定各个特征的条件概率，计算简单快速。

2. 能够处理小样本问题：由于贝叶斯分类器使用概率模型，可以在有限的样本情况下进行有准确性的估计。

3. 对缺失数据不敏感：贝叶斯分类器在估计条件概率时，对缺失数据不敏感，可以处理特征中存在缺失值的情况。

4. 适用于多分类问题：贝叶斯分类器可以直接应用于多分类问题，不需要额外的转换或修改。

缺点：
1. 对特征独立性的假设：贝叶斯分类器假设所有特征之间是独立的，即特征之间没有相互关系。

在实际应用中，这个假设并不总是成立，特征之间的依赖关系会影响分类准确性。

2. 数据较大时计算复杂：贝叶斯分类器需要计算每个特征的条件概率，当特征数量较大时，计算量会显著增加，导致计算复杂性提高。

3. 需要足够的训练样本：贝叶斯分类器的准确性依赖于训练数据，特别是在特征维度较高或数据噪声较大的情况下，需要足够的训练样本以获得可靠的概率估计。

4. 对输入数据分布的假设：贝叶斯分类器假设输入数据符合特
定的分布（如高斯分布），如果输入数据的分布与其假设不匹配，可能会导致较低的分类准确性。

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3.2 The First Stage of a Bayesian Model . . . . . . . . . . . . . 42 3.3 The Second Stage of the Bayesian Model: The Prior . . . . 55 3.4 Using the Data to Update the Prior: The Posterior Distri-
4.3 Posterior Predictive Distributions . . . . . . . . . . . . . . . 96
4.4 Case Study(cont.): Proportion of Heavy Sleepers . . . . . . 101
Previous Next First Last Back Forward
7.4 Case Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
Previous Next First Last Back Forward
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7.4.1 7.4.2
Example 1: The normal model for New York marathon data(See Section 4.2, Albert) . . . . . . . . . . . . . . 184 Example 2: A Multinomial Model for US election poll data . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
10.3 Generalized Linear Models . . . . . . . . . . . . . . . . . . . 367

心理科学进展 2018, Vol. 26, No. 6, 951–965 Advances in Psychological ScienceDOI: 10.3724/SP.J.1042.2018.00951951·研究方法(Research Method)·贝叶斯因子及其在JASP 中的实现胡传鹏1,2 孔祥祯3 Eric-Jan Wagenmakers 4 Alexander Ly 4,5 彭凯平1(1清华大学心理学系, 北京 100084) (2 Neuroimaging Center, Johannes Gutenberg University Medical Center, 55131 Mainz, Germany) (3 Language and Genetics Department, Max Planck Institute for Psycholinguistics, 6500 AH Nijmegen, The Netherlands) (4 Department of Psychological Methods, University of Amsterdam, 1018 VZ Amsterdam, The Netherlands) (5 Centrum Wiskunde & Informatica, 1090 GB Amsterdam, The Netherlands) 摘 要 统计推断在科学研究中起到关键作用, 然而当前科研中最常用的经典统计方法——零假设检验(Null hypothesis significance test, NHST)却因难以理解而被部分研究者误用或滥用。

有研究者提出使用贝叶斯因子(Bayes factor)作为一种替代和(或)补充的统计方法。

贝叶斯因子是贝叶斯统计中用来进行模型比较和假设检验的重要方法, 其可以解读为对零假设H 0或者备择假设H 1的支持程度。

贝叶斯统计方法贝叶斯统计方法是一种基于贝叶斯定理的统计分析方法，它在各个领域中被广泛应用。

本文将介绍贝叶斯统计方法的原理、应用以及优势。

一、贝叶斯统计方法的原理贝叶斯统计方法基于贝叶斯定理，该定理描述了如何根据已知的先验知识和新的数据进行推理和预测。

其基本公式如下：P(A|B) = (P(B|A) * P(A)) / P(B)其中，P(A|B)表示在已知B发生的前提下，A发生的概率；P(B|A)表示在已知A发生的前提下，B发生的概率；P(A)和P(B)分别表示A 和B分别独立发生的概率。

