将概率重写为布尔代数
我得到了三个二进制随机变量:X、Y 和 Z。我还得到了以下内容:
P(Z | X)
P(Z | Y)
P(X)
P(Y)
然后我应该确定是否可以找到 P(Z | Y, X)
P(Z | Y, X)代码>.我尝试以贝叶斯定理的形式重写解决方案,但一事无成。鉴于这些是布尔随机变量,是否可以用布尔代数重写系统?我知道条件可以映射到布尔含义(x -> y
或 !x + y
),但我不确定这将如何翻译我试图解决的总体问题。
(是的,这是一个家庭作业问题,但在这里我更感兴趣的是如何正式解决这个问题而不是解决方案是什么......我还认为这个问题对于 MathOverflow 来说太简单了)
I'm given three binary random variables: X, Y, and Z. I'm also given the following:
P(Z | X)
P(Z | Y)
P(X)
P(Y)
I'm then supposed to determine whether or not it is possible to find P(Z | Y, X)
. I've tried rewriting the solution in the form of Bayes' Theorem and have gotten nowhere. Given that these are boolean random variables, is it possible to rewrite the system in terms of boolean algebra? I understand that the conditionals can be mapped to boolean implications (x -> y
, or !x + y
), but I'm unsure how this would translate in terms of the overall problem I'm trying to solve.
(yes, this is a homework problem, but here I'm much more interested in how to formally solve this problem than what the solution is...I also figured this question would be entirely too simple for MathOverflow)
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我敢打赌有人已经做得更优雅了,但是......
在这种情况下,不,不可能确定 P(Z|Y,X)。一般来说,我认为可以从一组独立的“原子”概率开始,并在添加约束时消除它们。例如,查看 X 和 Y,我们从四个概率开始:
现在我们添加概率加起来必须为 1 的约束。我们可以消除一个变量,任何变量,例如 d:
现在假设我们也知道 P(X) =K:
等等。在上述问题中,我们可以消除原始八个概率中的五个,但随后我们会被要求提供仍然独立的两个概率的比率。
I'll bet someone has done this more elegantly, but...
In this case, no, it's not possible to determine P(Z|Y,X). In general I think it's possible to start with a set of independent "atomic" probabilities and eliminate them as we add constraints. For example, looking at X and Y, we start with four probabilities:
Now we add the constraint that the probs must add up to 1. We can eliminate one variable, any variable, say d:
Now suppose we also know P(X)=K:
And so on. In the stated problem we can eliminate five of the original eight probabilities, but then we're asked for the ratio of two that are still independent.