Information loss of a 9-input majority gate According to information theory, the logic gates AND, NAND, OR, NOR all lose 1.189 bits of information each with two bits of information at their inputs and with all inputs being independently and uniformly distributed.

When gates are used in combination with other gates in a circuit, to calculate the entropy loss for the entire circuit you have to use what are called "mixture probabilities" or "mixing distributions" to calculate the entropy loss, the loss for each gate being dependent on the entropy loss history of the preceding gates.

Can anyone calculate the entropy loss of a majority gate with 9 inputs, all equally likely $\{1,0\}$?
Let me first explain what the entropy loss is for a gate like AND. The input entropy is $2h(1/2) = 2$ (where $h(p) = -p\log_2p-(1-p)\log_2(1-p)$), while the output entropy is $h(1/4)$ (since the probability that the output is $1$ is $1/4$). Therefore the entropy loss is $2 - h(1/4) \approx 1.189$.

In the case of recursive majority, the output is uniform, so the output entropy is $1$. The input entropy, in contrast, is $9$, and so the entropy loss is $9-1=8$.