贝叶斯统计方法通过更新先验概率得到后验概率，从而更准确地估计参数或预测结果。

二、贝叶斯统计方法的应用1. 机器学习中的分类问题贝叶斯统计方法在机器学习中的分类任务中得到广泛应用。

通过构建贝叶斯分类器，可以根据已知的先验概率和数据集训练结果，对新的样本进行分类。

2. 自然语言处理中的文本分类贝叶斯统计方法在文本分类任务中也有着重要应用。

通过构建朴素贝叶斯分类器，可以根据文本的词频信息将其分类到不同的类别中。

3. 医学诊断中的预测贝叶斯统计方法在医学诊断中的预测也得到了广泛应用。

通过结合病人的先验信息和检测结果，可以计算患病的后验概率，从而辅助医生进行准确的诊断。

三、贝叶斯统计方法的优势1. 考虑先验知识贝叶斯统计方法通过引入先验知识，能够较好地处理具有先验信息的问题。

相比之下，频率统计方法仅根据样本数据进行推断，无法很好地利用已有的先验概率信息。

2. 灵活性高贝叶斯统计方法可以适应不同的问题和数据情况。

通过不同的先验分布和模型选择，可以灵活地对参数进行估计和预测。

3. 适用于小样本情况贝叶斯统计方法在小样本情况下仍能表现出良好的性能。

由于引入了先验知识，能够在样本量较小的情况下提供相对可靠的推断结果。

四、总结贝叶斯统计方法基于贝叶斯定理，通过更新先验概率得到后验概率，可用于各个领域中的数据分析、模型估计和预测问题。

贝叶斯回归公式贝叶斯回归是一种用于解决回归问题的统计模型。

它基于贝叶斯定理，通过计算后验概率来预测目标变量的值。

贝叶斯回归公式可以描述为：后验概率 = (先验概率 * 似然函数) / 证据在贝叶斯回归中，我们首先需要定义一个先验概率分布，它表示对目标变量的先前知识或信念。

然后，我们根据给定的数据集计算似然函数，它表示观测到的数据在不同参数值下的可能性。

最后，通过将先验概率与似然函数相乘，并除以证据（归一化常数），可以得到后验概率。

贝叶斯回归公式的应用非常广泛。

它可以用于解决各种回归问题，例如房价预测、销售量预测等。

与传统的最小二乘法不同，贝叶斯回归可以通过引入先验概率来处理过拟合问题，并提供更加准确的预测结果。

在贝叶斯回归中，先验概率的选择非常重要。

先验概率可以基于领域知识、经验或其他先前信息来确定。

如果没有先验信息可用，可以选择一个非具体的先验概率分布，如高斯分布。

然后，通过观察数据，根据贝叶斯定理来更新先验概率，得到后验概率。

贝叶斯回归还可以用于处理多个输入变量的情况。

在这种情况下，可以使用多元线性回归模型，并将贝叶斯回归公式推广到多维空间。

通过引入多个参数和对应的先验概率，可以建立一个更加灵活和准确的模型。

贝叶斯回归的优点之一是可以提供预测结果的不确定性估计。

通过计算后验概率分布，可以得到目标变量的概率分布，而不仅仅是一个点估计。

这对于决策制定者来说非常有价值，因为他们可以了解预测的可靠性，并相应地采取行动。

贝叶斯回归还可以进行模型选择和变量选择。

通过比较不同模型的后验概率，可以选择最合适的模型。

同样，通过比较不同变量组合的后验概率，可以选择最相关的变量。

虽然贝叶斯回归在理论上是非常有吸引力的，但在实践中也存在一些挑战。

首先，计算后验概率需要对参数空间进行积分，这在高维空间中是非常困难的。

为了克服这个问题，可以使用近似方法，如马尔可夫链蒙特卡洛（MCMC）方法。

贝叶斯回归的性能也受到先验概率的选择和参数设置的影响。

贝叶斯优化算法高斯过程贝叶斯优化算法和高斯过程在机器学习中被广泛应用，用于优化复杂函数的参数。

本文将介绍贝叶斯优化算法和高斯过程的基本原理、应用场景以及其优点和局限性。

一、贝叶斯优化算法的原理贝叶斯优化算法是一种基于贝叶斯统计和序列模型的优化方法。

它通过建立一个先验模型和一个观测模型来推断待优化函数的最优解。

具体来说，它通过不断地选择下一个样本点进行评估来逐步优化函数的参数，直到找到全局最优解或达到一定的停止准则。

二、高斯过程的原理高斯过程是一种概率模型，用于对随机变量的概率分布进行建模。

它假设任意有限个变量的线性组合服从多元高斯分布。

在贝叶斯优化算法中，高斯过程被用来建立待优化函数的先验模型。

通过观测已有的样本点，可以利用高斯过程进行预测，从而选择下一个最有可能是最优解的样本点进行评估。

三、贝叶斯优化算法的应用场景贝叶斯优化算法在很多领域都有广泛的应用。

例如，在超参数优化中，可以使用贝叶斯优化算法来选择最优的超参数组合，从而提高模型的性能。

在自动化机器学习中，贝叶斯优化算法可以自动选择合适的模型和算法，并进行参数调优。

此外，贝叶斯优化算法还可以应用于网络流量优化、物理实验设计等领域。

四、高斯过程在贝叶斯优化中的优点高斯过程作为一种非参数模型，具有很强的灵活性和适应性。

它可以根据观测数据自适应地调整模型的复杂度，并能够提供对未知函数的预测和不确定性的估计。

同时，高斯过程还具有数学上的优良性质，如可微性和闭式解等，使得贝叶斯优化算法更加高效和稳定。

五、贝叶斯优化算法的局限性虽然贝叶斯优化算法在很多问题上表现出色，但它也存在一些局限性。

首先，贝叶斯优化算法对待优化函数的光滑性和凸性有一定的要求。

当函数具有峰值或存在多个局部最优解时，贝叶斯优化算法可能无法找到全局最优解。

其次，贝叶斯优化算法在高维空间中的表现较差，因为样本点的评估成本很高，导致算法的收敛速度较慢。

六、总结贝叶斯优化算法和高斯过程是一对强力组合，在机器学习中被广泛应用于优化复杂函数的参数。

贝叶斯算法简单介绍贝叶斯算法是一种基于统计学的算法，主要用于机器学习与人工智能领域中的分类问题。

该算法是在 18 世纪由英国数学家托马斯·贝叶斯发明的，因此得名贝叶斯算法。

在机器学习领域中，贝叶斯算法被用于解决分类问题。

分类问题就是将一个实例归类到已有类别中的某一个类别中，如将一条邮件归类为垃圾邮件或非垃圾邮件。

贝叶斯算法的基本思想是：给定一个分类问题和一组特征，通过求解特征的条件概率来得到每个类别的概率，从而将实例分到概率最大的那个类别中。

在贝叶斯算法中，最重要的是先验概率和后验概率。

先验概率是指在没有任何与特征相关的信息时，每个类别的概率。

例如，在分类汉字的问题中，让我们假设“大” 字比“小” 字常见，这样我们就可以认为“大” 字的先验概率比“小” 字的先验概率高。

后验概率是基于输入数据的特征，通过学习得出的概率。

例如，当给出一个汉字时，通过学习得出该字是“大” 字的后验概率。

通过计算先验概率和后验概率，就得到了分类问题的最终概率。

下面我们来看一个具体的例子，假设我们要通过贝叶斯算法判断一个邮箱中的邮件是否是垃圾邮件。

我们可以将邮件的内容和标题等相关特征看成先验概率，将垃圾邮件和非垃圾邮件看成后验概率，应用贝叶斯公式进行计算。

具体步骤如下：首先，我们需要收集一些已知类别的邮件数据，将其分为两个类别：垃圾邮件和非垃圾邮件。

然后，我们需要对每个单词进行分析，看它们与垃圾邮件和非垃圾邮件的关系。

例如，“买药”这个词汇就与垃圾邮件有强关系，而“会议”这个词汇就与非垃圾邮件有强关系。

接下来，我们将每个单词与它们在垃圾邮件和非垃圾邮件中的出现次数进行记录。

这个过程中，我们需要使用平滑处理的技巧，避免数据稀疏问题。

之后，通过贝叶斯公式，我们可以得到该邮件为垃圾邮件的概率，也可以得到非垃圾邮件的概率。

根据这些概率，我们可以将邮件进行分类，并进行后续的处理。

当然，贝叶斯算法并不仅仅适用于垃圾邮件分类问题，还可以应用于医学诊断、自然语言处理、金融风险管理等领域。

贝叶斯公式的原理与应用1. 贝叶斯公式的原理贝叶斯公式是统计学中一种经典的概率计算方法。

它是由英国数学家托马斯·贝叶斯（Thomas Bayes）发现并发展起来的，被广泛应用于机器学习、自然语言处理、垃圾邮件过滤等领域。

贝叶斯公式的原理基于条件概率的定义，利用已知的信息来计算未知事件发生的概率。

贝叶斯公式的原理可以表示为：\[ P(A|B) = \frac{P(B|A) \cdot P(A)}{P(B)} \]其中，P(A)和P(B)分别表示事件A和事件B的概率，P(A|B)表示在事件B发生的条件下事件A发生的概率，P(B|A)表示在事件A发生的条件下事件B发生的概率。

