SURPRISE: it’s called SURPRISAL!
This is a well-known concept in information theory. It’s also called ‘information content‘.
Let’s see why. First, let’s remember the setup. We have a manifold
whose points are nowhere vanishing probability distributions on the set We have a function
called the Shannon entropy, defined by
For each point we define a cotangent vector by
As mentioned last time, this is the analogue of momentum in probability theory. In the second half of this post I’ll say more about exactly why. But first let’s compute it and see what it actually equals!
Let’s start with a naive calculation, acting as if the probabilities were a coordinate system on the manifold We get
so using the definition of the Shannon entropy we have
Now, the quantity is called the surprisal of the probability distribution at Intuitively, it’s a measure of how surprised you should be if an event of probability occurs. For example, if you flip a fair coin and it lands heads up, your surprisal is ln 2. If you flip 100 fair coins and they all land heads up, your surprisal is 100 times ln 2.
Of course ‘surprise’ is a psychological term, not a term from math or physics, so we shouldn’t take it too seriously here. We can derive the concept of surprisal from three axioms:
- The surprisal of an event of probability is some function of , say
- The less probable an event is, the larger its surprisal is:
- The surprisal of two independent events is the sum of their surprisals:
It follows from work on Cauchy’s functional equation that must be of this form:
for some constant We shall choose the base of our logarithms, to be We had a similar freedom of choice in defining the Shannon entropy, and we will use base for both to be consistent. If we chose something else, it would change the surprisal and the Shannon entropy by the same constant factor.
So far, so good. But what about the irksome “-1” in our formula?
Luckily it turns out we can just get rid of this! The reason is that the probabilities are not really coordinates on the manifold They’re not independent: they must sum to 1. So, when we change them a little, the sum of their changes must vanish. Putting it more technically, the tangent space is not all of but just the subspace consisting of vectors whose components sum to zero:
The cotangent space is the dual of the tangent space. The dual of a subspace
is the quotient space
The cotangent space thus consists of linear functionals modulo those that vanish on vectors obeying the equation
Of course, we can identify the dual of with in the usual way, using the Euclidean inner product: a vector corresponds to the linear functional
From this, you can see that a linear functional vanishes on all vectors obeying the equation
if and only if its corresponding vector has
So, we get
In words: we can describe cotangent vectors to as lists of n numbers if we want, but we have to remember that adding the same constant to each number in the list doesn’t change the cotangent vector!
This suggests that our naive formula
is on the right track, but we’re free to get rid of the constant 1 if we want! And that’s true.
To check this rigorously, we need to show
for all We compute:
where in the second to last step we used our earlier calculation:
and in the last step we used
Back to the big picture
Now let’s take stock of where we are. We can fill in the question marks in the charts from last time, and combine those charts while we’re at it.
|Classical Mechanics||Thermodynamics||Probability Theory|
What’s going on here? In classical mechanics, action is minimized (or at least the system finds a critical point of the action). In thermodynamics, entropy is maximized. In the maximum entropy approach to probability, Shannon entropy is maximized. This leads to a mathematical analogy that’s quite precise. For classical mechanics and thermodynamics, I explained it here:
These posts may give a more approachable introduction to what I’m doing now: now I’m bringing probability theory into the analogy, with a big emphasis on symplectic and contact geometry.
Let me spell out a bit of the analogy more carefully:
Classical Mechanics. In classical mechanics, we have a manifold whose points are positions of a particle. There’s an important function on this manifold: Hamilton’s principal function
What’s this? It’s basically action: is the action of the least-action path from the position at some earlier time to the position at time 0. The Hamilton–Jacobi equations say the particle’s momentum at time 0 is given by
Thermodynamics. In thermodynamics, we have a manifold whose points are equilibrium states of a system. The coordinates of a point are called extensive variables. There’s an important function on this manifold: the entropy
There is a cotangent vector at the point given by
The components of this vector are the intensive variables corresponding to the extensive variables.
Probability Theory. In probability theory, we have a manifold whose points are nowhere vanishing probability distributions on a finite set. The coordinates of a point are probabilities. There’s an important function on this manifold: the Shannon entropy
There is a cotangent vector at the point given by
The components of this vector are the surprisals corresponding to the probabilities.
In all three cases, is a symplectic manifold and imposing the constraint picks out a Lagrangian submanifold
There is also a contact manifold where the extra dimension comes with an extra coordinate that means
• action in classical mechanics,
• entropy in thermodynamics, and
• Shannon entropy in probability theory.
We can then decree that along with and these constraints pick out a Legendrian submanifold
There’s a lot more to do with these ideas, and I’ll continue next time.
For all my old posts on information geometry, go here: