royf comments on How to Be Oversurprised

[royf]

When the new memory state $M_t$ is generated by a Bayesian update from the previous one $M_{t-1}$ and the new observation $O_t$, it’s a sufficient statistic of these information sources for the world state $W_t$, so that $M_t$ keeps all the information about the world that was remembered or observed:

=I(W_t;(M_{t-1},O_t)))

As this is all the information available, other ways to update can only have less information.

The amount of information gained by a Bayesian update is

)-I(W_t;M_{t-1}))

}{\Pr(W_t)\Pr(M_{t-1},O_t)}-\log\frac{\Pr(W_t,M_{t-1})}{\Pr(W_t)\Pr(M_{t-1})}\right])

}{\Pr(W_t,M_{t-1})}-\log\frac{\Pr(M_{t-1},O_t)}{\Pr(M_{t-1})}\right])

-\log\Pr(O_t%7CM_{t-1})])

and because the observation only depends on the world

+S(O_t%7CM_{t-1})])