2. 贝叶斯公式的应用贝叶斯公式广泛应用于各个领域，包括机器学习、自然语言处理、垃圾邮件过滤等。

下面介绍一些实际应用案例。

2.1. 垃圾邮件过滤垃圾邮件过滤是贝叶斯公式的经典应用之一。

通过分析已知的垃圾邮件和非垃圾邮件的特征，可以计算出在给定的特征条件下，某封邮件是垃圾邮件的概率。

具体步骤如下：1.收集一组已知的垃圾邮件和非垃圾邮件，并提取它们的特征，比如邮件中的关键词、发件人等信息。

2.计算垃圾邮件和非垃圾邮件的概率P(Spam)和P(Non-spam)。

3.对于待分类的邮件，计算在垃圾邮件和非垃圾邮件的条件下，它是垃圾邮件的概率P(Spam|Email)和P(Non-spam|Email)。

4.根据计算得到的概率，将待分类的邮件判定为垃圾邮件或非垃圾邮件。

2.2. 文本分类贝叶斯公式在文本分类中也有广泛的应用。

文本分类是将一段给定的文本划分到某个预定义的类别中。

使用贝叶斯公式可以计算某个文本属于某个类别的概率，从而进行文本分类。

具体步骤如下：1.收集一组已知类别的文本样本，并提取它们的特征，比如词频和关键词等信息。

2.计算每个类别的先验概率P(C)，表示每个类别的出现概率。

3.计算每个特征在各个类别下的条件概率P(Feature|C)，表示在每个类别下特征出现的概率。

贝叶斯优化算法在强化学习中的应用贝叶斯优化是一种寻找全局最优解的优化算法。

它通过探索已知的样本点，并根据这些样本点来估计函数的后验概率分布，以找到最优解。

贝叶斯优化算法在强化学习中的应用越来越受到关注，可以用于处理高维、复杂的问题，例如机器人控制、自动驾驶、推荐系统等等。

在强化学习中，智能体通过与环境的交互来学习行为策略，以最大化累积奖励或最小化累积成本。

然而，很多强化学习问题中，环境的动态不确定，导致观察值和奖励信号都存在噪声，这使得任务更加困难。

在这种情况下，如何在有限的时间内找到最优策略是一个挑战。

贝叶斯优化可以很好地解决这个问题。

它使用高斯过程回归来建立模型，将已知的样本点与目标函数映射起来。

然后，它使用贝叶斯推断来估计函数的后验分布。

在每次迭代中，它根据估计的后验分布选择下一次样本点进行探索，以最大化收益。

这使得算法可以在尽可能快地收敛到全局最优解的同时，尽可能地减少探索代价。

一个经典的例子是在推荐系统中应用贝叶斯优化算法。

常见的做法是使用随机搜索或网格搜索来进行超参数调整。

但是，这些方法通常需要大量的计算和实验，并且无法保证找到全局最优解。

相比之下，贝叶斯优化算法可以在较少的实验次数内找到最优解，并且可以保证显著的性能提升。

一些研究者将贝叶斯优化算法和深度强化学习相结合，以解决高维、复杂的问题。

他们提出了一种基于贝叶斯优化的区域搜索算法，它可以自动选择最重要的状态变量并相应地调整值函数的分辨率。

这种算法在机器人控制和自动驾驶领域已经获得了一定的成功。

然而，贝叶斯优化算法也存在一些局限性。

首先，它需要合理的先验知识和正确的模型选择，否则可能会导致不良的推断结果。

其次，计算开销较大，在高维问题中可能需要大量的时间和计算资源。

最后，贝叶斯优化算法不太适用于动态环境下的任务，因为它难以处理环境变化带来的不确定性。

总之，贝叶斯优化算法在强化学习中的应用具有广泛的前景。

它可以在有限的时间和资源内找到最优解，同时还可以保证稳定性和可靠性。

⽣物信息学主要英⽂术语及释义（续完）These substitutions may be found in an amino acid substitution matrix such as the Dayhoff PAM and Henikoff BLOSUM matrices. Columns in the alignment that include gaps are not scored in the calculation. Perceptron（感知器，模拟⼈类视神经控制系统的图形识别机) A neural network in which input and output states are directly connected without intervening hidden layers. PHRED （⼀种⼴泛应⽤的原始序列分析程序，可以对序列的各个碱基进⾏识别和质量评价） A widely used computer program that analyses raw sequence to produce a 'base call' with an associated 'quality score' for each position in the sequence. A PHRED quality score of X corresponds to an error probability of approximately 10-X/10. Thus, a PHRED quality score of30 corresponds to 99.9% accuracy for the base call in the raw read. PHRAP （⼀种⼴泛应⽤的原始序列组装程序） A widely used computer program that assembles raw sequence into sequence contigs and assigns to each position in the sequence an associated 'quality score', on the basis of the PHRED scores of the raw sequence reads. A PHRAP quality score of X corresponds to an error probability of approximately 10-X/10. Thus, a PHRAP quality score of 30 corresponds to 99.9% accuracy for a base in the assembled sequence. Phylogenetic studies（系统发育研究） PIR （主要蛋⽩质序列数据库之⼀，翻译⾃GenBank） A database of translated GenBank nucleotide sequences. PIR is a redundant (see Redundancy) protein sequence database. The database is divided into four categories: PIR1 - Classified and annotated. PIR2 - Annotated. PIR3 -Unverified. PIR4 - Unencoded or untranslated. Poisson distribution（帕松分布） Used to predict the occurrence of infrequent events over a long period of time 143or when there are a large number of trials. In sequence analysis, it is used to calculate the chance that one pair of a large number of pairs of unrelated sequences may give a high local alignment score. Position-specific scoring matrix (PSSM)（特定位点记分矩阵，PSI-BLAST等搜索程序使⽤） The PSSM gives the log-odds score for finding a particular matching amino acid in a target sequence. Represents the variation found in the columns of an alignment of a set of related sequences. Each subsequent matrix column corresponds to the next column in the alignment and each row corresponds to a particular sequence character (one of four bases in DNA sequences or 20 amino acids in protein sequences). Matrix values are log odds scores obtained by dividing the counts of the residue in the alignment, dividing by the expected number of counts based on sequence composition, and converting the ratio to a log score. The matrix is moved along sequences to find similar regions by adding the matching log odds scores and looking for high values. There is no allowance for gaps. Also called a weight matrix or scoring matrix. Posterior (Bayesian analysis) A conditional probability based on prior knowledge and newly uated relationships among variables using Bayes rule. See also Bayes rule. Prior (Bayesian analysis) The expected distribution of a variable based on previous data. Profile（分布型） A matrix representation of a conserved region in a multiple sequence alignment that allows for gaps in the alignment. The rows include scores for matching sequential columns of the alignment to a test sequence. The columns include substitution scores for amino acids and gap penalties. See also PSSM. Profile hidden Markov model（分布型隐马尔可夫模型） A hidden Markov model of a conserved region in a multiple sequence alignment that includes gaps and may be used to search new sequences for similarity to the aligned sequences. Proteome（蛋⽩质组） The entire collection of proteins that are encoded by the genome of an organism. Initially the proteome is estimated by gene prediction and annotation methods but eventually will be revised as more information on the sequence of the expressed genes is obtained. Proteomics （蛋⽩质组学） Systematic analysis of protein expression_r of normal and diseased tissues that involves the separation, identification and characterization of all of the proteins in an organism. Pseudocounts Small number of counts that is added to the columns of a scoring matrix to increase the variability either to avoid zero counts or to add more variation than was found in the sequences used to produce the matrix. 144PSI-BLAST （BLAST系列程序之⼀） Position-Specific Iterative BLAST. An iterative search using the BLAST algorithm. A profile is built after the initial search, which is then used in subsequent searches. The process may be repeated, if desired with new sequences found in each cycle used to refine the profile. Details can be found in this discussion of PSI-BLAST. (Altschul et al.) PSSM （特定位点记分矩阵） See position-specific scoring matrix and profile. Public sequence databases （公共序列数据库，指GenBank、EMBL和DDBJ） The three coordinated international sequence databases: GenBank, the EMBL data library and DDBJ. Q20 (Quality score 20) A quality score of > or = 20 indicates that there is less than a 1 in 100 chance that the base call is incorrect. These are consequently high-quality bases. Specifically, the quality value "q" assigned to a basecall is defined as: q = -10 x log10(p) where p is the estimated error probability for that basecall. Note that high quality values correspond to low error probabilities, and conversely. Quality trimming This is an algorithm which uses a sliding window of 50 bases and trims from the 5' end of the read followed by the 3' end. With each window, the number of low quality (10 or less) bases is determined. If more than 5 bases are below the threshold quality, the window is incremented by one base and the process is repeated. When the low quality test fails, the position where it stopped is recorded. The parameters for window length low quality threshold and number of low quality bases tolerated are fixed. The positions of the 5' and 3' boundaries of the quality region are noted in the plot of quality values presented in the" Chromatogram Details" report. Query （待查序列/搜索序列） The input sequence (or other type of search term) with which all of the entries in a database are to be compared. Radiation hybrid (RH) map （辐射杂交图谱） A genome map in which STSs are positioned relative to one another on the basis of the frequency with which they are separated by radiation-induced breaks. The frequency is assayed by analysing a panel of human–hamster hybrid cell lines, each produced by lethally irradiating human cells and fusing them with recipient hamster cells such that each carries a collection of human chromosomal fragments. The unit of distance is centirays (cR), denoting a 1% chanceof a break occuring between two loci Raw Score （初值，指最初得到的联配值S） The score of an alignment, S, calculated as the sum of substitution and gap scores. Substitution scores are given by a look-up table (see PAM, BLOSUM). Gap scores are typically calculated as the sum of G, the gap opening penalty 145and L, the gap extension penalty. For a gap of length n, the gap cost would be G+Ln. The choice of gap costs, G and L is empirical, but it is customary to choose a high value for G (10-15)and a low value for L (1-2). Raw sequence （原始序列/读胶序列） Individual unassembled sequence reads, produced by sequencing of clones containing DNA inserts. Receiver operator characteristic The receiver operator characteristic (ROC) curve describes the probability that a test will correctly declare the condition present against the probability that the test will declare the condition present when actually absent. This is shown through a graph of the tesls sensitivity against one minus the test specificity for different possible threshold values. Redundancy （冗余） The presence of more than one identical item represents redundancy. In bioinformatics, the term is used with reference to the sequences in a sequence database. If a database is described as being redundant, more than one identical (redundant) sequence may be found. If the database is said to be non-redundant (nr), the database managers have attempted to reduce the redundancy. The term is ambiguous with reference to genetics, and as such, the degree of non-redundancy varies according to the database manager's interpretation of the term. One can argue whether or not two alleles of a locus defines the limit of redundancy, or whether the same locus in different, closely related organisms constitutes redundency. Non-redundant databases are, in some ways, superior, but are less complete. These factors should be taken into consideration when selecting a database to search. Regular expression_rs This computational tool provides a method for expressing the variations found in a set of related sequences including a range of choices at one position, insertions, repeats, and so on. For example, these expression_rs are used to characterize variations found in protein domains in the PROSITE catalog. Regularization A set of techniques for reducing data overfitting when training a model. See also Overfitting. Relational database（关系数据库）Organizes information into tables where each column represents the fields of informa-tion that can be stored in a single record. Each row in the table corresponds to a single record. A single database can have many tables and a query language is used to access the data. See also Object-oriented database. Scaffold （⽀架，由序列重叠群拼接⽽成） The result of connecting contigs by linking information from paired-end reads from plasmids, paired-end reads from BACs, known messenger RNAs or other sources. The contigs in a scaffold are ordered and oriented with respect to one another. 146 Scoring matrix（记分矩阵） See Position-specific scoring matrix. SEG （⼀种蛋⽩质程序低复杂性区段过滤程序） A program for filtering low complexity regions in amino acid sequences. Residues that have been masked are represented as "X" in an alignment. SEG filtering is performed by default in the blastp subroutine of BLAST 2.0. (Wootton and Federhen) Selectivity (in database similarity searches)（数据库相似性搜索的选择准确性） The ability of a search method to locate members of a protein family without making a false-positive classification of members of other families. Sensitivity (in database similarity searches)（数据库相似性搜索的灵敏性） The ability of a search method to locate as many members of a protein family as possi-ble, including distant members of limited sequence similarity. Sequence Tagged Site （序列标签位点） Short cDNA sequences of regions that have been physically mapped. STSs provide unique landmarks, or identifiers, throughout the genome. Useful as a framework for further sequencing. Significance（显著⽔平） A significant result is one that has not simply occurred by chance, and therefore is prob-ably true. Significance levels show how likely a result is due to chance, expressed as a probability. In sequence analysis, the significance of an alignment score may be calcu-lated as the chance that such a score would be found between random or unrelated sequences. See Expect value. Similarity score (sequence alignment) （相似性值） Similarity means the extent to which nucleotide or protein sequences are related. The extent of similarity between two sequences can be based on percent sequence identity and/or conservation. In BLAST similarity refers to a positive matrix score. The sum of the number of identical matches and conservative (high scoring) substitu-tions in a sequence alignment divided by the total number of aligned sequence charac-ters. Gaps are usually ignored. Simulated annealing A search algorithm that attempts to solve the problem of finding global extrema. The algorithm was inspired by the physical cooling process of metals and the freezing process in liquids where atoms slow down in movement and line up to form a crystal. The algorithm traverses the energy levels of a function, always accepting energy levels that are smaller than previous ones, but sometimes accepting energy levels that are greater, according to the Boltzmann probability distribution. Single-linkage cluster analysis An analysis of a group of related objects, e.g., similar proteins in different genomes to identify both close and more distant relationships, represented on a tree or dendogram. The method joins the most closely related pairs by the neighbor-joining algorithm by representing these pairs as outer branches on 147the tree. More distant objects are then pro-gressively added to lower tree branches. The method is also used to predict phylogenet-ic relationships by distance methods. See also Hierarchical clustering, Neighbor-joining method. Smith-Waterman algorithm（Smith-Waterman算法） Uses dynamic programming to find local alignments between sequences. The key fea-ture is that all negative scores calculated in the dynamic programming matrix are changed to zero in order to avoid extending poorly scoring alignments and to assist in identifying local alignments starting and stopping anywhere with the matrix. SNP （单核苷酸多态性） Single nucleotide polymorphism, or a single nucleotide position in the genome sequence for which two or more alternative alleles are present at appreciable frequency (traditionally, at least 1%) in the human population. Space or time complexity（时间或空间复杂性） An algorithms complexity is the maximum amount of computer memory or time required for the number of algorithmic steps to solve a problem. Specificity (in database similarity searches)（数据库相似性搜索的特异性） The ability of a search method to locate members of one protein family, including dis-tantly related members. SSR （简单序列重复） Simple sequence repeat, a sequence consisting largely of a tandem repeat of a specific k-mer (such as (CA)15). Many SSRs are polymorphic and have been widely used in genetic mapping. Stochastic context-free grammar A formal representation of groups of symbols in different parts of a sequence; i.e., not in the same context. An example is complementary regions in RNA that will form sec-ondary structures. The stochastic feature introduces variability into such regions. Stringency Refers to the minimum number of matches required within a window. See also Filtering. STS （序列标签位点的缩写） See Sequence Tagged Site Substitution （替换） The presence of a non-identical amino acid at a given position in an alignment. If the aligned residues have similar physico-chemical properties the substitution is said to be "conservative". Substitution Matrix （替换矩阵） A substitution matrix containing values proportional to the probability that amino acid i mutates into amino acid j for all pairs of amino acids. such matrices are constructed by assembling a large and diverse sample of verified pairwise alignments of amino acids. If the sample is large enough to be statistically significant, the resulting matrices should reflect the true probabilities of mutations occuring through a period of evolution. 148Sum of pairs method Sums the substitution scores of all possible pair-wise combinations of sequence charac-ters in one column of a multiple sequence alignment. SWISS-PROT （主要蛋⽩质序列数据库之⼀） A non-redundant (See Redundancy) protein sequence database. Thoroughly annotated and cross referenced. A subdivision is TrEMBL. Synteny The presence of a set of homologous genes in the same order on two genomes. Threading In protein structure prediction, the aligning of the sequence of a protein of unknown structure with a known three-dimensional structure to determine whether the amino acid sequence is spatially and chemically compatible with that structure. TrEMBL （蛋⽩质数据库之⼀，翻译⾃EMBL） A protein sequence database of Translated EMBL nucleotide sequences. Uncertainty（不确定性） From information theory, a logarithmic measure of the average number of choices that must be made for identification purposes. See also Information content. Unified Modeling Language (UML) A standard sanctioned by the Object Management Group that provides a formal nota-tion for describing object-oriented design. UniGene （⼈类基因数据库之⼀） Database of unique human genes, at NCBI. Entries are selected by near identical presence in GenBank and dbEST databases. The clusters of sequences produced are considered to represent a single gene. Unitary Matrix （⼀元矩阵） Also known as Identity Matrix.A scoring system in which only identical characters receive a positive score. URL（统⼀资源定位符） Uniform resource locator. Viterbi algorithm Calculates the optimal path of a sequence through a hidden Markov model of sequences using a dynamic programming algorithm. Weight matrix See Position-specifc scoring matrix.。

