When deriving the probability distribution for the isolated system, we were working in the dark – we assigned the probabilities devoid of information about the system. It is natural to ask: were we endowed with some data, how could we use it to revise our set of probabilities?

Let’s suppose that with each microstate we associate a value . The quantity is the value of some observable in the state . The physical meaning of can be anything you like, so long as it can be measured numerically. If is the probability that the system is in state , then the probability that has the value is . So the *mean value* of is

by definition of the probabilities . *The mean value of the observable ** is that which we measure in the lab. *If we can say with confidence that we know some property of the system through experimental evidence, we place a further constraint on the through the form of equation given above. *We convey our knowledge about the system through constraints.*

At last we’re ready to continue.

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