artificial-intelligence - 人工智能一种现代方法 - 随时间推移的概率推理

Question

我目前正在阅读 Peter Norvig 的《人工智能一种现代方法》第 15 章随时间推移的概率推理，但我无法遵循过滤和预测的推导（第 572 页）。

给定过滤到时间 t 的结果，代理需要根据新证据e _t+1计算 t + 1 的结果，

P ( X _t+1 | e _1:t+1 ) = f( e _t+1 , P ( X _t | e _1:t ))，

对于某些函数 f。这称为递归估计。我们可以将计算视为由两部分组成：首先，当前状态分布从 t 向前投影到 t + 1；然后使用新证据e _t+1对其进行更新。当重新排列公式时，这个由两部分组成的过程非常简单：

P ( X _t+1 | e _1:t+1 ) = P ( X _t+1 | e _1:t , e _t+1 ) （划分证据）
= α P ( e _t+1 | X _{t+ 1} , e _1:t ) P ( X _t+1 | e _1:t ) （使用贝叶斯规则）

使用贝叶斯规则如何导致上一个论坛？难道不应该

α P ( e _1:t , e _t+1 | X _t+1 ) P ( X _t+1 )

标签： artificial-intelligenceprobabilitybayesianreasoning

Answer 1

They are both correct, except that the α is not identical between the two. The trick is that in Norvig's application of Bayes', the conditioning on e_1:t is kept constant throughout. Just imagine that it wasn't there to begin with. You would still have all the identities. Then apply that conditioning to every part of the equation, and all the identities will hold.

Alternatively, the derivation can also be done without explicitly using Bayes', but instead, just using the definitions of joint and conditional probability (i.e., P(A|B)P(B)=P(A,B)=P(B|A)P(A)).

P(X_t+1|e_1:t, e_t+1) P(e_t+1,e_1:t) = P(e_1:t, e_t+1|X_t+1) P(X_t+1)

Expanding the joint probabilities on both sides gives

P(X_t+1|e_1:t, e_t+1) P(e_t+1|e_1:t) P(e_1:t) = P(e_t+1|X_t+1,e_1:t) P(e_1:t|X_t+1) P(X_t+1)

The last two terms on the RHS can be rewritten as

P(X_t+1|e_1:t, e_t+1) P(e_t+1|e_1:t) P(e_1:t) = P(e_t+1|X_t+1,e_1:t) P(X_t+1|e_1:t) P(e_1:t)

Now the P(e_1:t) cancels (assuming nonzero), and we have

P(X_t+1|e_1:t, e_t+1) = α P(e_t+1|X_t+1,e_1:t) P(X_t+1|e_1:t)

where α = 1 / P(e_t+1|e_1:t).