贝叶斯判别法一、引言贝叶斯判别法（Bayesian Discriminant Analysis）是一种基于贝叶斯定理的统计学习方法。

它的核心思想是利用样本数据来估计各个类别的先验概率和条件概率密度函数，然后根据贝叶斯定理计算后验概率，从而实现分类。

二、基本原理1. 贝叶斯定理贝叶斯定理是统计学中一个重要的公式，它描述了在已知先验概率的情况下，如何根据新的观测数据来更新对事件发生概率的估计。

具体地说，设A和B是两个事件，则：P(A|B) = P(B|A) * P(A) / P(B)其中P(A|B)表示在已知事件B发生的前提下，事件A发生的条件概率；P(B|A)表示在已知事件A发生的前提下，事件B发生的条件概率；P(A)和P(B)分别为事件A和事件B的先验概率。

2. 贝叶斯判别法贝叶斯判别法是一种基于贝叶斯定理进行分类的方法。

假设有K个类别C1,C2,...,CK，每个类别Ci对应一个条件概率密度函数f(x|Ci)，其中x为样本特征向量。

给定一个新的样本x，我们需要将其归为某个类别中。

根据贝叶斯定理，可以计算出后验概率P(Ci|x)，即在已知样本特征向量x的前提下，该样本属于类别Ci的概率。

具体地说：P(Ci|x) = P(x|Ci) * P(Ci) / P(x)其中P(x|Ci)表示在已知类别Ci的前提下，样本特征向量x的条件概率密度函数；P(Ci)表示类别Ci的先验概率；P(x)表示样本特征向量x的边缘概率密度函数。

根据贝叶斯判别法，将新样本x归为后验概率最大的那个类别中，即：argmax(P(Ci|x)) = argmax(P(x|Ci)*P(Ci))三、分类器构建1. 参数估计贝叶斯判别法需要估计各个类别的先验概率和条件概率密度函数。

