Building the Mechanism

Bivalent, Exhaustive Mediators for Front-Door Identification with Bayesian SEMs

Nathaniel Forde

Data Science @ PandaDoc

and Open Source Contributor @ PyMC

2026-11-12

Preliminaries

Who am I?

I’m a data scientist at PandaDoc
- Bayesian statistician,
- Reformed philosopher and logician.
Website: https://nathanielf.github.io/

Link to my website

Code or it didn’t Happen

The companion notebook, with every model and sensitivity analysis in this talk, is here. The worked example is drawn from the SEM chapter of Bayesian Causal Inference (Vincent & Forde, in preparation).

The Pitch

The front-door criterion identifies a causal effect through unmeasured confounding, with no instrument required.

So why does almost nobody use it?

It asks the mediator to carry everything: every directed path from \(X\) to \(Y\).
Any effect that leaks around the mediator breaks identification.
Read as a demand on a single observed node, almost no real mediator qualifies.

The argument: the mediator should be built, not found. Identification is a piece of information architecture.

Agenda

Act I: The Demand of the Front Door. Why a single mediator node almost never qualifies, and why identification is an architecture rather than a discovery

Act II: Building the Mechanism. A bivalent mediator set for job satisfaction, encoded in two matrices of a Bayesian SEM

Act III: Auditing the Architecture. Relaxing the zeros, recovering the parameters, and springing a leak

Act I: The Demand of the Front Door

The Front-Door Criterion

Pearl (1995): \(M\) identifies the effect of \(X\) on \(Y\) when

\(M\) intercepts every directed path from \(X\) to \(Y\)
There is no unblocked back-door path from \(X\) to \(M\)
Every back-door path from \(M\) to \(Y\) is blocked by \(X\)

Then, despite the unmeasured \(U\):

\[P(y \mid do(x)) = \sum_m P(m \mid x) \sum_{x'} P(y \mid m, x') P(x')\]

Two well-identified pieces, \(X \to M\) and \(M \to Y \mid X\), are chained together.

Why It Is Rarely Used

A bare DAG node is a single indicator: loading fixed at 1, no measurement error.
Asked to exhaust a pathway in social, marketing or organisational data, it fails.

The pieces exist: Pearl admits a mediator set; Kuroki (2000) and Kuroki & Cai (2004) treat front-door in linear SEMs; multi-indicator latents are standard practice.

What is missing is the construction step.

This is the hinge of the talk. On the left, the way we usually read the criterion: go and find a node M. One observed variable, loading fixed at one, no error, and we must believe it carries the whole effect. Any leak, the dashed arc, and identification is gone. Morgan and Winship’s complaint applies here: explanations that stop at “X raises Y” without a mechanism aren’t good enough, yet the front-door criterion demands a mechanism so complete that no single node can supply it. On the right, the alternative: build the mechanism. A set of latent constructs, each measured by its own indicators, chosen because together they plausibly exhaust the pathway. None of the pieces are new. What is usually left implicit is the construction step itself, and that is what I want to make explicit.

Identification as Information Architecture

Every credible design decides where variation is allowed to flow. None of these plans is found in the data.

Design	Unwanted variation is…	The argument you owe
Propensity score weighting	rebalanced: reweighting builds a pseudo-population where treatment is independent of the observed covariates	every confounder is observed, and overlap holds
Double ML	residualised away: flexible nuisance models partial it out of \(T\) and \(Y\)	every confounder is observed
Mundlak device	absorbed by group means: cluster means of the covariates soak up the correlation between group effects and treatment	the unobserved confounding lives at the group level
Control-function IV	modelled explicitly: the first-stage residual \(\hat v\) enters the outcome equation	relevance and exclusion of the instrument
Front-door SEM	quarantined: confounding may live in \(\text{Cov}(\zeta_X, \zeta_Y)\), and nowhere else	the mechanism is exhaustive and unconfounded

A structural equation model is a blueprint of permitted covariances. The front-door design is one floor plan: the effect must walk through the mechanism, the confounding must stay in its room.

