Sample space, events and probability Conditional probability Random variables and probability distributions Joint probability distribution Independence Conditional probability distribution Bayes’ Theorem P(A¬) = 1 − P(A) Feng Li (SDU) GDA, NB and EM September 27, 2023 5 / 122 Conditional Probability Definition of conditional probability: Fraction of worlds in which event A is true given event B is true A2, A1)P(A3 | A2, A1)P(A2 | A1)P(A1) Feng Li (SDU) GDA, NB and EM September 27, 2023 6 / 122 Conditional Probability (Contd.) Real valued random variable is a function of the outcome of a ran- domized

P(A | B) = P(B | A)P(A) P(B) (1) where P(A | B) is the conditional probability of event A given event B happens, P(B | A) is the conditional probability of event B given A is true, and P(A) and P(B) mass function (PMF) as pY (y; ψ) = P(Y = y) = ψy(1 − ψ)1−y (5) • A2: X | Y = 0 ∼ N(µ0, Σ): The conditional probability of continuous random variable X given Y = 0 is a Gaussian distribution parameterized 0) = 1 (2π)n/2|Σ|1/2 exp � −1 2(x − µ0)T Σ−1(x − µ0) � (6) • A3: X | Y = 1 ∼ N(µ1, Σ): The conditional probability of continuous random variable X given Y = 1 is a Gaussian distribution parameterized

Given an input feature vector x, we have The conditional probability of Y = 1 given X = x Pr(Y = 1 | X = x; θ) = hθ(x) = 1 (1 + exp(−θTx) The conditional probability of Y = 0 given X = x Pr(Y = 0 |

distribution N(0, �2). The density of "(i) is given by f(✏) = 1 p 2⇡� exp ✓ � ✏2 2�2 ◆ Hence, the conditional probability density function of y given x is defined by p(y | x; ✓) = 1 p 2⇡� exp ✓ �(y � ✓T

distribution N(0, σ2) The density of ϵ(i) is given by f (ϵ) = 1 √ 2πσ exp � − ϵ2 2σ2 � The conditional probability density function of y y | x; θ ∼ N(θTx, σ2) Feng Li (SDU) Linear Regression September

Recognition and Machine Learning[M]. Springer,2006. [9] Rosenberg A, Hirschberg J. V-Measure: A conditional entropy-based external cluster evaluation[C]// Conference on Emnlp-conll. DBLP, 2007. [10] Campello

an objective function yields a value given a set of hyperparameters . They are represented as a conditional probability distribution . The surrogates are cheaper to compute in comparison to the blackbox

the surrounding words (context). Mathematically, we want to find a word , which maximizes the conditional probability , where the size of the sliding window of context is ( words on each side of the masked

时发生的可能性不大于A = a或是B = b单独发生的可能性。 条件概率 联合概率的不等式带给我们一个有趣的比率：0 ≤ P (A=a,B=b) P (A=a) ≤ 1。我们称这个比率为条件概率（conditional probability），并用P(B = b | A = a)表示它：它是B = b的概率，前提是A = a已发生。 贝叶斯定理 使用条件概率的定义，我们可以得出统计学中最有用的方程 A)P(A) P(B) . (2.6.1) 请注意，这里我们使用紧凑的表示法：其中P(A, B)是一个联合分布（joint distribution），P(A | B)是一个条 件分布（conditional distribution）。这种分布可以在给定值A = a, B = b上进行求值。 边际化 为了能进行事件概率求和，我们需要求和法则（sum rule），即B的概率相当于计算A的所有可能选择，并将

9.7.4 生成数据 通过生成模型在原有数据上进行训练，学习到真实数据的分布，从而利用生成模型获 得新的样本，这种方式也可以在一定程度上提升网络性能。如通过条件生成对抗网络 (Conditional GAN,简称 CGAN)可以生成带标签的样本数据，如图 9.36 所示。 图 9.36 CGAN 网络生成的手写数字图片 9.7.5 其它方式 除了上述介绍的典型图片数据