其中先验概率可以通过训练集中各个类别出现次数占总数比例来估计。

而条件概率密度函数则需要根据训练集中各个类别对应的样本特征向量来进行估计。

常见的条件概率密度函数包括高斯分布、多项式分布和伯努利分布等。

多层结构方程模型贝叶斯估计下载温馨提示：该文档是我店铺精心编制而成，希望大家下载以后，能够帮助大家解决实际的问题。

文档下载后可定制随意修改，请根据实际需要进行相应的调整和使用，谢谢!并且，本店铺为大家提供各种各样类型的实用资料，如教育随笔、日记赏析、句子摘抄、古诗大全、经典美文、话题作文、工作总结、词语解析、文案摘录、其他资料等等，如想了解不同资料格式和写法，敬请关注!Download tips: This document is carefully compiled by the editor. I hope that after you download them, they can help yousolve practical problems. The document can be customized and modified after downloading, please adjust and use it according to actual needs, thank you!In addition, our shop provides you with various types of practical materials, such as educational essays, diary appreciation, sentence excerpts, ancient poems, classic articles, topic composition, work summary, word parsing, copy excerpts,other materials and so on, want to know different data formats and writing methods, please pay attention!多层结构方程模型（MSEM）是一种结合了结构方程模型（SEM）和层次模型的统计分析方法，可以同时考虑不同层次的变量之间的关系。

bic贝叶斯信息准则贝叶斯信息准则（Bayesian Information Criterion, BIC）是一种经常用来评估模型的选择的统计量。

它基于贝叶斯定理，是对模型复杂度和模型涵盖数据的平衡考虑，可以用来比较不同复杂度和参数数量的模型对数据的拟合效果。

BIC在模型选择的学科领域中很受欢迎，因为它既考虑了模型的拟合程度，又考虑了模型的复杂性，从而对于在数据集上通常表现最好的简洁模型进行特别的鼓励。

因此，BIC可以解决模型选择中的粘滞问题，它可以在进行多模型比较的时候提供一个更客观的准则，同时提供了了解不同模型的相对质量的量化方法。

不同于像Akaike信息准则（AIC）这样的信息准则，BIC对于模型中参数数量有更严格的惩罚因素，因此，相对于AIC而言，BIC更喜欢参数少而精度高的模型。

BIC和AIC的计算公式十分相似，但是BIC具有更大的加权指数，这使得BIC更能够从模型选择中提取有信义而有价值的信息。

在模型比较的时候，具有更小的BIC值的模型更加适合于数据。

BIC除了用于模型选择外，还可以应用于时间序列分析，过程估计以及线性回归分析等等。

在时间序列模型中，BIC可以用来选择最优的自回归滞后项（AR）或自回归移动平均的滞后项（ARMA）。

在使用BIC模型选择时，需要注意的是，BIC并不是适用于所有情况的最优选择方法，对于数据特征复杂且自变量较多的情况，BIC不一定能够完全反映模型和数据的适配程度，因此需要结合实际情况进行判断。

