Interaction I: Individual-based learning of a candidate (ij)-particle from a field (kl)-particle with learning variables u⁎ and u⁎, respectively. The transition probability density is supposed to depend on both the learning variables and the distribution functions of the FSs to which candida...
bino′mial distribu′tion n. a statistical distribution giving the probability of obtaining a specified number of successes in a finite set of independent trials in which the probability of a success remains the same from trial to trial. ...
In our example of a sequence of bits, the distribution function gives the probability that the sequence will have k or fewer bits in error, and its formula is (9.4)Fk=∑n=0kmnpnqm−nwhere Σ is the symbol for summation. The example which we used up to now involves a discrete random...
Swanson (IEEE Tran. Inform. Theory,\nvol.IT-32, p.701-3, 1986) offered an upper bound on PE\n(u), the decoder error probability given u\nsymbol errors occur. In the present study, by using a combinatoric\ntechnique such as the principle of inclusion and exclusion, an exact\nformula...
A subsetEof\(\Omega \subset \mathbb R^n\)is dense in\(\Omega \)if,\(\forall x\in \Omega \),\(\forall \epsilon >0\),\(\exists y \in E\)such that\(|x-y|\le \epsilon \). 30. This theorem implies that the intersection of countable collection of open dense sets\((U_{k}...
一般来说,任何内积(第3.2节)都可以在(6.107)中使用,为了具体说明,我们在这里使用标准的点乘 ( ⟨ θ , ϕ ( x ) ⟩ = θ ⊤ ϕ ( x ) ) \left(\langle\boldsymbol{\theta}, \boldsymbol{\phi}(\boldsymbol{x})\rangle=\boldsymbol{\theta}^{\top} \boldsymbol{\phi}(\boldsymbol{x...
The meaning of such aconditionwill become clearer later on. For the moment it suffices to understand a condition as a restriction specified by a function\(f(\boldsymbol{\omega })\), which implies that not all subsets of sample points belong toA. Such a condition, for example, is a scor...
Distribution nameDistribution symbolProbability mass function (pmf)MeanVariance fx(k) = P(X=k) k = 0,1,2,...E(x)Var(x) Binomial X ~ Bin(n,p) np np(1-p) Poisson X ~ Poisson(λ) λ≥ 0 λ λ Uniform X ~ U(a,b) Geometric X ~ Geom(p) Hyper-geometric X ~ HG(N,K,n)...
对于这种参数化形式,参数\boldsymbol{θ}被称为自然参数(natural parameters)。乍一看,指数族似乎是一个普通的变换,只是通过将点积代入指数函数。然而,基于我们可以从\phi(\boldsymbol{x})得到有关数据的信息这一事实,我们可以实现方便的建模和高效的计算。 例6.13 高斯分布作为指数族 考虑一元高斯分布\mathcal{N}...
The Greek symbol μ is pronounced “mew,” and the notation E[X] denotes the expected value of the random variable X. The summation sign Σ means that we add up the products XiP[Xi]. Suppose, for example, that X is equal to the number of heads that occur when three coins are flippe...