## 数学代写|信息论作业代写information theory代考|INFM130

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• Advanced Probability Theory 高等概率论
• Advanced Mathematical Statistics 高等数理统计学
• (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

## 数学代写|信息论作业代写information theory代考|Systems with Four Interacting Spins

In this and in the next few sections we study the MI among larger number of spins. As we have noted in Chap. 4 of Ben-Naim [1], there are several possible generalizations of the MI. In Chap. 4 of Ben-Naim [1] we discussed mainly two generalizations of the MI.

The first was referred to as the total MI denoted TI. This is defined for any number of random variables (or experiments) by:
$$T I\left(X_1 ; \ldots ; X_n\right)=\sum p\left(x_1, \ldots x_n\right) \log g\left(x_1, \ldots, x_n\right)$$
where $p\left(x_1, \ldots, x_n\right)$ is the probability of finding the event $\left{X_1=x_1, X_2=\right.$ $\left.x_2, \ldots X_n=x_n\right)$ and $g\left(x_1, \ldots, x_n\right)$ is the correlation function, defined by:
$$g\left(x_1, \ldots, x_n\right)=\frac{P\left(x_1, \ldots, x_n\right)}{\prod_{i=1}^n P\left(x_i\right)}$$
The second generalization was referred to as the conditional MI, and denoted $\mathrm{CI}$. There are several possible definitions of CI. Here, we use the one chosen by Matsuda [3]. This is not the most informative form of $\mathrm{CI}$, however, we use this particular one to compare our results with Matsuda’s results. This definition is:
$$C I\left(X_1 ; \ldots ; X_n\right)=\sum_{k=1}^n(-1)^{k-1} \sum_{\left(i_1, \ldots, i_k\right)} H\left(X_1, \ldots X_n\right)$$
Here, the sum on the right hand side of (3.43) is over-all possible sets of indices $\left(i_1, i_2, \ldots, i_k\right)$, with $: 1<i_1<i_2<\ldots i_k<k$.
As we have noted several times, this quantity may be negative.

## 数学代写|信息论作业代写information theory代考|Four-Spin Systems; Perfect Square

Figure 3.23 shows three possible arrangements of the four spins system: (a) is a regular square with equal edges. (b) is a parallelogram in which the distance between 1 and 3 is the same between 1 and 3 and (c) a rectangle with two short and two twice longer edges.

For the arrangement of Fig. 3.20a the total number of configurations is $2^4=16$. Again, we assign the value of $(+1)$ to the “up” and $(-1) \ldots$

In Sect. 4.10 of Ben-Naim [1] we presented some details about the various SMI, the pair and triplet MI, etc. Here, we proceed directly to discuss only the total MI and the conditional MI.

Figure 3.24 shows TI and $\mathrm{CI}$ for the square arrangement. For $\beta J=0$ (either $J=0$ or $T \rightarrow \infty$ ) there is no correlation between the spins. Hence, both TI and CI

are zero. For larger $\beta J$ (either positive or negative) the behavior is similar for both TI and CI.
For the TI, we have in this limit
\begin{aligned} T I\left(X_1 ; X_2 ; X_3 ; X_4\right) & =-H\left(X_1, X_2, X_3, X_4\right)+4 H\left(X_1\right) \ & =-1+4=3 \end{aligned}
The reason for this result is clear for very strong interactions there are only two configurations for the four spins with equal probabilities. Iherefore, the SMI for the four spins is one, as well as for each individual spin.

Unlike the case of three spins for which the behavior of $\mathrm{CI}$ is different for positive and negative $\beta J$, here, we have the same behavior for both $\beta J \rightarrow \pm \infty$. The actual value is $\log _2 2=1$, which may be calculated from equation:

\begin{aligned} & C I\left(X_1 ; X_2 ; X_3 ; X_4\right)=4 H\left(X_1\right)-4 H\left(X_1, X_2\right) \ & \quad-2 H\left(X_1, X_3\right)+4 H\left(X_1, X_2, X_3\right)-H\left(X_1, X_2, X_3, X_4\right) \ & \quad \rightarrow 4-4-2+4-1=1 \end{aligned}
Note that unlike the case of triangle we do not have frustration in this case. As we noted earlier Matsuda attributed the negative value of the $\mathrm{CI}$ to the frustration effect.

# 信息论代写

## 数学代写|信息论作业代写information theory代考|Systems with Four Interacting Spins

$$T I\left(X_1 ; \ldots ; X_n\right)=\sum p\left(x_1, \ldots x_n\right) \log g\left(x_1, \ldots, x_n\right)$$
andg\left(x_1, \dots, $\left.\mathrm{x} _\mathrm{n} \backslash \mathrm{right}\right)$ isthecorrelation function, de finedby :
$g\left(x_1, \ldots, x_n\right)=\frac{P\left(x_1, \ldots, x_n\right)}{\prod_{i=1}^n P\left(x_i\right)}$
Thesecondgeneralizationwasre ferredtoastheconditionalMI, anddenoted $\backslash$ mathrm ${\mathrm{Cl}}$ .ThereareseveralpossibledefinitionsofCI. Here, weusetheonechosenby Matsuda[3]. T Imathrm ${\mathrm{Cl}}$
, however, weusethisparticularonetocompareourresultswith Matsuda’sresults. Thisde $C I\left(X_1 ; \ldots ; X_n\right)=\sum_{k=1}^n(-1)^{k-1} \sum_{\left(i_1, \ldots, i_k\right)} H\left(X_1, \ldots X_n\right)$

## 数学代写|信息论作业代写information theory代考|SMI is not a probability

SMI 与概率的关联可能是由于布里渊 [6]。在他的《科学与信息论》一书的第 120 页，我们发现：

## 有限元方法代写

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## MATLAB代写

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