此外，BIC也不能解决过度拟合的问题。

总之，BIC是一种非常有用的模型选择工具，它给出了一个客观评估模型的标准，帮助我们在不同模型之间进行选择，从而提高模型的预测能力和实用性。

同时，BIC的应用领域也在不断拓展，使得它成为了多个学科的重要工具之一。

Bayesian Human Segmentation in Crowded Situations Tao Zhao Ram NevatiaUniversity of Southern CaliforniaInstitute for Robotics and Intelligent SystemsLos Angeles,CA90089-0273taozhao nevatia@AbstractProblem of segmenting individual humans in crowded sit-uations from stationary video camera sequences is exacer-bated by object inter-occlusion.We pose this problem as a “model-based segmentation”problem in which human shape models are used to interpret the foreground in a Bayesian framework.The solution is obtained by using an efficient Markov chain Monte Carlo(MCMC)method which uses do-main knowledge as proposal probabilities.Knowledge of var-ious aspects including human shape,human height,camera model,and image cues including human head candidates, foreground/background separation are integrated in one the-oretically sound framework.We show promising results and evaluations on some challenging data.1Introduction and MotivationSegmentation and tracking of humans in video sequences are important for a number of tasks such as video surveillance and event inference as humans are the principal actors in daily activities of interest.In the situation of stationary cameras, background subtraction is a widely used technique to extract the moving pixels(foreground).If objects are sparse in the scene,each connected component of the foreground(blob) usually corresponds to an object,though the blobs may be fragmented due to low contrast or partial occlusion by scene objects or may contain non-human pixels caused by shadow. When the density of the objects increases(such as in the exam-ple shown in Fig.1),it is common that several objects form one big blob.Therefore the blobs do not directly provide the ob-ject level description that is needed for human event inference. Our goal in this paper is to segment the foreground(a binary mask)into individual human objects which may overlap with each other.1.1Previous workSome work has been done to segment or track multiple overlapping humans.In[5],peaks in the vertical histogram of the blob are used to help locate the positions of the heads. In[16]and[11],vertical peaks on the foreground boundary 1This research was supported,in part,by the Advanced Research and De-velopment Activity of the ernment under contract No.MDA-908-00-C-0036.(a)(b)Figure1:A sample input frame(a)and its foreground from standard background subtraction(b).are used to locate the positions of the heads.[16]also em-ploys an iterative processing to handle the case when the heads are not on the foreground boundary,assuming that the occlu-sion from each other is slight.In[3],humans are assumed to be isolated as they enter the scene so that a human spe-cific color model can be initialized for segmentation when oc-clusion occurs.[16]and[11]also use the initialized human specific model to help in tracking through occlusion.All of the above techniques either are based on some heuristics for segmentation or rely on initialized human models before oc-clusion occurs.They are not likely to be effective in crowded situations.In[10],the segmentation problem is solved using region-based stereo with input from multiple(up to16)cameras. However it is limited to applications of small spatial areas.[12]and[8]propose work to track multiple humans us-ing particlefilters.Due to the possible inter-occlusions,joint states(the parameters of all the humans in the scene)are used in the tracking.Performance of particlefilters is limited by the dimentionality of the state space,therefore,extending these approaches to track a large number of humans may be diffi-cult.Color segmentation is not likely to segment individual hu-mans.Motion segmentation also may not give satisfactory re-sult due to the non-rigid human motion and the similarity of the motion of individuals in a group.Face detection may not be effective when the humans don’t face the camera or when the image size of the human is small.Direct human detection (e.g.,[9])has been limited to restricted viewpoints(frontal or back).1.2Our approachWe take a3D model-based approach and use human shape models to interpret the foreground.The problem is defined as a model-based segmentation problem in a Bayesian framework. The solution is defined to be the number of human objects and their associated parameters maximizing the posterior proba-bility.The posterior probability reflects how well the solution reconstructs the foreground by the image likelihood while pre-ferring small number of objects by the prior.We use MCMC with jump and diffusion dynamics to pur-sue the best solution by traversing the solution space.Markov chain Monte Carlo(MCMC)is a tool to sample a probabilistic distribution.Recently DDMCMC(data-driven MCMC)has been proposed to solve computer vision problems.It improves the efficiency of traditional MCMC by incorporating domain knowledge to compute proposal probabilities of the Markov chain especially when the state is defined on a complex space. Promising results have been shown on object recognition[18], image segmentation[14]and range image segmentation[7].Domain knowledge including head candidates computed from foreground boundaries,head candidates computed from intensity edges and analysis of foreground residue map direct the creation of new human hypotheses which improves the ef-ficiency of the MCMC significantly.The described work builds on our earlier work reported in [17],The differences include more expressive human models to capture the human shape variations in mid-range data(i.e. where major limb articulations can be seen,stochastic diffu-sion which both improves the speed and the accuracy of local-ization and other refinements that result in more robust perfor-mance.We have performed experiments on datasets of different ob-ject densities and under different imaging conditions.The pro-posed approach gives good results with affordable computa-tion time as described later in the paper.2A Bayesian Formulation of the Problem We formulate the segmentation problem as computing the maximum a posteriori(MAP)estimation such that(1)where is the number of human objects and their parameters and is the foreground mask.Following Bayes rule,the pos-terior probability is decomposed into a likelihood term and a prior term:(2) The human shape model,the prior and the likelihood model are described in this section.2.13D human shape modelHuman body shape is highly articulated.To model it pre-cisely,a kinematics model with over20DOF and a mass dis-tribution model are needed.However,in our applicationthe Figure2:A number of3D human models are used to capture the gross shape of standing and walking humans.First row: standing models(with orientation and);second/third row:walking models with left/right leg forward(with orienta-tion,,and).NOTE:they are2D projection of3D models under the camera model of Seq.2in the result section.human motion is mostly limited to standing or walking and we do not attempt to capture the detailed parameters of the human body.We can thus use a number of low dimensional models to capture the gross shape of human bodies at different pose/viewpoint combinations.We model human shape by four ellipsoids corresponding to head,torso and two legs.An ellipsoidfits human body parts well and has the property that its projection is an ellipse with a convenient form[6].Each ellipsoid is controlled by two pa-rameters called length and fatness.The length parameter also determines the width by using afixed ratio;fatness parameter defines depth besides the proportional change.We consider only three articulations:both legs together,left leg forward and right leg forward.These models are sufficient to capture the gross shape variations of most humans in the scene for mid-resolution images.We assume that the humans move on a ground plane,there-fore,besides height and fatness,the parameters of the model also include position on the ground plane and orientation.The orientations of the models are quantized for computation ef-ficiency as follows:the standing model has two orientations (,frontal and,from side)and each of the two walking models has six orientations(,,and).Since both the model type and the orientation are discrete,we com-bine them into one parameter model/orientation label in one of the14combinations(Fig.2).Therefore,the parameters of each human object are which are model/orientation label,position,height and fatness respec-tively.We assume that the camera model and the ground plane are known as in[16].The camera model and the3D shape model automatically take care of the change in2D size and shape due to the change in position and viewpoint.This is advantageous to2D shape models such as in[4].2.2The solution spaceThe solution to the model-based segmentation problem in-cludes the number of objects in the scene and their asso-ciated parameters.It can be written in the form of.where is the subspace of exactly objects,is the enu-merated space of the model/orientation()and is the space for position and shape parameters().Since we don’t know in advance how many objects are present in a given scene,the solution space contains sub-spaces of varying dimensions.2.3Prior distributionsWe assume the prior probability of the state is the product of the prior probabilities of all the objects.The prior probability of an object is made up of the prior probability on its image size()and the prior probability on its parameters().(3) Thefirst term penalizes large total object sizes which avoids unnecessary overlapping.The second term penalizes objects with small image sizes since they are more likely to be due to image noise(is the distribution of the noise blob size).is a coefficient which controls the maximum overlap allowed in thefinal segmentation and is related to the noise level of the images.We also experimented with prior proba-bilities of the number of objects()but found it to be not effective in scenes which contain human objects with large image size variations.We set so that to penalize the complexity(more ellipsoid)of the walking models.is a uniform distribution in the image.is a Gaussian distribution(,)truncated in the range of and is Gaussian distribu-tion(,)truncated in the range of .Therefore:2.4Multi-object joint likelihoodSince multiple humans may occlude each other,the image likelihood cannot be decomposed into the product of image likelihoods individual human hypotheses.Given a state,we compute joint likelihood based on the formation of the fore-ground asfollows.(a)(b)(c)Figure3:The likelihood based on the number of wrongly clas-sified pixels.(a)the foreground;(b)the region of the solu-tion;(c)four kinds of pixels.NOTE:ellipse model is used for illustration.We assume that the foreground is formed by human objects in the scene.Denote as the image foreground and as the union of image regions of all human objects in a solution(see Fig.3for,and their relationship).is the probability that a pixel in an human object results in a foreground pixel, is the probability that a pixel in an human object does not result in a foreground pixel,is the probability that a pixel outside human objects results in a foreground pixel,is the probability that a pixel outside human objects does not create a foreground pixel.and.Assuming the pixels are independent,we have the following likelihood:where is a constant which absorbs the terms independent of ,are two coefficients depending on and and is the intersection of two regions.The likelihood only depends on and,which are the difference of the foreground and the solution region.The parameters and are estimated by examples.By combining the prior(Equ.3)and the above likelihood, the posterior probability(Equ.2)becomes:3Efficient MAP ComputationWe want tofind the segmentation that maximizes the pos-terior probability defined in the previous section.However, as stated in Sec.2.2,the solution space contains subspaces of varying dimensions and may contain many local minimums.Markov chain Monte Carlo(MCMC)methods combined with jump-diffusion dynamics provides a way to sample the poste-rior probability in such a complex solution space to search for the maximum.The basic idea of MCMC is as follows.A Markov chaincan be designed to sample a probability distribution(we use for clarity).At each iteration,we samplea candidate state according to from a proposal distri-bution(in simple words,what new state should the Markov chain go to from the previous state.).The candidatestate is accepted with the following probability:If the candidate state is accepted,,otherwise, .This is the well-known Metropolis-Hasting algo-rithm.It can be proven that the Markov chain constructed this way has its stationary distribution equal to,independent of the choice of the proposal probability and the initial state[13].However,the choice of the proposal probability can affect the efficiency of the MCMC significantly.A ran-dom proposal probability will lead to very slow convergence rate while a proposal probability designed with domain knowl-edge([18][14][7])will make the Markov chain traverse the solution space more efficiently.If the proposal probability is informative enough so that each sample can be thought of as a hypothesis,then the MCMC approach can be thought of as a stochastic version of the hypothesize and test approach([18]). Sampling the proposal probability results in certain dynam-ics of the Markov chain.Jump means that the structure(e.g. dimension)of the state is change while diffusion means that the structure does not change but the values of the parameters change.The crucial point in this problem is where to add a human object.We incorporate three types of domain knowledge in proposing the positions of new human objects.3.1Hypothesizing human objects3.1.1Head candidates from foreground boundaryThis method detects the heads which are on the boundary of the foreground[16].The basic idea is tofind the local vertical peaks of the boundary.The peaks are furtherfiltered by check-ing if there are enough foreground pixels below it according to the human height range and the camera model.This de-tector has a high detection rate and is also effective when the human is small and image edges are not reliable;however,it cannot detect the heads in the interior of the foreground blobs. Fig.4.