The Savage Representation

The Savage Representation

As is understood since Savage, proper scoring rules are in one-to-one correspondence with strictly convex functions via the so-called Savage representation

As is understood since Savage, proper scoring rules are in one-to-one correspondence with strictly convex functions via the so-called Savage representation

Seeing that Savage worked with Von Neumann seems to explain quite a few things: the game-theoretic ideas and convex duality arguments and ideas seem very connected to Von Neumann's works in that field

Seeing that Savage worked with Von Neumann seems to explain quite a few things: the game-theoretic ideas and convex duality arguments and ideas seem very connected to Von Neumann's works in that field

$s(\vec \pi,i)=G(\vec \pi)+\lang\delta_i,\nabla G(\vec \pi)\rang$

This is assuming that $G$ is smooth, which we do

This is assuming that $G$ is smooth, which we do

The proof of this is simple and beautiful

The proof of this is simple and beautiful

We take the Savage function $S$, represented the expected return assuming $i$ is sampled according to the "reality" $\vec p$

We take the Savage function $S$, represented the expected return assuming $i$ is sampled according to the "reality" $\vec p$

$S(\vec\pi,\vec p)=G(\vec \pi) + \lang\vec p,\nabla G(\pi)\rang$

The point is then that this is an affine function of the reality $\pi$ (like any expectation is), and we can understand the optimization problem via the angle that since $G$ is convex, as a function of $\vec p$, we have

The point is then that this is an affine function of the reality $\pi$ (like any expectation is), and we can understand the optimization problem via the angle that since $G$ is convex, as a function of $\vec p$, we have

$S(\vec \pi,\vec p)\leq G(\vec p)$ for all $\vec p$

$S(\vec \pi,\vec p)\leq G(\vec p)$ for all $\vec p$

This follows since $\vec p \mapsto S(\vec \pi,\vec p) $ is affine and tangent to $G$ at $\vec p=\vec \pi$

This follows since $\vec p \mapsto S(\vec \pi,\vec p)$ is affine and tangent to $G$ at $\vec p=\vec \pi$

Since $G$ is convex, we have that it is the sup of all tangents to it, so $\sup_{\vec \pi} S(\vec\pi,\vec p)=G(\vec p)$, which is exactly to say that the scoring rule is proper

Since $G$ is convex, we have that it is the sup of all tangents to it, so $\sup_{\vec \pi} S(\vec\pi,\vec p)=G(\vec p)$, which is exactly to say that the scoring rule is proper

How are things if $G$ is piecewise affine?

How are things if $G$ is piecewise affine?

The converse: any proper scoring rule is Savage

The converse: any proper scoring rule is Savage

It simply hinges on understanding that $G(\vec p)$ is the sup over all $S(\vec \pi,\vec p)$, and since $S(\vec \pi,\vec p)$ is affine in $\vec p$, we find that its value can be deduced from $G$ at the tangent point via the Savage representation

It simply hinges on understanding that $G(\vec p)$ is the sup over all $S(\vec \pi,\vec p)$, and since $S(\vec \pi,\vec p)$ is affine in $\vec p$, we find that its value can be deduced from $G$ at the tangent point via the Savage representation

And the concrete scoring rule is obtained by "collapsing $\vec p$" to a "deterministic reality $i$"

And the concrete scoring rule is obtained by "collapsing $\vec p$" to a "deterministic reality $i$"

This is probably an interesting thing to look at!

This is probably an interesting thing to look at!