This is the frame I want you to leave with. These designs are familiar because they are presented as estimators, but each is an architecture: a decision about which variation is nuisance and where it is allowed to go. Read the table top to bottom and the confounding gets harder to see. Propensity weighting rebalances it away, and DML residualises it away, but both need every confounder observed. The Mundlak device tolerates unobserved confounding provided it lives at the group level, where the cluster means can absorb it. The control function gives unobserved confounding a named term, paid for with an instrument. The front-door SEM quarantines it: treatment-outcome confounding is allowed to exist, but only in one covariance cell, and the mechanism is walled off from it. In each case the identifying argument is about the design, not about anything the data can certify.

Act II: Building the Mechanism

The Job Satisfaction Constructs

Houghton & Jinkerson (2007): 12 indicators, 4 constructs; 263 observations simulated from their published correlations.

CTS, constructive thought strategies (treatment): EBA, ST, MI
DTS, dysfunctional thought: DA1–DA3
WB, well-being: UF1, UF2, FOR
Satisfaction (outcome): JW1–JW3

The mechanism is bivalent: CTS suppresses a negative construct and raises a positive one.

The front-door claim: these two routes exhaust CTS’s influence: a premise to argue, not a finding.

One Covariance Form, Two Matrices

Both models share the marginal covariance

\[\Sigma_y = \Lambda(I - \color{blue}{B})^{-1}\color{blue}{\Psi_\zeta}(I - \color{blue}{B})^{-\top}\Lambda^\top + \Psi\]

and the mechanism is encoded entirely in \(\color{blue}{B}\) and \(\color{blue}{\Psi_\zeta}\).

\[\color{blue}{B} = \begin{bmatrix} \overbrace{0}^{Sat} & \overbrace{\beta_{sw}}^{WB} & \overbrace{\beta_{sd}}^{DTS} & \overbrace{\color{red}{0}}^{CTS} \\ 0 & 0 & \color{red}{0} & \beta_{wc} \\ 0 & 0 & 0 & \beta_{dc} \\ 0 & 0 & 0 & 0 \end{bmatrix}\]

Direct edge CTS → Sat and the path DTS → WB are removed.

\[\color{blue}{\Psi_\zeta} = \begin{bmatrix} \psi_{s} & 0 & 0 & \color{blue}{\psi_{sc}} \\ 0 & \psi_{w} & \color{blue}{\psi_{wd}} & 0 \\ 0 & \color{blue}{\psi_{wd}} & \psi_{d} & 0 \\ \color{blue}{\psi_{sc}} & 0 & 0 & \psi_{c} \end{bmatrix}\]

Exactly two covariances stay free: treatment–outcome confounding \(\psi_{sc}\), and structure inside the mediator set \(\psi_{wd}\).

Every other zero is the formal content of “the mechanism explains everything.”

The Mechanism in Code

FRONTDOOR_PATHS = [
    ("dysfunctional", "constructive"),  # CTS -> DTS
    ("well being", "constructive"),     # CTS -> WB
    ("satisfaction", "well being"),     # WB  -> Sat
    ("satisfaction", "dysfunctional"),  # DTS -> Sat
]
FRONTDOOR_COVS = [
    ("satisfaction", "constructive"),   # confounding we survive
    ("well being", "dysfunctional"),    # inside the mechanism
]
with pm.Model(coords=coords) as frontdoor_model:
    Lambda, Psi = measurement_model(sample_df)
    Psi_zeta = make_Psi_zeta(FRONTDOOR_COVS)
    betas = pmd.Normal("mu_betas", 0, 1, dims="frontdoor_paths")
    B = set_edges(pt.zeros((4, 4)), FRONTDOOR_PATHS, betas.values)

    inv = pt.linalg.solve(pt.eye(4) - B, pt.eye(4))
    Sigma_y = Lambda @ inv @ Psi_zeta @ inv.T @ Lambda.T + Psi
    pm.MvNormal("likelihood", mu=0, cov=Sigma_y, observed=observed_data)

The edge list is the mechanism.
The permitted covariances are the walls.
Everything else is a zero we have to defend.
The total effect is read off \((I - B)^{-1}\).

