# Core Counterfactual Question Logic
# P(Y0=0, Y1=1 | X) = Probability customer would NOT churn WITHOUT treatment,
# but WOULD churn WITH treatment ("Sleeping Dogs")

# 1. Estimate uplift (CATE) for the individual
cate = uplift_model.predict_proba(X_customer)[:, 1] - uplift_model.predict_proba(X_customer)[:, 0]

# 2. Estimate baseline risk from control group model
p_y0 = control_model.predict_proba(X_customer)[:, 1]

# 3. Bound the joint probability using the Frechet-Hoeffding bounds
lower_bound = max(0, p_y0 + cate - 1)
upper_bound = min(p_y0, cate)

# The true joint probability lies within [lower_bound, upper_bound]
# Use domain knowledge or stronger assumptions to pinpoint.

The Uplift vs. Counterfactual Gap

Uplift modeling asks: "What is the *difference* in outcome if I treat this person?" It gives you a net effect, like a +5% chance of retention. It's powerful for targeting.

Counterfactual identification asks a harder, richer question: "What is the *joint probability* of both potential outcomes for this specific person?" For example: "What's the chance this customer would have churned without our offer (Y1=1) BUT would have stayed if we had given it (Y0=0)?"

That joint probability is the holy grail. It tells you not just the effect, but the underlying nature of the individual.

The Four Archetypes of Every Audience

With full counterfactual knowledge, you can segment any population into four precise groups:

• Sure Things: Will buy/convert/stay regardless. Don't waste your treatment on them.
• Lost Causes: Will not convert no matter what. Don't waste resources.
• Persuadables: Will convert ONLY if treated. This is who uplift models target.
• Sleeping Dogs: Will convert ONLY if left *untreated*. Your intervention actually annoys or dissuades them.

Traditional uplift finds Persuadables. Counterfactual identification finds all four groups. Missing Sleeping Dogs can be catastrophic—your "helpful" nudge drives them away.

How The New Method Works (The Synergy)

The arXiv research shows you don't need to start from scratch. You can build on existing uplift models.

The key is moving from estimating a difference (uplift) to estimating a bivariate distribution. The provided code uses the Frechet-Hoeffding bounds to give a range for that joint probability based on what your uplift and control models already tell you.

It's a pragmatic bridge. You use your reliable uplift estimate (CATE) and baseline risk (p_y0) to constrain the possible values of the counterfactual joint probability. With additional, reasonable assumptions (like monotonicity), you can pinpoint it.

Why This Matters Now

Privacy changes and signal loss make broad-brush marketing inefficient. Precision is everything. Wasting a discount on a "Sure Thing" customer isn't just lost revenue—it trains them to wait for discounts.

In healthcare, it's the difference between knowing a drug works on average and knowing which patients it helps, harms, or does nothing for. This is personalized medicine at a causal level.

The tools are here. Libraries like CausalML and EconML provide the uplift models. This research provides the mathematical framework to layer on the counterfactual insight. The first teams to implement this will stop guessing and start knowing.