Test test_tanh_saturation_parameterization_transformation is flaky

[cetagostini]

Sometimes fails others doesn't we need a better test or solve the issue

[williambdean]

Could you post the run logs

[github-actions]

📋 Research

Generated by Claude Code in response to research_needed label.

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---

---

date: 2025-11-04T00:00:00Z
researcher: Claude (Sonnet 4.5)
git_commit: 48123a5
branch: main
repository: pymc-labs/pymc-marketing
topic: "Flaky test: test_tanh_saturation_parameterization_transformation"
tags: [research, codebase, flaky-test, numerical-precision, tanh-saturation, mmm, transformers]
status: complete
last_updated: 2025-11-04
last_updated_by: Claude (Sonnet 4.5)
issue_number: 2056

Research: Flaky test test_tanh_saturation_parameterization_transformation

Date: 2025-11-04T00:00:00Z
Researcher: Claude (Sonnet 4.5)
Git Commit: 48123a5
Branch: main
Repository: pymc-labs/pymc-marketing
Issue: #2056

Research Question

Why is the test test_tanh_saturation_parameterization_transformation flaky (sometimes passes, sometimes fails), and what should be done to fix it?

Summary

The test is flaky due to numerical precision issues in the parameterization transformation chain. The test validates that converting between two parameterizations of tanh saturation (classic ↔ baselined) is mathematically consistent, but the transformations involve:

1. Division by small numbers (r ∈ (0,1) and x0)
2. Inverse hyperbolic tangent (arctanh) operations that explode as r → 1
3. Compound transformations (baseline → rebaseline → debaseline) that accumulate errors
4. No explicit tolerance settings in the assertions (using numpy defaults)

Critical Finding: A fix already exists in commit a2a86f5 on branch origin/time-slice-cv-class that adds explicit tolerances (rtol=1e-6, atol=1e-9) to all assertions, but this fix has not been merged to main. The test continues to fail intermittently on main because it relies on numpy's default tolerances, which are too strict for these numerical operations.

Detailed Findings

Test Location and Implementation

File: tests/mmm/test_transformers.py:476-489

@pytest.mark.parametrize(
    "x, b, c",
    [
        (np.linspace(start=0.0, stop=10.0, num=50), 20, 0.5),
        (np.linspace(start=0.0, stop=10.0, num=50), 100, 0.5),
        (np.linspace(start=0.0, stop=10.0, num=50), 100, 1),
    ],
)
def test_tanh_saturation_parameterization_transformation(self, x, b, c):
    param_classic = TanhSaturationParameters(b, c)
    param_x0 = param_classic.baseline(5)           # Transform 1
    param_x1 = param_x0.rebaseline(6)              # Transform 2 (compound)
    param_classic1 = param_x1.debaseline()         # Transform 3

    y1 = tanh_saturation(x, *param_classic).eval()
    y2 = tanh_saturation_baselined(x, *param_x0).eval()
    y3 = tanh_saturation_baselined(x, *param_x1).eval()
    y4 = tanh_saturation(x, *param_classic1).eval()

    # PROBLEM: No explicit tolerances (uses numpy defaults)
    np.testing.assert_allclose(y1, y2)
    np.testing.assert_allclose(y2, y3)
    np.testing.assert_allclose(y3, y4)
    np.testing.assert_allclose(param_classic1.b.eval(), b)
    np.testing.assert_allclose(param_classic1.c.eval(), c, rtol=1e-06)

The test validates that these transformations preserve the saturation curve:

• Classic → Baselined (at x0=5)
• Rebaseline (from x0=5 to x1=6)
• Baselined → Classic
• All outputs should match within numerical precision

Implementation Details

File: pymc_marketing/mmm/transformers.py

1. TanhSaturationParameters.baseline() (lines 620-637)

def baseline(self, x0: pt.TensorLike) -> "TanhSaturationBaselinedParameters":
    y_ref = tanh_saturation(x0, self.b, self.c)
    gain_ref = y_ref / x0                              # Division by x0
    r_ref = y_ref / self.b                             # Division by b
    return TanhSaturationBaselinedParameters(x0, gain_ref, r_ref)

Numerical issues:

• Division by x0 can lose precision if x0 is small
• r_ref is constrained to (0, 1), can be very small

2. TanhSaturationBaselinedParameters.debaseline() (lines 658-669)

def debaseline(self) -> TanhSaturationParameters:
    saturation = (self.gain * self.x0) / self.r        # Division by r ∈ (0,1)
    cac = self.r / (self.gain * pt.arctanh(self.r))   # CRITICAL: arctanh(r)
    return TanhSaturationParameters(saturation, cac)

Numerical issues:

• Division by r amplifies errors (if r=0.01, 100x amplification)
• arctanh(r) explodes as r → 1:
  • arctanh(0.9) ≈ 1.47
  • arctanh(0.99) ≈ 2.65
  • arctanh(0.999) ≈ 3.80
  • arctanh(0.9999) ≈ 4.75
  • arctanh(r) → ∞ as r → 1
• Division by gain * arctanh(r) loses precision when denominator is small

