Clipped Inverse Propensity Weighting

Background

Click-through training data is biased by where each impression appeared: a slot-1 impression has a much higher prior probability of being clicked than a slot-10 impression. Inverse propensity weighting (IPW) corrects for this by weighting each example by $1/p_k$ where $p_k$ is the prior click probability at the example's slot. Clipped IPW caps the weight at $w_{\max}$ so a single rare-slot example doesn't dominate the gradient.

Problem statement

Implement clipped_ipw_weights(slot_indices, slot_priors, w_max=5.0):

$$w_i \;=\; \min\!\Bigl(\frac{1}{p_{\text{slot}(i)}},\ w_{\max}\Bigr)$$

Input

• slot_indices - sequence of int slot positions for each example.
• slot_priors - sequence of float > 0, slot prior click probability indexed by slot position.
• w_max - float > 0, clip ceiling (default $5$).

Output

• np.ndarray of weights, same length as slot_indices.

Examples

Input: slot_indices=[0, 1, 9], slot_priors=[1.0, 0.5, 0.05]
       (where slot 0 always clicks, slot 9 rarely)
Output: [1.0, 2.0, 5.0]   # 1/0.05 = 20 clipped to 5

Constraints

• $p > 0$ required; ValueError otherwise.
• Slot index out of bounds raises IndexError.

Notes

• Why clip. Without clipping, variance grows like $1/p$, so a slot-10 example with $p = 0.1$ has 10x the variance of a slot-1 example. Clipping at $w_{\max} = 5$ is the canonical production default.