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数学代写|信息论代写Information Theory代考|ECE1502 Three Regions on a Board

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数学代写|信息论代写Information Theory代考|Three Regions on a Board

In the example of Sect. 1.5.1 we had three coins, or two spins, each of which could be in one of two states, “up” or “down.” We saw that there is no way of representing either the SMI or the MI in a Venn diagram.

In the next example we replace the three coins by three regions on a board. We throw a dart on the board of unit area. We know that the dart hit the board. The events are: “the dart is in region A” (or B, or C). We shall treat this system in two languages. First, as events having probabilities and represented in a Venn diagram. Second, as random variables, having SMIs and MIs which cannot be represented by a Venn diagram.
The system discussed in this section is shown in Fig. 1.17.
It is an extension of the system discussed in Sect. 1.4.2. Instead of two overlapping regions, we have here three overlapping regions, only in pairs, not in triplets. We assume that a dart was thrown on a board of unit area. Each of the regions A, B, and C have the same area chosen as $q=0.1$, hence, the probability of finding the dart in any of these areas is 0.1.

We denote by $d$ the area of overlapping between $\mathrm{A}$ and $\mathrm{B}$, and between $\mathrm{A}$ and $\mathrm{C}$. We denote by $x$ the overlapping area between $\mathrm{B}$ and $\mathrm{C}$. We start by listing the triplet probabilities which can be read from Fig. 1.17, These are:
\begin{aligned} & P(1,1,1)=0 \text { (no triple overlapping) } \ & P(0,0,0)=1-3 q+2 d+x \end{aligned}
(this is the area of the whole board minus the area of $A \cup B \cup C$ )
\begin{aligned} & P(1,0,0)=q-2 d \ & P(0,1,0)=q-d-x \ & P(0,0,1)=q-d-x \ & P(1,1,0)=d \ & P(1,0,1)=d \ & P(0,1,1)=x \end{aligned}

数学代写|信息论代写Information Theory代考|A Caveat to the Caveat on Frustration

In this section, we showed two examples of three-random variables for which we found negative values of the triple-conditional MI. In the first example (coins with embedded magnets) we have frustration as defined for a three-spin system, and discussed in Chap. 4 of Ben-Naim [1], However, there is not even a hint as to how the values of SMI may be represented as areas in a Venn diagram, and the MI as overlapping areas measuring the extent of dependence.

In the second example (with areas on a board on which a dart hit) we can describe the various events on a Venn diagram, but this description is valid only when we treat events and their probabilities, not when we treat random variables and the dependence between them. Once we move from events to random variables, and the corresponding SMI and MI, there is no way to describe the SMI and the extent of dependence on a Venn diagram.

The second question is whether or not a negative value of $\mathrm{CI}$ may be considered as a measure of frustration in the system (without any reference to a description by a Venn diagram). The answer to this question depends on which of the (equivalent) definition we use for the $\mathrm{CI}$.

In general, we can safely say that none of the definitions of the CI offer any interpretation as a measure of frustration. Again, one should be careful here about the distinction between the treatment on the “level” of single events, their probabilities, and the extent of dependence on one hand, and the treatment on the “level” of random variable, their SMIs, and the extent of dependence, on the other hand.

Consider the following story which might be interpreted as frustration on one level, but not in general, on the other level. [A more detailed story may be found in Ben-Naim [9]].

数学代写|信息论代写Information Theory代考|Three Regions on a Board

$$P(1,1,1)=0 \text { (no triple overlapping) } \quad P(0,0,0)=1-3 q+2 d+x$$
(这是整个板的面积减去面积 $A \cup B \cup C$ )
$$P(1,0,0)=q-2 d \quad P(0,1,0)=q-d-x P(0,0,1)=q-d-x \quad P(1,1,0)=d P(1,0,1)=d \quad P(0,1,1)=x$$

MATLAB代写

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