组合联合概率
我正在尝试计算概率分布的表达式（与生物信息学相关），并且在组合来自两个不同来源的随机变量的信息时遇到困难。 本质上，场景如下： 有 3 个离散随机变量 X、A 和 A。 B. X 依赖于 A 和 B。A 和 B 仅通过 X 相关，即给定 X 时 A 和 B 是独立的。现在，我导出了以下表达式： P(X, A) 和 P(X, B)。 我需要计算 P(X, A, B) - 这不是链式法则的简单应用。
因为 P(A) 可用，所以我可以从第一个表达式导出 P(X | A)。 B 永远不会独立于 A 被观察到，P(B) 不容易获得 - 充其量我可以通过边缘化 A 来近似它，但表达式 P(A, B) 没有闭合形式，因此积分很棘手。
关于如何在不丢弃信息的情况下导出 P(X, A, B) 有什么想法吗？ 提前谢谢了。
阿米特
I am trying to work out the expression for a probability distribution (related to bioinformatics), and am having trouble combining the information about a random variable from two different sources. Essentially, here is the scenario:
There are 3 discrete random variables X, A & B. X depends on A and B. A and B are related only through X, i.e. A and B are independent given X. Now, I have derived the expressions for:
P(X, A) and P(X, B). I need to calculate P(X, A, B) - this is not a straightforward application of the chain rule.
I can derive P(X | A) from the first expression since P(A) is available. B is never observed independently of A, P(B) is not readily available - at best I can approximate it by marginalizing over A, but the expression P(A, B) does not have a closed form so the integration is tricky.
Any thoughts on how P(X, A, B) can be derived, without discarding information? Many thanks in advance.
Amit
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您在这里处理的是一个无向无环图。 给定X，A有条件地独立于B，但X取决于（我直接假设）A和B。我对你的问题的性质有点困惑，即你的概率分布的指定形式，但你可以看看信念传播。
What you're dealing with here is an undirected acyclic graph. A is conditionally independent of B given X, but X depends (I assume directly) on A and B. I'm a little confused about the nature of your problem, i.e. what form your probability distributions are specified in, but you could look at belief propagation.
好吧，自从我做联合概率以来已经很长了，所以对此持保留态度，但考虑到 A 和 B 是正交的，我首先要开始寻找的是类似这样的表达式：
P(X, A, B) = P(X,A) + (P(X,B) * (1-P(X,A)));
再说一次，这只是为了给你一个探索的想法，因为我已经很长时间没有做过这类工作了！
Ok, it has been a long time since I've done joint probabilities so take this with a big grain of salt but the first place I would start looking, given that A and B are orthogonal, is for an expression something like:
P(X, A, B) = P(X,A) + (P(X,B) * (1-P(X,A)));
Again, this is just to give you an idea to explore as it has been a very long time since I did this type of work!
就您观察到的内容和未知的内容而言，您的问题非常不清楚。 似乎您明确陈述的唯一事实是 A 和 B 是独立的，给定 X。也就是说，
假设： P(A,B|X)=P(A|X)P(B|X)
因此： P(A ,B,X)=P(A,B|X)P(X)=P(A|X)P(B|X)P(X)=P(A,X)P(X)=P(B ,X)P(X)
选择因式分解。
Your question is very unclear in terms of what you observe and what are unknowns. It seems like the only fact that you state clearly is A and B are independent given X. That is,
Assumption: P(A,B|X)=P(A|X)P(B|X)
Hence: P(A,B,X)=P(A,B|X)P(X)=P(A|X)P(B|X)P(X)=P(A,X)P(X)=P(B,X)P(X)
Take your pick of factorizations.