## Surprise index

Warren Weaver [1] introduced what he called the surprise index to quantify how surprising an event is. At first it might seem that the probability of an event is enough for this purpose: the lower the probability of an event, the more surprise when it occurs. But Weaver’s notion is more subtle than this.

Let *X* be a discrete random variable taking non-negative integer values such that

Then the surprise index of the *i*th event is defined as

Note that if *X* takes on values 0, 1, 2, … *N*−1 all with equal probability 1/*N*, then *S*_{i} = 1, independent of *N*. If *N* is very large, each outcome is rare but not surprising: because all events are equally rare, no specific event is surprising.

Now let *X* be the number of legs a human selected at random has. Then *p*_{2} ≈ 1, and so the numerator in the definition of *S*_{i} is approximately 1 and *S*_{2} is approximately 1, but *S*_{i} is large for any value of *i* ≠ 2.

The hard part of calculating the surprise index is computing the sum in the numerator. This is the same calculation that occurs in many contexts: Friedman’s index of coincidence, collision entropy in physics, Renyi entropy in information theory, etc.

## Poisson surprise index

Weaver comments that he tried calculating his surprise index for Poisson and binomial random variables and had to admit defeat. As he colorfully says in a footnote:

I have spent a few hours trying to discover that someone else had summed these series and spent substantially more trying to do it myself; I can only report failure, and a conviction that it is a dreadfully sticky mess.

A few years later, however, R. M. Redheffer [2] was able to solve the Poisson case. His derivation is extremely terse. Redheffer starts with the generating function for the Poisson

and then says

Let *x* = *e*^{iθ}; then *e*^{−iθ}; multiply; integrate from 0 to 2π and simplify slightly to obtain

The integral on the right is recognized as the zero-order Bessel function …

Redheffer then “recognizes” an expression involving a Bessel function. Redheffer acknowledges in a footnote at a colleague M. V. Cerrillo was responsible for recognizing the Bessel function.

It is surprising that the problem Weaver describes as a “dreadfully sticky mess” has a simple solution. It is also surprising that a Bessel function would pop up in this context. Bessel functions arise frequently in solving differential equations but not that often in probability and statistics.

## Unpacking Redheffer’s derivation

When Redheffer says “Let *x* = *e*^{iθ}; then *e*^{−iθ}; multiply; integrate from 0 to 2π” he means that we should evaluate both sides of the equation for the Poisson generating function equation at these two values of *x*, multiply the results, and average the both sides over the interval [0, 2π].

On the right hand side this means calculating

This reduces to

because

i.e. the integral evaluates to 1 when *m* = *n* but otherwise equals zero.

On the left hand side we have

Cerrillo’s contribution was to recognize the integral as the Bessel function *J*_{0} evaluated at -2*i*λ or equivalently the modified Bessel function *I*_{0} evaluated at -2λ. This follows directly from equations 9.1.18 and 9.6.16 in Abramowitz and Stegun.

Putting it all together we have

Using the asymptotic properties of *I*_{0} Redheffer notes that for large values of λ,

[1] Warren Weaver, “Probability, rarity, interest, and surprise,” The Scientific Monthly, Vol 67 (1948), p. 390.

[2] R. M. Redheffer. A Note on the Surprise Index. The Annals of Mathematical Statistics, Mar., 1951, Vol. 22, No. 1 pp. 128ndash;130.