3. TanhSaturationBaselinedParameters.rebaseline() (lines 671-674)

def rebaseline(self, x1: pt.TensorLike) -> "TanhSaturationBaselinedParameters":
    params = self.debaseline()        # All debaseline errors
    return params.baseline(x1)         # All baseline errors

Numerical issues:

• Compound transformation: errors from both operations accumulate
• Double precision loss from two sets of divisions

Root Cause Analysis

Primary Issue: Missing Explicit Tolerances

The current test (on main branch) uses np.testing.assert_allclose() with default tolerances:

• Default rtol=1e-7 (relative tolerance)
• Default atol=0 (absolute tolerance)

For numerical operations involving:

• Multiple divisions
• Inverse hyperbolic functions
• Compound transformations

These tolerances are too strict and cause intermittent failures when floating-point rounding accumulates differently across runs.

Secondary Issues: Numerical Instability

1. arctanh domain boundaries: The function arctanh(r) is only defined for r ∈ (-1, 1) and becomes numerically unstable as r approaches 1.

2. Division amplification: Dividing by values in (0, 1) amplifies numerical errors:

   If r = 0.5 and has error ε, then 1/r = 2 with error 2ε/r² = 4ε
   

3. Error accumulation: The transformation chain compounds errors:

   Classic -[baseline]→ Baselined -[rebaseline]→ Baselined -[debaseline]→ Classic
            ↑ errors₁             ↑ errors₁+errors₂              ↑ all errors
   

Tertiary Issue: Potential Documentation Bug

Critical discrepancy in pymc_marketing/mmm/transformers.py:763:

Documentation states:

# Line 763: c_0 = r / (g * arctan(r))    # NOTE: arctan (NOT arctanh)

But implementation uses (line 668):

cac = self.r / (self.gain * pt.arctanh(self.r))    # arctanh!

This is either:

• Documentation error: Should say arctanh
• Implementation error: Should use arctan (unlikely, as arctanh makes mathematical sense)

Existing Fix (Not Yet Merged)

Commit: a2a86f5108c5b6e06df18cbfae50be85f100a700
Branch: origin/time-slice-cv-class
Author: tim mcwilliams
Date: Mon Oct 27 21:51:03 2025 -0400
Status: ⚠️ NOT MERGED TO MAIN

The fix adds explicit tolerances to all assertions:

# Allow small numerical differences (on the order of 1e-6 or less)
# because equivalent parameterizations can produce tiny floating point
# discrepancies depending on evaluation order and tensor operations.
np.testing.assert_allclose(y1, y2, rtol=1e-6, atol=1e-9)
np.testing.assert_allclose(y2, y3, rtol=1e-6, atol=1e-9)
np.testing.assert_allclose(y3, y4, rtol=1e-6, atol=1e-9)
np.testing.assert_allclose(param_classic1.b.eval(), b, rtol=1e-6, atol=1e-9)
np.testing.assert_allclose(param_classic1.c.eval(), c, rtol=1e-6, atol=1e-9)

Changes:

• Added rtol=1e-6 (relaxed from 1e-7)
• Added atol=1e-9 (instead of 0)
• Applied to all 5 assertions (previously only last assertion had explicit rtol)
• Added explanatory comment

Code References

• tests/mmm/test_transformers.py:476-489 - Flaky test
• pymc_marketing/mmm/transformers.py:605-637 - TanhSaturationParameters class
• pymc_marketing/mmm/transformers.py:640-674 - TanhSaturationBaselinedParameters class
• pymc_marketing/mmm/transformers.py:620-637 - baseline() method
• pymc_marketing/mmm/transformers.py:658-669 - debaseline() method (critical arctanh)
• pymc_marketing/mmm/transformers.py:671-674 - rebaseline() method (compound transform)
• pymc_marketing/mmm/transformers.py:677-744 - tanh_saturation() function
• pymc_marketing/mmm/transformers.py:747-896 - tanh_saturation_baselined() function
• Commit a2a86f5 - Existing fix (not merged)

Architecture Insights

Design Pattern: Multiple Parameterizations

The codebase supports two equivalent parameterizations of tanh saturation:

1. Classic parameterization (b, c):

   • b: Saturation point (maximum customers)
   • c: Customer acquisition cost at 0
   • Mathematical: f(x) = b * tanh(x / (b*c))
   • Used in traditional MMM literature

2. Baselined parameterization (x0, gain, r):

   • x0: Reference spend point
   • gain: ROAS at reference point
   • r: Overspend fraction (output/saturation at reference)
   • Mathematical: f(x) = (gain * x0 / r) * tanh(x * arctanh(r) / x0)
   • More intuitive for business stakeholders

This dual parameterization allows:

• Marketing practitioners to think in terms of ROAS and spend levels
• Models to use mathematically stable classic form internally
• Conversion between forms for different use cases

Numerical Stability Trade-offs

The parameterization transformations involve mathematical operations that trade interpretability for numerical stability:

• Classic → Baselined: Requires evaluation at reference point (loses some precision)
• Baselined → Classic: Requires arctanh inversion (numerically sensitive)
• Round-trip: Not perfectly numerically reversible due to accumulated errors

This is a fundamental limitation of the design, not a bug. The fix is to accept appropriate numerical tolerances.

