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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 统计代写|随机控制代写Stochastic Control代考|s-Irrelevance and s-independence

In a recent paper (Doria, 2007) the new definitions of s-irrelevance and s-independence with respect to upper and lower conditional probabilities assigned by outer and inner Hausdorff measures have been proposed. They are based on the fact that epistemic independence and irrelevance, introduced by Walley, must be tested for events $A$ and $B$, such that they and their intersection $A B$, have the same Hausdorff dimension. The concept of epistemic independence (Walley, 1991) is based on the notion of irrelevence; given two events $A$ and $B$, we say that $B$ is irrelevant to $A$ when $\bar{P}(A \mid B)=\bar{P}\left(A \mid B^{c}\right)=\bar{P}(A)$ and $\underline{P}(A \mid B)=\underline{P}\left(A \mid B^{c}\right)=\underline{P}(A)$.

The events $A$ and $B$ are epistemically independent when $B$ is irrelevant to $A$ and $A$ is irrelevant to $B$. As a consequence of this definition we can obtain that the factorization property $P(A B)=P(A) P(B)$, which constitutes the standard definition of independence for events, holds either for $P=\bar{P}$ and $P=\underline{P}$. In a continuous probabilistic space $(\Omega, F, P)$, where $\Omega$ is equal to $[0,1]^{n}$ and the probability is usually assumed equal to the Lebesgue measure on $\Omega$, we have that the finite, countable and fractal sets (i.e. the sets with Hausdorff dimension non integer) have probability equal to zero. For these sets the standard definition of independence, given by the factorization property, is always satisfied since both members of the equality are zero. In Theorem 6 of this Section we prove that an event $B$ is always irrelevant, according to the definition of Walley, to an event $A$ if $\operatorname{dim}{H}(A)<\operatorname{dim}{H}(B)<\operatorname{dim}_{H}(\Omega)$ and $A$ and $B$ have positive and finite Hausdorff outer measures in their dimensions; moreover if $A$ and $B$ are disjoint then they are epistemically independent. Nevertheless $B$ is not s-irrelevant to $A$.
To avoid these problems the notions of s-irrelevance and s-independence with respect to upper and lower conditional probabilities assigned by a class of Hausdorff outer and inner measures are proposed to test independence. The definitions of s-independence and s-irrelevance are based on the fact that epistemic independence and irrelevance, must be tested for events $A$ and $B$, such that they and their intersection $A B$, have the same Hausdorff dimension. According to this approach to independence, sets that represent events can be imagined divided in different layers; in each layer there are sets with the same Hausdorff dimension; two events $A$ and $B$ are s-independent if and only if the events $A$ and $B$ and their intersection $A B$ belong to the same layer and they are epistemically independent.

## 统计代写|随机控制代写Stochastic Control代考|s-Independence for curves filling the space

In this section the notions of s-irrelevance and s-independence for events $A$ and $B$ that are represented by curves filling the space are analyzed.In particular Peano curve, Hilbert curve and Peano-Sierpinski curve are proven to be s-independent. Curves filling the space Sagan (1994) can be defined as the limit of a Cauchy sequence of continuous functions $f_{n}$, each mapping the unit interval into the unit square. The convergence is uniform so that the limit is a continuous function, i.e. a curve. The definition of irrelevance given by Walley, which is condition $2 \mathrm{~s}$ ) of s-irrelevance, holds when the two events $A$ and $B$ are not the trivial events $(\Omega, \oslash)$. If the conditioning event $B$ is represented by a curve filling the space, we have that the complement of $B$ is the empty-set and so in this case the notion of irrelevance becomes $\bar{P}(A \mid B)=\bar{P}(A)$; and $\underline{P}(A \mid B)=\underline{P}(A)$. If $A$ and $B$ are represented by curves filling the space we obtain the following definition of s-independence.

Definition 5 Let $(\Omega, d)$ be a metric space and let $A$ and $B$ be two curves filling the space $\Omega$. Then $A$ and $B$ are s-independent if the following conditions hold

• 1s) $\operatorname{dim}{H}(A B)=\operatorname{dim}{H}(B)=\operatorname{dim}_{H}(A)$
• 2s) $\bar{P}(A \mid B)=\bar{P}(A)$ and $\underline{P}(A \mid B)=\underline{P}(A)$;
• 3s) $\bar{P}(B \mid A)=\bar{P}(B)$ and $\underline{P}(B \mid A)=\underline{P}(B)$;
Moreover $B$ is s-irrelevant to $A$ if conditions $1 \mathrm{~s}$ ) and $2 \mathrm{~s}$ ) are satisfied.
Theorem 9. Let $\Omega=[0,1]^{n}$ and let $\bar{P}$ and $\underline{P}$ be the upper and lower conditional probabilities defined as in Theorem 4. If $A$ and $B$ are two curves filling the space then $A$ and $B$ are s-independent.

## 统计代写|随机控制代写Stochastic Control代考|s-Independence for curves filling the space

• 1s) $\operatorname{dim} H(A B)=\operatorname{dim} H(B)=\operatorname{dim}_{H}(A)$
• 2s) $\bar{P}(A \mid B)=\bar{P}(A)$ 和 $\underline{P}(A \mid B)=\underline{P}(A)$;
• 3s) $\bar{P}(B \mid A)=\bar{P}(B)$ 和 $\underline{P}(B \mid A)=\underline{P}(B)$;
而且 $B$ 与 $\mathrm{s}$ 无关 $A$ 如果条件 $1 \mathrm{~s})$ 和 $2 \mathrm{~s})$ 满意。
定理 $9 .$ 让 $\Omega=[0,1]^{n}$ 然后让 $\bar{P}$ 和 $P$ 是定理 4 中定义的上限和下限条件概率。如果 $A$ 和 $B$ 那么是两条曲线填充空间 $A$ 和 $B$ 是 $s$ 独立的。

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assignmentutor™您的专属作业导师
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