Hierarchical models

Idea: A short record next to a long one

Venue K: $3$ days averaging $14$. Venue L: $36$ days, also $14$. One number for each?

• Each venue alone takes lucky days at face value
• One pooled number ignores real differences
• A hierarchical model sits between

Idea: Randomness in layers

Each venue has its own edge ${\theta }_i$, drawn from a shared population.

• Between venues: ${\theta }_i\sim N(\mu ,{\tau }^2)$
• Within a venue: ${\bar{x}}_i\mid {\theta }_i\sim N\left({\theta }_i,\frac{{\sigma }^2}{n}\right)$
• The shared layer lets a thin record borrow

Formula: Partial pooling

$E[{\theta }_i\mid {\bar{x}}_i]=(1-B){\bar{x}}_i+B\mu$

The estimate slides from the venue’s own average toward μ by the shrinkage factor B.

Formula: The shrinkage factor

$B=\frac{\frac{{\sigma }^2}{n}}{\frac{{\sigma }^2}{n}+{\tau }^2}$

Noise of the venue’s average over noise plus real spread. A noisy average gives B near 1.

Example: Venue K after 3 days

Take $\mu =2$, ${\tau }^2=4$, ${\sigma }^2=36$. Venue K has $n=3$ and averages $14$. Estimate its edge.

1. Noise of the average: $\frac{36}{3}=12$
2. $B=\frac{12}{12+4}=\frac{3}{4}$
3. $\frac{1}{4}\times 14+\frac{3}{4}\times 2=5$

Quick check: Venue L after 36 days

Same population. Venue L has $n=36$ and also averages $14$. What is its estimate?

Idea: Does it help?

Venue K’s raw average has squared error $12$. The shrunk estimate has $12\times \frac{4}{12+4}=3$.

• A small pull toward $\mu$ buys a large cut in variance

Common mistake: Shrinkage as a discount

Estimates move toward $\mu$, not toward zero, so bad records improve too.

Instead A venue at $-10$ after $3$ days moves up to $\frac{1}{4}\times (-10)+\frac{3}{4}\times 2=-1$.

Idea: Where the top level comes from

With equal $n$, the spread of the averages estimates ${\tau }^2+\frac{{\sigma }^2}{n}$.

• Spread $10$, $n=9$, ${\sigma }^2=36$: $10-\frac{36}{9}=6$
• Negative? Set ${\tau }^2$ to zero and pool fully
• Few groups: put priors on $\mu$ and ${\tau }^2$ too

Idea: News travels through the top level

A good run from one signal raises the chance of a good regime, and so the estimate for another signal.

• Even for a signal that has never traded
• The data moved up to the top level and back down