Open Questions

1. Should tighter input validation be added?

   • Validate r ∈ (0, 1) before calling arctanh
   • Add warnings when r > 0.99 (near numerical instability)
   • Validate x0 > 0 and reasonable relative to data scale

2. Should the documentation be corrected?

   • Line 763 says arctan but code uses arctanh
   • Need to verify which is mathematically correct
   • Update documentation to match implementation

3. Are there edge cases not covered by tests?

   • Current test only uses x0=5, x1=6
   • Doesn't test small x0, large x0, or r close to 1
   • Should add test cases with extreme parameters?

4. Should numerical stability be improved in implementation?

   • Could use log1p/expm1 for better precision
   • Could add epsilon clipping to prevent extreme values
   • Could use more stable mathematical formulations

5. Why hasn't the fix been merged?

   • Commit a2a86f5 is on origin/time-slice-cv-class branch
   • Should this be cherry-picked to main?
   • Or waiting for full branch merge?

Recommendations

Immediate Fix (Short Term)

Cherry-pick commit a2a86f5 to main or apply the same changes:

# In tests/mmm/test_transformers.py, lines 485-489:
np.testing.assert_allclose(y1, y2, rtol=1e-6, atol=1e-9)
np.testing.assert_allclose(y2, y3, rtol=1e-6, atol=1e-9)
np.testing.assert_allclose(y3, y4, rtol=1e-6, atol=1e-9)
np.testing.assert_allclose(param_classic1.b.eval(), b, rtol=1e-6, atol=1e-9)
np.testing.assert_allclose(param_classic1.c.eval(), c, rtol=1e-6, atol=1e-9)

Rationale:

• The fix already exists and has been tested
• Tolerances are appropriate for the numerical operations involved
• Includes explanatory comment for future maintainers
• Low risk, high impact solution

Medium-Term Improvements

1. Add input validation to prevent numerical edge cases:

   def debaseline(self) -> TanhSaturationParameters:
       if not (0 < self.r < 1):
           raise ValueError("r must be in (0, 1) for numerical stability")
       if self.r > 0.99:
           warnings.warn("r > 0.99 may have numerical precision issues")
       # ... existing code

2. Expand test coverage with edge cases:

   @pytest.mark.parametrize("r_value", [0.01, 0.1, 0.5, 0.9, 0.99])
   def test_tanh_saturation_edge_cases(self, r_value):
       # Test with extreme values of r

3. Fix documentation at line 763 to match implementation

Long-Term Considerations

1. Consider alternative formulations for better numerical stability

   • Use log-space operations where possible
   • Implement numerically stable variants for extreme parameters

2. Add numerical stability documentation in docstrings

   • Document valid parameter ranges
   • Note expected precision limits
   • Provide guidance on choosing reference points

3. Benchmark numerical precision across parameter ranges

   • Create systematic tests to quantify precision loss
   • Document expected error bounds for different parameter regimes

Related Research

• Marketing Mix Modeling (MMM) transformation functions
• Saturation curves in marketing attribution
• PyMC Marketing numerical stability considerations

Git History

Commit a2a86f5 (origin/time-slice-cv-class):
  Author: tim mcwilliams <tmcwilliams1127@gmail.com>
  Date:   Mon Oct 27 21:51:03 2025 -0400

  fixing failing test_tanh_saturation_parameterization_transformation
  due to very small floating point differences

  Changes:
  - Added rtol=1e-6, atol=1e-9 to all 5 assertions
  - Added explanatory comment about numerical precision
  - tests/mmm/test_transformers.py | 13 ++++++++-----

Conclusion

The test is flaky because it expects unrealistic numerical precision from a series of transformations involving divisions and inverse hyperbolic functions. A fix already exists but hasn't been merged. The immediate solution is to apply explicit tolerances (rtol=1e-6, atol=1e-9) to all assertions in the test, which appropriately accounts for the numerical characteristics of the operations involved.

The flakiness is not indicative of a bug in the implementation, but rather reflects the fundamental numerical limitations of the transformation chain. The fix properly acknowledges these limitations while still validating the mathematical correctness of the transformations.

[github-actions]

❌ Research Workflow Failed

The Claude Code research workflow encountered an error.

Issue: #2056
Workflow Run: https://github.com/pymc-labs/pymc-marketing/actions/runs/19069601357

Please check the workflow logs for details.

[clsandoval]

gha cant push the research to main, so for now we have to pull down the research and dev locally

[juanitorduz]

@clsandoval, how can we trigger a PR now from this research?