互信息的计算

发布于 2024-10-16 03:41:22 字数 403 浏览 10 评论 0原文

假设 M 是一组对象 m，每个对象都具有属性 X 和 Y。现在，如果给定 m，X 和 Y 只能有一个值（即 X,Y 是随机变量，其中 P(X=x_i|M=m_i)，P( Y=y_i|M=m_i))，可以计算X和Y的互信息。但是如果X可以同时有多个结果呢？即对于 m_3 X={x1,x2} - 通常 X 的结果是所有可能结果的子集。在这种情况下，可以衡量互信息或其他依赖程度吗？

是否可以将 X 拆分为二进制随机变量 X_1, X_2 等，其中 X_1=1 iff X 包含 x1, X_1=0 否则，然后计算所有组合 i,j 的 I(X_i,Y_j) 并按顺序汇总信息得到 I(X,Y)？

谢谢。

例子：

m_1: X={a,b}, Y={x,y}; m_2: X={c}, Y={z,x}

原文

Suppose M is a set of objects m each having attributes X and Y. Now if X and Y can have only one value for given m (i.e. X,Y are random variables with P(X=x_i|M=m_i), P(Y=y_i|M=m_i)), it's possible to calculate mutual information of X and Y. But what if X can have multiple outcomes at once? I.e. for m_3 X={x1,x2} - generally outcome of X is subset of all possible outcomes. Can mutual information or some other measure of dependence be measured in such a case?

Is it possible to split X into binary random variables X_1, X_2, etc where X_1=1 iff X contains x1, X_1=0 otherwise and then compute I(X_i,Y_j) for all combinations i,j and sum up the information in order to get I(X,Y)?

Thanks.

Example:

m_1: X={a,b}, Y={x,y}; m_2: X={c}, Y={z,x}

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扛起拖把扫天下 2024-10-23 03:41:22

如果我没记错的话，你设置的前提是：

If M is a set of objects { m1, m2, ... },
and each mi has two attributes X, Y,
and X, Y can be a set of { x1, x2, ... } , { y1, y2, ... } respectively

那么你想定义

*(X, Y) based on each mi's X, Y

好吧，这在计算方面显着增加了问题的复杂性，但你仍然可以做相同类型的相关，除了不是关联两个值 X和 Y，您正在关联两个子集 X 和 Y。

If I'm not wrong, the premise you set:

If M is a set of objects { m1, m2, ... },
and each mi has two attributes X, Y,
and X, Y can be a set of { x1, x2, ... } , { y1, y2, ... } respectively

then you want to define

*(X, Y) based on each mi's X, Y

Well, this increases the complexity of the problem significantly in terms of computation, but you can still do the same type of correlation, except instead of correlating two values X and Y, you are correlating two subsets X and Y.

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