Self-information, one event half as likely than another event conveys twice the amount of information?

Question

I was reading the following:

"If one event is half as likely as another, then learning about the former event shouldconvey twice as much information as the latter"

I know it should be easy to verify, but I just wanted to confirm with the community, that statement is false, right?

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This is what I have:

Let b half as likely as a. Then:

$$\frac{P_X(a)}{2} = P_X(b)$$

So we should have according to the comment:

$$I(b) = 2I(a) = I(a) + I(a)$$

So lets apply the definition of $I(x) = -\log(P_X(x))$

$I(b) = -\log(\frac{P_X(a)}{2}) = -\log(1/2) + -\log(P_X(a))$

$I(b) = 1 + I(a)$

Which implies that it only added one more bit of information rather than adding one addition $I(a)$ units of information (i.e. doubling $I(a)$)

This makes me belief either there must be a incredibly smart trick to make this work, or I am wrong or I did a super embarrassing mistake or the statement was wrongly expressed.

Answer 1

Perhaps we need more context. By itself, and using the common definition of information, the statement is false. For example, take: $P(A)=1/2$, and $P(B)=1$. The amount of information of event $B$ is zero. It makes no sense to say that $A$ gives usa twice information than that.