The Payoff

	Total effect CTS → Sat	94% HDI	Posterior SD	SRMR
Saturated	+0.19	[−1.25, +1.62]	0.77	0.054
Front-door	+0.14	[+0.01, +0.27]	0.07	0.053

The saturated model is not identified: its interval is mostly prior. The front-door model is identified, and its interval just excludes zero. Global fit is unchanged: the restrictions trade one-for-one for identification, so fit cannot test them.

What the Second Mediator Buys

CTS raises a positive force (well-being: +0.14 × +0.69 = +0.09) and suppresses a negative one (dysfunctional thinking: −0.16 × −0.31 = +0.05).
Find only well-being: the suppression route becomes a leak, and the front door misses about a third (35%) of the effect.

Two different gains from building:

Several indicators per construct handle measurement error.
Several constructs, of opposite valence, handle exhaustiveness: that is what closes the pathway.

This is what “built, not found” buys, in numbers, and it’s where the bivalence becomes visible. On the left are the four links. Constructive thinking raises well-being, and well-being raises satisfaction: two positives. It also suppresses dysfunctional thinking, which itself drags satisfaction down: two negatives. Both routes therefore raise satisfaction, one by adding a good force, the other by removing a bad one. In a linear SEM the total effect is the sum of the two routes: about 0.09 through well-being and 0.05 through suppression. Neither route on its own has an interval clear of zero; only the two together do. Now suppose we had gone looking for a mediator, found well-being, and stopped. A front door through well-being alone treats the dysfunctional-thinking route as a path around the mediator, a leak, and as we’ll see in Act Three, a leak biases the front door by exactly its size. So the single-mediator analysis would report only the well-being route and miss about a third of the effect. Two different things come from building. Multiple indicators per construct deal with measurement error. Multiple constructs, chosen to cover opposite valences, deal with exhaustiveness, which is what the front door actually needs.

Act III: Auditing the Architecture

Relaxing the Zeros

Muthén & Asparouhov (2012): replace each exact zero with \(\text{Normal}(0, \sigma)\) and watch the conclusion.

The mean barely moves (≈ 0.14–0.18): the data are nearly silent about the direct edge.
The interval does all the moving: at \(\sigma = 0.05\) the HDI already reaches zero; at \(\sigma = 0.5\) it is almost as wide as the saturated model’s.
Re-admitting DTS → WB changes nothing. One zero, CTS → Sat, carries the precision.

Simulating from the Mechanism

Can the model recover itself?

# Fix every parameter at the fitted values,
# plus a direct leak the front-door model forbids
with pm.do(generator, {**truth, "leak": [delta]}):
    y = pm.sample_prior_predictive(draws=1)[
        "prior"]["likelihood"]  # one synthetic survey, n = 263
# Condition a fresh front-door model on it
with pm.observe(build_frontdoor(observed=None),
                {"likelihood": y}):
    idata = pm.sample()

With \(\delta = 0\) the model recovers what generated the data: the total-effect HDI covers the truth in 8 / 8 replicates. But the estimates are noisy at \(n = 263\), because the CTS → mediator paths are small.

Springing a Leak

In a linear SEM a front door recovers only what flows through its mediators, so a leak of size \(\delta\) biases it by exactly \(-\delta\) (dotted line). The simulations track this, noisily: five replicates per \(\delta\); coverage of the true effect falls to zero by \(\delta = 0.3\).
The leak is absorbed as confounding: \(\rho(\zeta_{CTS}, \zeta_{Sat})\) rises from 0.10 to 0.27, and SRMR never moves.