(a)shows the on the exampleframe.(a)(b)(c)(d)Figure4:Head detectors.(a)Head candidates(crosses)from foreground boundaries();(b)Distance trans-formation on Canny edge detection result;(c)The head-shoulder()model:dark contour-head and shoulder,light line-normals;(d)Head candidates(crosses)from intensity edges().Read Sec.3.1for detail.3.1.2Head candidates from intensitywe also use a head detector based on image intensity edges which is also effective for the heads in the interior of the blobs [17].First,a Canny edge detector[2]is applied to the dilated foreground region of the input image.A distance transforma-tion[1]is then computed on the edge map.Fig.4.(b)shows the exponential edge map where(is the distance to the closest edge point and is a factor to control the responsefield and set to.).Besides, the coordinates of the closest pixel point are also recorded as .The unit image gradient vector is only com-puted at edge pixels.The“”shape of head and shoulder contour(Fig.4.(c)) is easily derived from our human model.The head-shoulder contour is generated from the projected ellipses by taking the whole head and the upper quarter torso as the shoulder.The normals of the contour points are also computed.The size of the human model is determined by the camera calibration as-suming a known height(1.6,1.7,1.8meters are used and the maximum response is recorded).Denote and as the positions and the unit normals of the model points respectively when head top is at.The model is matched with the image in the following way.A head candidate map is constructed by evaluatingon every pixel in the dilated foreground region.After smooth-ing it,wefind all the peaks above a threshold selected to give a very high detection rate but may also result in a high falsealarm rate.An example is shown in Fig.4.(d).The false alarmstend to happen in the areas of rich texture where there are abun-dant edges of various orientations.3.1.3Residue analysisWe denote the foreground map with the already formed hy-potheses removed as the foreground residue map.The fore-ground may initially contain some blobs of almost separated human objects.After some human objects are hypothesizedand removed from the foreground,the residue map may showmore separated objects.Morphological open operation can help isolate the objects that are only connected by thin bridgesand remove small/thin residues.We generate human candi-dates from the foreground residue map as follows.Given foreground and the region corresponding to thecurrent solution,compute the foreground residue map .Perform open operation with a vertically elongated structural element and compute connected components.Fromeach connected component,three human candidates can be generated assuming:1)the centroid of the is aligned with the center of human body;2)the top center point of is aligned with the human head;and3)the bottom center point of is aligned with the human feet.3.2Markov chain dynamicsDenote the state at iteration as.The following Markov chain dynamics are applied to which results in.The dynamics corre-spond to sampling the proposal probability:1.Human hypothesis addition:Randomly select a methodfrom the three techniques described in Sec.3.1to gener-ate a new hypothesis.Assume that the hypothesized po-sition is at,sample position in a Gaus-sian density.Rest of the pa-rameters()are sampled from their respective prior distributions.A new human hypothesis is assem-bled as..2.Human hypothesis removal:Randomly select an ex-isting human hypothesis to remove..3.Model/orientation switch:switch the model/orientationlabel of a human hypothesis.Randomly select an exist-ing human hypothesis and randomly switch the model/orientation label to another one.All other parameters are inherited.4.Stochastic diffusion of model parameters:update theparameters of a human hypothesis in the direction of theirgradients plus random noise.Randomly select an existinghuman hypothesis,update according to:where,is a step coefficient andis a Gaussian noise.The noise helps avoid local maxi-mums.The parameters are also bound to their minimumand maximum allowed values.5.Head position switch:switch the head position of a hu-man hypothesis.Randomly select an existing human hy-pothesis and randomly switch the head position to some other head candidate around the original position.It is equivalent tofirst removing this human andthen adding the new one.However,removing may re-sult in a big decrease of the posterior probability so that it has small chance to be accepted.This is similar to build-ing a bridge on a valley in the search space.Thefirst two are referred to as jump dynamics and the restare referred to as diffusion dynamics.It is guaranteed that the Markov chain designed this way is ergodic(i.e.,any state is reachable from any other state withinfinite number of itera-tions)and aperiodic(i.e.,the Markov chain does not oscil-late in afixed pattern)since all of moves are stochastic[14]. Furthermore,redundant dynamics(e.g.,head position switch) is added for ease of traversal in the solution space.Multiple ways of adding human hypotheses increase the robustness of the system.3.3Incremental computationIn one iteration our algorithm only changes one object. Thus the new likelihood can be computed more efficiently by incrementally computing it only within the neighborhood of the area associated with this object and those overlapping with it.Furthermore,with the help of storing number of object layers at each pixel(),the incremental likelihood com-putation is only needed within the region of the object being changed.When a human hypothesis is added,for each pixel in the region of the added object,if AND;if AND;;When a human hypothesis is removed,for each pixel in the region of the removed object,;if AND;if AND;In case of diffusion,and can be simply computedby a removal followed by an addition.The foreground residue map also gets updated incrementally at the same time.Although a joint state and joint likelihood are used,the computation of each iteration is reduced to the region of an object(in a general case,to the region of an object’s neighbor-hood)through the incremental computation.This is in contrast to particlefilters where the evaluation of each particle(joint state)needs the computation of the full joint likelihood.4Experiments and EvaluationsWe have tested the approach described above on a number of data sets and the results are very stable.Due to the space limit,we will mainly show the results and the performance evaluation on two sequences and mention the results of some other scenarios1.We perform background subtraction using a standard tech-nique([15])and feed the foreground into the segmentation al-gorithm after morphology close operation.In our experiments, the parameters arefixed as the following:, ,,and.Besides,we apply a hardconstraint(25pixels)on the minimum image height of human object.The Markov chain starts from a null stateand we assume.The results are shown in input and output pairs.The outputs are the human models overlaid on the original images.Due to the small im-age size,we have explicitly marked the errors on the images when they occur:false alarms in black arrows and missed de-tections in white arrows.Seq.1is a605-frame sequence captured from a camera on secondfloor with the camera tilt angle around.A group of22humans walked through the scene.The image sizes of human objects in the scene have a large variation(more than 10times in area).The dense edges of the tree branch shadows at the far side result in high false positives for head candidates. There are50to100head candidates per frame.Result on the example frame in Fig.1is given in Fig.5.We compare our proposed approach with a random(uni-form over the image)proposal probability for adding new hu-man hypotheses in5000-interation runs on the same frame. The log posterior probability histories of the two cases are shown in Fig.5.(b).The log posterior probability of our ap-proach climbs quickly and then stays close to maximum value forfine adjustments while the log posterior probability of the random proposal probability improves significantly slower.Our proposed approach is not sensitive to the initial state. We show the log posterior probability histories of1000-iteration runs from a null initial state and an initial state con-taining20random humans on the foreground in Fig.5.(c).Al-though showing some difference at the beginning,they show little difference after400iterations.Results on some more frames of Seq.1are shown in Fig.7. The results were obtained by2000-iteration runs.Seq.2is a900-frame sequence captured from a camera above a building gate with the camera tilt angle.A large tilt angle results in significant perspective effect on hu-man shape in images.33humans passed by the scene with 23going out of and10going in the building.Results on a few frames are shown in Fig.8.They were obtained by1000-iteration runs.To evaluate the accuracy of the method,we compare the1The results of all frames are available in video format at /˜taozhao/papers/CVPR03/CVPR03.html.(a)(b)(c)Figure5:Experiment result of the image in Fig.1.(a)Theresult.False alarms are marked with dark arrows and miss de-tections are marked with white arrows(same for the followingfigures);(b)Compare convergence()with uniform proposal probability for adding human hypotheses;(c)Com-pare convergence from different initial states.Read text fordetail.result of each frame with ground truth derived from hand an-notation.If a human object in the solution has an over50%overlap with a human object in the ground truth,a match is declared.One-to-one mapping of the objects is enforced.Theunmatched objects in the solution are declared as false alarms and the unmatched objects in the ground truth are declared asmiss-detections.The humans whose bodies are partially out-side the scene are not counted as mathches or errors(i.e.,they are“don’t cares”.).The results of the evaluation are summa-rized in Tab.1.To make the evaluation results more meaningful,we char-acterize the complexity of the dataset for human segmen-tation by the number of human objects per u-ally more humans per blob implies greater challenge to the segmentation algorithm.Fig.6shows the histogram of theoccurrence that a human object is in a blob of( maximum number of objects in one blob)human objects. In Seq.1,50%of the humans appear in a blob containing5or more humans and30%of the humans appear in a blob con-taining9or more humans.In Seq.2,about30%of the humans appear in a blob containing5or more humans.The two se-quences have similar detection rate.Seq.1has a higher false alarm rate due to the foreground introduced by the morphol-ogy close operation when the humans are small(at the far side) and close to each other.Besides the errors marked on the output,we show some other errors in Fig.9.Most of the missed detections are due to the failure of the background subtraction algorithm when the(a)(b)Figure6:The histogram of the number of human objects perblob in Seq.1(a)and Seq.2(b).Table1:Results of performance evaluations on Seq.1andSeq.2.Seq.1Seq.2valid humans84666726correct detections78816243missed-detections585483false alarms29112detection rate93.09%92.82%false alarm rate 3.43%0.18%human’s clothing has color similar to the background or theoverlapping humans form ambiguous shape or a combinationof the two.The false alarms are mainly due to the inclusionof large(relative to a human’s size)non-human regions in theforeground.These errors can be reduced by utilizing other im-age cues or temporal cues(e.g.tracking).Some of the selectedmodel/orientations are not accurate.They are mainly due tothe error of the foreground around the feet.From the experiments we performed(not all shown here),we found that our proposed method is insensitive to blob frag-mentation in cases with a significant amount of foreground in-dicating the presence of a human/humans.It also works ro-bustly when there are significant amount of noises in the fore-ground.The computation is affected by the complexity of the scene.More iterations are needed for a scene containing more hu-mans and more occlusion.As an example,a1000-iterationrun on the above reported dataset requires about0.5secondsof CPU time on a Pentium IV2.7G Hz PC,with un-optimizedC++code.If there are a small number of people in the scene,the system runs in real-time.5Conclusion and Future WorkWe have presented an approach to segmenting individualhumans in a crowded scene acquired from a static camera.Theproblem is formulated as a Bayesian MAP estimation problemand the solution is pursued using an efficient Markov chainMonte Carlo approach.The quality of the results is depen-dent on the definition of posterior probability according to im-age formation.Efficiency is obtained by incorporating do-main knowledge as the proposal probability of theMarkovFigure7:More results on the Seq.1.Left column:input;rightcolumn:output.chain.Experiments and evaluations on challenging real-lifedata show promising results.We feel that due to the Bayesianformulation,the described approach is more robust and effec-tive in a wider range of scenarios(e.g.crowded situations asreported here)compared to earlier work(e.g.[16]).The work described here could be improved or extended inseveral ways.Currently the likelihood is based only on a bi-nary foreground mask,other image cues such as edge or colorcould be used to reduce some ambiguities but at an increasein computation cost.Non-human objects(e.g.cars)should beadded to the model set for more versatility but this too willincrease computation and possibly result in more misclassifi-cations.Finally,the work should be extended to include track-ing:tracking information will provide important temporal pri-ors which will both resolve some ambiguities of single frameanalysis and reduce the computation.Tracking is also neededas an input for event inference algorithms.AcknowledgementThe authors would like to thank Dr.Song-Chun Zhu andDr.Zhuowen Tu for helpful discussion and the anonymousreviewers for their helpful comments.。

