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I am reading Jaynes' "probability theory: the logic of science"

He uses a notation that I do not understand.

He says that if $A_i$ and $A_j$ are two mutually exclusive events, then:

$p(A_i A_j |B) = p(A_i |B)δ_{ij} $

How am I to understand this notation? Jayne uses Boolean logic notation, so $A_i A_j$ means the event that both $A_i$ and $A_j$ are true. However if these two events are mutually exclusive, wouldn't that mean that:

$p(A_i A_j |B) = 0$

So why the weird notation with $\delta _{ij}$?

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The notation $\delta_{i,j}$, sometimes called Kronecker delta, is defined as $\delta_{i,j}=0$ for $i \neq j$ and $\delta_{i,j}=1$ for $i = j$.

Thus, for $i\neq j$ the expression is indeed $0$ as you said. The point is that the expression you quote from the book is also valid for the case $i=j$, where it is not necessarily $0$.

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    $\begingroup$ As quid said. The Kronecker delta is a partial function, often used in vector, matrix, and tensors algebra. $$\begin{align}p(A_iA_j\mid B) ~&=~ p(A_i\mid B)~\delta_{i,j} \\[1ex] ~&=~ \begin{cases} p(A_i\mid B) &:& i=j\\0&:&i\neq j\end{cases}\end{align}$$ $\endgroup$ – Graham Kemp Dec 24 '16 at 16:06
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$$P(A_iA_j)=\left.\begin{cases}P(A_iA_i)=P(A_i), &\text{if } i=j\\ P(A_iA_j)=P(\emptyset)=0, &\text{if } i\neq j\end{cases}\right\}=P(A_iA_j)δ_{ij}$$ where $δ_{ij}$ is called Kronecker delta and has the purpose to indicate the event $i=j$. That is $$δ_{ij}=\begin{cases}1, &i=j\\0, &i\neq j\end{cases}$$

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