Position-Based Click Model

Definition

Position-Based Click Model (PBM)

The Position-Based Click Model is the simplest practical click model. It assumes a user clicks a document $d$ shown at rank $k$ iff the user examines rank $k$ and the document is relevant — and crucially that examination depends only on the rank $k$, never on the document, the query, or the surrounding results. It is the operational instantiation of the Examination Hypothesis in which the examination probability is a per-rank constant.

Intuition

Think of each result position as having a fixed “visibility” determined purely by where it sits on the page. Rank 1 is almost always looked at; rank 8 is looked at far less. The PBM bakes this into a single number per rank, the propensity $P({\text{Exam}}_k)$, and then treats whether the user clicks given that they looked as a clean measurement of relevance.

This factorization is what makes the model so useful: the two latent causes of a click — did they see it? (position) and did they like it? (relevance) — are assumed independent. Once you know the per-rank examination probabilities, an observed click becomes a noisy but unbiased-after-reweighting signal of relevance, which is exactly what IPW exploits.

The contrast to keep in mind: PBM says examination of rank $k$ is the same regardless of what is above it. The cascade model / Cascading Position Bias says the opposite — examination depends on the relevance of everything above $k$.

Mathematical Formulation

A click random variable $C_{d,k}\in \{0,1\}$ for document $d$ at rank $k$ factors into two independent Bernoulli events:

$P(C_{d,k}=1)=\underset{\text{examination (position only)}}{\underbrace{P(E_k=1)}}\cdot \underset{\text{relevance (position-free)}}{\underbrace{P(R_d=1\mid q)}}$

where:

• $C_{d,k}$ — observed click on document $d$ displayed at rank $k$
• $E_k\in \{0,1\}$ — latent examination event for rank $k$; $P(E_k=1)$ is the propensity ${\theta }_k$
• $R_d\in \{0,1\}$ — latent relevance of $d$ to query $q$; $P(R_d=1\mid q)={\gamma }_d$
• ${\theta }_k=P(E_k=1)$ — depends only on $k$ (the defining PBM assumption), monotonically decreasing in $k$
• ${\gamma }_d=P(R_d=1\mid q)$ — depends only on $(d,q)$, never on $k$

So the per-(document, rank) click probability is simply the product ${\theta }_k{\gamma }_d$. The latent variables $E_k$ are unobserved; only $C_{d,k}$ is observed, which is what forces an EM-style inference.

Likelihood and EM estimation

Given a click log of sessions $s$, each presenting document $d_s$ at rank $k_s$ with observed click $c_s$, the data log-likelihood is:

$\mathcal{L}(\theta ,\gamma )={\sum }_s[c_s\log ({\theta }_{k_s}{\gamma }_{d_s})+(1-c_s)\log (1-{\theta }_{k_s}{\gamma }_{d_s})]$

Because $E_k$ is latent, parameters are fit by Expectation-Maximization:

• E-step — for a non-click ($c_s=0$) we infer the posterior that the rank was nonetheless examined (the click failed because the doc was irrelevant): $P(E_{k_s}=1\mid c_s=0)=\frac{{\theta }_{k_s}(1-{\gamma }_{d_s})}{1-{\theta }_{k_s}{\gamma }_{d_s}},P(R_{d_s}=1\mid c_s=0)=\frac{(1-{\theta }_{k_s}){\gamma }_{d_s}}{1-{\theta }_{k_s}{\gamma }_{d_s}}$ (for a click $c_s=1$ both $E_{k_s}=1$ and $R_{d_s}=1$ are certain).
• M-step — re-estimate each parameter as the average of its inferred posterior over the relevant sessions: ${\theta }_k\leftarrow \frac{{\sum }_{s:k_s=k}[c_s+(1-c_s)P(E_k=1\mid c_s=0)]}{|\{s:k_s=k\}|},{\gamma }_d\leftarrow \frac{{\sum }_{s:d_s=d}[c_s+(1-c_s)P(R_d=1\mid c_s=0)]}{|\{s:d_s=d\}|}$

Iterating E and M to convergence yields the propensities ${\theta }_k$ used downstream.

Key Properties / Variants

• Two latent factors, one product — the entire model is $P(C_{d,k})={\theta }_k{\gamma }_d$; everything else (EM, IPW) is bookkeeping on top of this factorization.
• Propensities for counterfactual LTR — the fitted ${\theta }_k$ are exactly the inverse weights used by Inverse Propensity Weighting: a click at rank $k$ counts as $1/{\theta }_k$ units of relevance evidence, debiasing Counterfactual Learning to Rank objectives.
• Estimating ${\theta }_k$ without full EM — propensities can also be recovered by result randomization (swap a document across ranks and watch how its click rate scales) or intervention harvesting from naturally occurring rank changes, avoiding the joint EM fit.
• Identifiability caveat — if documents rarely change rank, the data cannot separate “clicked because examined” from “clicked because relevant”; multiple $(\theta ,\gamma )$ explain the log equally well. Randomization breaks this degeneracy.
• Position-only assumption is the weakness — PBM ignores that earlier results affect later examination. When users scan-and-stop, the cascade model / Cascading Position Bias is the correct alternative.
• Does not model trust or outlier effects — top ranks attracting extra clicks (Trust Bias) and visually distinctive items grabbing attention (Outlier Bias) both violate PBM’s clean factorization and require extended models.

Algorithm: PBM Parameter Estimation via EM
──────────────────────────────────────────────
Input: click log {(d_s, k_s, c_s)} over sessions s
Initialize θ_k, γ_d ∈ (0,1) for all ranks k, docs d
 
Repeat until convergence:
  # E-step: posteriors over latent E, R for non-clicks
  For each session s:
    if c_s == 1:
      P(E=1) ← 1 ;  P(R=1) ← 1
    else:                                  # c_s == 0
      denom    ← 1 - θ_{k_s} * γ_{d_s}
      P(E=1)   ← θ_{k_s} * (1 - γ_{d_s}) / denom
      P(R=1)   ← (1 - θ_{k_s}) * γ_{d_s} / denom
 
  # M-step: re-estimate as posterior averages
  For each rank k:
    θ_k ← mean over {s : k_s = k} of [ c_s + (1-c_s)*P(E=1)_s ]
  For each doc d:
    γ_d ← mean over {s : d_s = d} of [ c_s + (1-c_s)*P(R=1)_s ]
 
Return propensities {θ_k}  →  feed as 1/θ_k weights to IPW

Connections

• Instantiates: Examination Hypothesis (examination probability made a per-rank constant)
• Member of: Click Models (simplest member of the family)
• Quantifies: Position Bias via the propensities ${\theta }_k$
• Feeds: Inverse Propensity Weighting and Counterfactual Learning to Rank / Unbiased Learning to Rank
• Contrasted with: cascade model / Cascading Position Bias (examination depends on items above)
• Violated by: Trust Bias, Outlier Bias, Surrounding Item Bias
• Robust alternative when assumptions break: Doubly Robust Estimation

Appears In

• Examination Hypothesis
• Inverse Propensity Weighting
• Outlier Bias
• Unbiased Learning to Rank
• IR-L11 - Unbiased Learning to Rank