Bits Per Reading in the Long Run

A cleaning robot wanders between three rooms — Kitchen, Hall, Study — and its room at the next minute depends only on the room it is in now. The transition probabilities are:

Currently in	$\to$ Kitchen	$\to$ Hall	$\to$ Study
Kitchen	$0.6$	$0.3$	$0.1$
Hall	$0.3$	$0.5$	$0.2$
Study	$0.2$	$0.4$	$0.4$

A logger records which room the robot is in once every minute, forever. After the chain has settled into its long-run behaviour, the room recorded on a given minute behaves like a draw from the chain's stationary distribution.

You want the average surprisal of a single logged reading, judged on its own and ignoring which room preceded it — the Shannon entropy of that stationary distribution.

Base-2 logarithms (you will not need all of them):

$v$	$0.1$	$0.2$	$0.3$	$0.4$	$0.5$	$0.6$
$\log_2 v$	$-3.3219$	$-2.3219$	$-1.7370$	$-1.3219$	$-1.0000$	$-0.7370$

What is that entropy, in bits? Round your answer to 3 decimal places.

Bits Per Reading in the Long Run

Hints

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Markov Chain Simulator

Transition Rates