Plausibility-Constrained Generative
Scenario Design

for Explaining a Predict–Optimize Portfolio Pipeline

Batuhan Ataş

MPhil Business Data Science · Tinbergen Institute / University of Amsterdam

SUPERVISOR

Prof. dr. S.I. (İlker) Birbil · University of Amsterdam, Faculty of Economics and Business

COMMITTEE

Prof. dr. J.A.S. (Joaquim) Gromicho · University of Amsterdam, Faculty of Economics and Business

dr. E. (Ece) Karakoyun · Erasmus School of Economics, Econometric Institute

Research Master Pre-Defense

7 July 2026 · University of Amsterdam, Roeterseiland Campus

Portfolio decision pipelines

PTO : predict then optimize
PAO : predict and optimize

Markowitz (1952) · Ceria & Stubbs (2006) · Gu, Kelly & Xiu (2020)

one macro state m travels the machine · running: neural pipeline
PAO training · through the optimizer to the score · neural only PTO training · fit the forecast (MSE) · both classes inputs characteristics ca,t ∈ ℝ140 macro state mt ∈ ℝ9 features X(m) x = (c, c ⊗ m) ∈ ℝ1,400 return predictor fθ neural · smooth PTO / PAO tree · steps PTO only predicted returns μˆ(m) robust mean–variance layer
$\max_{w}\ \underbrace{\hat{\mu}^{\top}w}_{\text{reward}}-\underbrace{\kappa\sqrt{w^{\top}\Omega w}}_{\text{distrust}}-\underbrace{\tfrac{\lambda}{2}w^{\top}\Sigma w}_{\text{risk}}$
$w\in\Delta:\ \mathbf{1}^{\top}w=1,\ w\ge 0$ · long-only
 portfolio w(m) long-only weights on Δ score S(w, ·) · optional decision pipeline π · θ*, κ, λ frozen after training · only m varies
$\pi:\ \textcolor{#E24B4A}{m}\ \longmapsto\ X(\textcolor{#E24B4A}{m})\ \longmapsto\ \hat{\mu}(\textcolor{#E24B4A}{m})\ \longmapsto\ w_\pi(\textcolor{#E24B4A}{m})\in\Delta$

The gap: three literatures, each one ingredient short

Ribeiro et al. (2016) · Lundberg & Lee (2017) · Gu, Kelly & Xiu (2020) · Ataş, Aydın, Kıral & Birbil (2026)

ML in asset pricing trees and neural nets predict best importance ranks the forecast but not the optimizer's decision explanation methods attribution explains one forecast; counterfactuals invert to one input one point, no notion of plausibility scenario generation generates plausible market scenarios to stress-test portfolios but never inverts a pipeline's decision no single literature explains a trained predict-optimize pipeline, as a distribution over inputs, under an economically credible notion of plausibility the coauthored framework answers this, walking with gradients: the tree pipeline stays out of reach

Two research questions

Ataş, Aydın, Kıral & Birbil (2026)

RQ1 can a non-differentiable pipeline be explained at all? reach the behavior, stay realistic, without ever asking for a slope RQ2 do the two model classes tell the same economic story? one behavior, explained twice: through the tree, and through the neural network

Reversing the question: from a target behavior to macro states

probing framework: Ataş, Aydın, Kıral & Birbil (2026) · arXiv:2606.25808

m ∈ ℝ9 the pipeline, frozen X(m) fθ* κ, λ wπ(m) a chosen behavior at the decision level
hit a benchmark bPOE framework
which states bring the return to b?
$G^{\mathrm{bench}}_{\pi}(\textcolor{#E24B4A}{\vec m})=\big(\vec w_\pi(\textcolor{#E24B4A}{\vec m})^{\top}\vec r_{\bar t+1}-b\big)^{2}$
concentrate or diversifyPOE framework
which states pile up or spread the book?
$G^{\mathrm{conc}}_{\pi}(\textcolor{#E24B4A}{\vec m})=\pm H\big(\vec w_\pi(\textcolor{#E24B4A}{\vec m})\big)$
PTO vs PAOthis thesis
where does training choice change the decision?
$G^{\mathrm{div}}_{\pi}(\textcolor{#E24B4A}{\vec m})=\mp\big\lVert \vec w_{\mathrm{PTO}}-\vec w_{\mathrm{PAO}}\big\rVert_{1}$
Gπ(m) m-space valleys of Gπ the behavior is reached

each question becomes a probing function Gπ on macro states: low exactly where the behavior holds

The explanation: reach the behavior, stay plausible

variational formulation: Ataş, Aydın, Kıral & Birbil (2026)

Gπ(m) m-space valleys of Gπ the behavior is reached pπ,τ(m) m-space mass in the valleys the behavior, reachable the prior p0 reins it in explanations stay plausible
$$p_{\pi,\tau}\;=\;\operatorname*{arg\,min}_{p}\ \Big\{\ \underbrace{\mathbb{E}_p[G_\pi]}_{\textcolor{#0F6E56}{\text{reach the behavior}}}\;+\;\tau\,\underbrace{\mathrm{KL}(p\,\Vert\,p_0)}_{\textcolor{#185FA5}{\text{stay plausible}}}\ \Big\}$$
$$\text{unique solution:}\ \ p_{\pi,\tau}(m)\;=\;\frac{e^{-G_\pi(m)/\tau}\,p_0(m)}{\textcolor{#BA7517}{Z_\tau}},\quad \textcolor{#BA7517}{Z_\tau}=\int_{\mathcal{M}} e^{-G_\pi/\tau}\,p_0\,d\mu$$
τ small τ: sit on the behavior large τ: relax toward the prior

