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Hierarchical Bayesian Models

You want to evaluate 20 traders with varying track record lengths. Two obvious approaches, both wrong:

Complete pooling: treat everyone as identical. Ignores real differences.

No pooling: estimate each independently. The trader with 15 trades gets an estimate driven entirely by noise.

Hierarchical models take the middle path, and it is not a compromise so much as the correct answer.

Partial pooling

Assume individual parameters are drawn from a common distribution:

A hierarchical model
θi∼Beta(α,β),yi∼Binomial(ni,θi)\theta_i \sim \text{Beta}(\alpha, \beta), \qquad y_i \sim \text{Binomial}(n_i, \theta_i)

Each unit gets its own parameter and those parameters share a distribution. That sharing is the borrowing of strength.

Traders differ (each has their own θi\theta_i) and are related (all drawn from a shared population). The hyperparameters α,β\alpha, \beta describe the population and are themselves estimated from the data.

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