R贝叶斯包分类介绍(R task view ofBayesian)=========一般模型==================arm包: 包括使用lm,glm,mer,polr等对象进行贝叶斯推断的R函数BACCO: 随机函数的贝叶斯分析. 包含3个子包: emulator, calibrator, and approximator, 进行贝叶斯估计和评价计算机程序.bayesm: 市场与微经济分析模型的许多贝叶斯推断函数. 模型包括线性回归, 多项式logit, 多项式probit, 多元probit, 多元混合normals(包括聚类), 密度估计-使用有限混合正态模型与Dirichlet先验过程, 层次线性模型, 层次多元logit, 层次负二项回归模型, 线性工具变量模型(linear instrumental variable models). bayesSurv: 生存回归模型的贝叶斯推断.DPpackage: 贝叶斯非参数和半参数模型. 现在还包括密度估计, ROC曲线分析, 区间一致数据, 二项回归模型, 广义线性模型和IRT类型模型的半参数方法. MCMCpack: 特定模型的MCMC模拟算法, 广泛用于社会和行为科学. 拟合很多回归模型的R函数. 生态学模型推断. 还包括一个广义Metropolis采样器, 适合任何模型.mcmc: 随机行走Metropolis算法, 对于连续随机向量.==========特殊模型和方法=============AdMit: 拟合适应性混合t分布拟合目标密度使用核函数.bark: 实现(Bayesian Additive Regression Kernels)BayHaz: 贝叶斯估计smooth hazard rates, 通过Compound Poisson Process (CPP) 先验概率.bayesGARCH: 贝叶斯估计GARCH(1,1) 模型, 使用t分布.BAYSTAR: 贝叶斯估计threshold autoregressive modelsBayesTree: implements BART (Bayesian Additive Regression Trees) by Chipman, George, and McCulloch (2006).BCE: 从生物注释数据中估计分类信息.bcp: a Bayesian analysis of changepoint problem using the Barry and Hartigan product partition model.BMA:BPHO: 贝叶斯预测高阶相互作用, 使用slice 采样技术.bqtl: 拟合quantitative trait loci (QTL) 模型.可以估计多基因模型, 使用拉普拉斯近似. 基因座内部映射(interval mapping of genetic loci).bim: 贝叶斯内部映射, 使用MCMC方法.bspec: 时间序列的离散功率谱贝叶斯分析cslogistic: 条件特定的logistic回归模型(conditionally specified logistic regression model)的贝叶斯分析.deal: 逆运算网络分析: 当前版本覆盖离散和连续的变量, 在正态分布下.dlm: 贝叶斯与似然分析动态信息模型. 包括卡尔曼滤波器和平滑器的计算, 前向滤波后向采样算法.EbayesThresh: thresholding methods 的贝叶斯估计. 尽管最初的模型是在小波下开发的, 当参数集是稀疏的, 用户也可以受益.eco: 使用MCMC方法拟合贝叶斯生态学推断in two by two tables evdbayes: 极值模型的贝叶斯分析.exactLoglinTest: log-linear models 优度拟合检验的条件P值的MCMC估计. HI: transdimensional MCMC 方法几何途径, 和随机多元Adaptive Rejection Metropolis Sampling.G1DBN: 动态贝叶斯网络推断.Hmisc内的gbayes()函数, 当先验和似然都是正态分布, 导出后验(且最优)分布, 且当统计量来自2-样本问题.geoR包的krige.bayes()函数地理统计数据的贝叶斯推断, 允许不同层次的模型参数的不确定性.geoRglm 包的binom.krige.bayes() 函数进行贝叶斯后验模拟, 二项空间模型的空间预测.MasterBayes: MCMC方法整合家谱数据(由分子和形态数据得来的)lme4包的mcmcsamp()函数信息混合模型和广义信息混合模型采样.lmm: 拟合信息混合模型, 使用MCMC方法.MNP: 多项式probit模型, 使用MCMC方法.MSBV AR: 估计贝叶斯向量自回归模型和贝叶斯结构向量自回归模型.pscl: 拟合item-response theory 模型, 使用MCMC方法, 且计算beta分布和逆gamma分布的最高密度区域RJaCGH: CGH微芯片的贝叶斯分析, 使用hidden Markov chain models. 正态数目的选择根据后验概率, 使用reversible jump Markov chain Monte Carlo Methods 计算.sna: 社会网络分析, 包含函数用于从Butt's贝叶斯网络精确模型, 使用MCMC方法产生后验样本.tgp: 实现贝叶斯treed 高斯过程模型: 一个空间模型和回归包提供完全的贝叶斯MCMC后验推断, 对于从简单线性模型到非平稳treed高斯过程等都适合. Umacs: Gibbs采样和Metropolis algorithm的贝叶斯推断.vabaye1Mix: 高斯混合模型的贝叶斯推断, 使用多种方法.=Post-estimation tools=====BayesValidate: 实现了对贝叶斯软件评估的方法.boa: MCMC序列的诊断, 描述分析与可视化. 导入BUGS格式的绘图. 并提供Gelman and Rubin, Geweke, Heidelberger and Welch, and Raftery and Lewis 诊断. Brooks and Gelman 多元收缩因子.coda: (Convergence Diagnosis and Output Analysis) MCMC的收敛性分析, 绘图等. 可以轻松导入WinBUGS, OpenBUGS, and JAGS 软件的MCMC输出. 亦包括Gelman and Rubin, Geweke, Heidelberger and Welch, and Raftery and Lewis 诊断. mcgibbsit: 提供Warnes and Raftery MCGibbsit MCMC 诊断. 作用于mcmc对象上面.ramps: 高斯过程的贝叶斯几何分析, 使用重新参数化和边际化的后验采样算法. rv: 基于模拟的随机变量类, 后验模拟对象可以方便的作为随机变量来处理. scapeMCMC: 处理年龄和时间结构的人群模型贝叶斯工具. 提供多种MCMC诊断图形, 可以方便的修改参数===========学习贝叶斯的包===================BaM: Jeff Gill's book, "Bayesian Methods: A Social and Behavioral Sciences Approach, Second Edition" (CRC Press, 2007). 伴随的包Bolstad: 此书的包. Introduction to Bayesian Statistics, by Bolstad, W.M. (2007). 的包LearnBayes: 学习贝叶斯推断的很多的函数. 包括1个,2个参数后验分布和预测分布, MCMC算法来描述分析用户定义的后验分布. 亦包括回归模型, 层次模型. 贝叶斯检验, Gibbs采样的实例.贝叶斯包一般模型拟合Bayesian packages for general model fitting1.The arm package contains R functions for Bayesianinference using lm, glm, mer and polr objects. arm package 包含了用于使用lm,glm,mer 和polr对象的贝叶斯推理的R函数Install.packages(“arm”)Library(“arm”)Help(package=”arm”) Documentation for package …arm‟ version 1.5-08 DESCRIPTION file.Help PagesFunctions to compute the balance statistics函数来计算平衡统计balanceFunctions to compute the balance statistics函数来计算平衡统计balance-classbayesglm-class Bayesian generalized linear models. 贝叶斯广义线性模型。