Now dial the leak up. The analytic answer comes first: in a linear model the front door recovers only what flows through its mediators, so the bias is exactly minus the leak. The dotted line is that prediction, and the simulations follow it, noisily, with only five replicates per setting. Where does the leak go? Into the one covariance the design leaves open, the treatment-outcome confounding term. A path around the mechanism is observationally equivalent to the confounding we built the design to tolerate, so with a positive leak the model reads part of the real effect as confounding and under-reports it. And global fit is flat throughout. Nothing in the data warns you. This is why the argument for exhaustiveness has to be made on the theory, and why the relaxation curve belongs next to the point estimate.

A Design Pattern for People Analytics

Same DAG, easier premises to defend

The recipe, applied

Build: after a thought-strategy course, measure the change in CTS, well-being, dysfunctional thinking and perceived support.
Argue: the course reaches satisfaction only through them. Who gets sent is the confounding the design survives.
Encode: the argument as zeros in \(B\) and \(\Psi_\zeta\).
Audit: relax the zeros, simulate leaks, recover parameters.

If enrolment can be randomised, just compare the groups. The front door is for selected enrolment, the usual HR case.
Given the DAG, the cross-sectional SEM is already a causal argument: conditioning stands in for intervention. A dated course makes the DAG’s premises easier to defend: time order, and a named confounder.

Here is the study I would actually want, and it’s the one people analytics teams can run. The treatment is attending a thought-strategy course: a dated intervention, not a personality trait. After the course you measure the mechanism: the change in constructive thinking, well-being, dysfunctional thinking. Satisfaction comes later. Time order is now built in. The confounding the front door survives is concrete: who gets sent on the course. Keen employees opt in; managers nominate their engaged people; and those same people are more satisfied anyway. That’s why you can’t just compare attendees with everyone else. The obvious leak in this design is that being sent on a course can make people feel invested in, which raises satisfaction whatever happens to their thinking. The recipe’s answer is not to assume that away but to measure it, perceived organisational support, and build it into the mechanism as its own route. If you can randomise enrolment, do that and skip all of this; the front door is for the much more common case where you can’t. And to be clear about what this does and doesn’t change. Pearl’s point stands: if the DAG is right, conditioning does the work of intervention, so the cross-sectional SEM we fitted is already a valid causal argument, conditional on the graph. A course doesn’t change that logic. It makes the graph easier to believe: the time order is built in, the treatment is a dated event rather than a trait, and the confounder has a name you can argue about.

Conclusion

Identification is Built

“SEM is an inference engine that takes in two inputs, qualitative causal assumptions and empirical data, and derives two logical consequences of these inputs: quantitative causal conclusions and statistical measures of fit for the testable implications of the assumptions… Fitting the data does not ”prove” the causal assumptions, but it makes them tentatively more plausible.”

– Bollen & Pearl, Eight Myths about Causality and Structural Equation Models

The front-door restrictions are not even tentatively tested by fit: they are paid for in identification, one-for-one.

So the warrant for the estimate lives in the construction: the choice of constructs, their measurement, and the argument that they exhaust the pathway.

Bayesian SEMs make that warrant auditable: soften the zeros, simulate the leaks, and report how much of the answer the architecture supplies.

“The mediator is not found.

It is built.”

Resources

Resources

Companion notebook: github.com/NathanielF/…/frontdoor_sem_sensitivity.ipynb
PyMC: pymc.io
Pearl (1995), “Causal Diagrams for Empirical Research,” Biometrika
Kuroki (2000), “Selection of Post-Treatment Variables for Estimating Total Effect,” J. Japan Stat. Soc.
Kuroki & Cai (2004), “Selection of Identifiability Criteria for Total Effects,” UAI
Muthén & Asparouhov (2012), “Bayesian SEM: A More Flexible Representation of Substantive Theory,” Psych. Methods
Houghton & Jinkerson (2007), “Constructive Thought Strategies and Job Satisfaction,” J. Business & Psychology
Bollen & Pearl (2013), “Eight Myths about Causality and SEMs”