Glooow的博客

1. Sum-product(Message passing) on trees目的是为了计算边缘分布，相比于 elimination 的优势在于可以用较少的计算次数计算所有随机变量的边缘分布，关键在于复用 messagealgorithmStep 1: Compute messagesmi→j(xj)=∑xiϕi(xi)ψij(xi,xj)∏k∈N(i)\{j}mk→i(xi...

1. Elimination algorithm主要目的是为了计算边缘分布Reconstituted graph: 若消去的随机变量为 xkx_kxk​，则所有与 xkx_kxk​ 连接的随机变量会形成一个新的 clique复杂度Brute-force marginalization：O(∣X∣N)O\left(|\mathcal{X}|^N\right)O(∣X∣N)Z...

1. Undirected graphical models(Markov random fields)节点表示随机变量，边表示与节点相关的势函数px(x)∝φ12(x1,x2)φ13(x1,x3)φ25(x2,x5)φ345(x3,x4,x5)p_{\mathbf{x}}(\mathbf{x}) \propto \varphi_{12}\left(x_{1}, x_{2}\right)...

1.Bayesian ApproachConsider a nested sequence of model classesP1⊂P2⊂P3⊂⋯\mathcal{P}_{1} \subset \mathcal{P}_{2} \subset \mathcal{P}_{3} \subset \cdotsP1​⊂P2​⊂P3​⊂⋯ML decision rule:m^=arg⁡max...

1. 一些定理Markov inequality: r.v.  x≥0r.v. \ \ \mathsf{x}\ge0r.v.  x≥0P(x≥μ)≤E[x]μ\mathbb{P}(x\ge\mu)\le \frac{\mathbb{E}[x]}{\mu}P(x≥μ)≤μE[x]​Proof: omit…Weak law of large num...

1. Modeling problemformulationa set of distributionsP={py(⋅;x)∈Py:x∈X}\mathcal{P}=\left\{p_{\mathrm{y}}(\cdot ; x) \in \mathcal{P}^{y}: x \in \mathcal{X}\right\}P={py​(⋅;x)∈Py:x∈X}approxim...

1. EM-ML algorithmformulationcomplete data : z=[y,w]\mathsf{z=[y,w]}z=[y,w]observation : y\boldsymbol{y}yhidden variable : w\boldsymbol{w}westimation : x\mathcal{x}xDerivation期望获得 ML 估计，但...

1. Generalized Bayesian decisionFormulationSoft decision: qx(⋅∣y)q_x(\cdot|y)qx​(⋅∣y)Cost function: C(x,qx)C(x,q_x)C(x,qx​)Cost functionproper: px∣y(⋅∣y)=arg⁡min⁡{qx≥0:∑aq(a)=1}E[C(x,qx(⋅)...

1. Exponential familyDefinitionPDF: p(y;x)=exp⁡(λ(x)Tt(y)−α(x)+β(y))p(y;x)=\exp(\lambda(x)^T t(y)-\alpha(x)+\beta(y))p(y;x)=exp(λ(x)Tt(y)−α(x)+β(y))y∼ε(x;λ(⋅),t(⋅),β(⋅))y\sim \varepsilon(x;\lamb...

1. Bayesian parameter estimationFormulationPrior distribution px(⋅)p_{\mathsf{x}}(\cdot)px​(⋅)Observation py∣x(⋅∣⋅)p_{\mathsf{y|x}}(\cdot|\cdot)py∣x​(⋅∣⋅)Cost C(a,a^)C(a,\hat a)C(a,a^)Solut...

1. Binary Bayesian hypothesis testing1.0 Problem SettingHypothesisHypothesis space H={H0,H1}\mathcal{H}=\{H_0, H_1\}H={H0​,H1​}Bayesian approach: Model the valid hypothesis as an RV HPrior P0=p...

今天我们来讲一下粒子滤波算法，这也是我在学习过程中的一个笔记，由于有很多的个人理解，有不对的地方欢迎大家批评指正。另：个人博客 Glooow 开业啦！欢迎各位大驾光临文章目录贝叶斯滤波1. 预测2.更新总结卡尔曼滤波大数定律粒子滤波简单实例1. 预测2. 更新3. 递推重采样粒子滤波原理推导总结Reference贝叶斯滤波贝叶斯滤波是我们理解粒子滤波的基础。假设我们有如下图的隐马尔科夫...