Plausibility is the prior: a VAR(1) tilted to the anchor

prior construction: Ataş, Aydın, Kıral & Birbil (2026) · Hamilton (1994)

“stay plausible” meant $\mathrm{KL}(p \,\Vert\, \textcolor{#185FA5}{p_0})$: plausible according to what? a law learned from the macro series itself

the plane of macro states $m$
$\mu^*$ · the long-run economy
$m_{\bar t}$ · the anchor month
the prior $p_0$: midway, and tighter
$$\vec m_{t+1}=\Phi\,\vec m_t+\vec c+\vec\varepsilon_t\ \ \Longrightarrow\ \ \text{stationary law }\ \mathcal N(\vec\mu^*,\Sigma^*)$$
$$\vec\mu^*=(I-\Phi)^{-1}\vec c\qquad\quad \Sigma^*=\Phi\,\Sigma^*\Phi^{\top}+\Sigma_\varepsilon$$
$$p_0\;=\;\mathcal N\!\Big(\tfrac{\vec\mu^*+\rho\,\vec m_{\bar t}}{1+\rho},\;\tfrac{\Sigma^*}{1+\rho}\Big)\qquad \textcolor{#E24B4A}{\rho=1}:\ \text{midway mean, half the covariance}$$

We cannot compute the explanation density, but we can walk it

Metropolis et al. (1953) · Hastings (1970)

the explanation density pπ,τ(m) prior p0 m-space pπ,τ(m) pπ,τ(m′) m m′ past visits
$$\text{keep }m'\text{ with chance}\quad \alpha \;=\; \min\Big\{1,\ \underbrace{\tfrac{p_{\pi,\tau}(m')}{p_{\pi,\tau}(m)}}_{\textcolor{#0F6E56}{\text{just a ratio: }\textcolor{#BA7517}{Z_\tau}\text{ cancels}}}\cdot\underbrace{\,c(m',m)\,}_{\textcolor{#185FA5}{\text{correction for how we propose}}}\Big\}$$
the walker's visits are the generated scenarios
preconditioned MALA same walk · different propose step c gradient-free ensemble

Sampling the explanation: preconditioned MALA on the Gibbs target

Roberts & Tweedie (1996) · Girolami & Calderhead (2011) · Ataş, Aydın, Kıral & Birbil (2026)

accepted 0 · acceptance –
realized return +1.50% b median generated return – b Metropolis-Hastings test accept chance –
target  $p_{\pi,\tau}\,\propto\,e^{-G_\pi/\tau}\,p_0$
$\widetilde G_\pi = G_\pi-\tau\log p_0$  ·  the prior folded into the landscape
 illustrative geometry target prior anchor target median chain gradient Σ*-proposal target level b
generated historical anchor running median
$$m' = m - \underbrace{\eta\,\Sigma^{*}\nabla\widetilde G_\pi(m)}_{\text{gradient drift}} + \underbrace{\sqrt{2\tau\eta}\,\big(\Sigma^{*}\big)^{1/2}\xi}_{\text{exploration noise}}$$
$$\text{accept with }\;\alpha=\min\Big\{1,\;\underbrace{\tfrac{p_{\pi,\tau}(m')}{p_{\pi,\tau}(m)}}_{\text{target ratio}}\cdot\underbrace{\tfrac{q(m\mid m')}{q(m'\mid m)}}_{\text{the correction }c(m',m)}\Big\}$$

Trees give the sampler no slope

gradient-based sampler: Ataş, Aydın, Kıral & Birbil (2026)

smooth · neural steps · tree m-space Gπ(m) tree pipeline inside a leaf cell flat ∇Gπ = 0 no slope to follow at the splits: ∇Gπ undefined
$$m' \;=\; m \;\underbrace{-\;\eta\,\Sigma^{*}\,\textcolor{#E24B4A}{\cancel{\textcolor{#4a5a6a}{\nabla\widetilde G_\pi(m)}}}}_{\textcolor{#E24B4A}{\nabla G_\pi=0\ \text{a.e., undefined at the splits}}}\;+\;\underbrace{\sqrt{2\tau\eta}\,\big(\Sigma^{*}\big)^{1/2}\xi}_{\text{a blind random walk: all that is left}}$$

This thesis: sample the same target without a gradient

sampler: Goodman & Weare (2010) · framework: Ataş, Aydın, Kıral & Birbil (2026)

a crowd of walkers reads direction and scale from its own spread: same target, no gradient anywhere

$$p_{\pi,\tau}\;\propto\;e^{-G_\pi/\tau}\,p_0\qquad\text{the target does not change: only the walk is new}$$
tree pipeline · no slope Gπ: the probe to sample the same target walkers sample it a crowd of walkers the coauthored framework Ataş, Aydın, Kıral & Birbil (2026) · arXiv:2606.25808 · the Gibbs target and the variational theory behind it · the plausibility prior: a VAR(1) tilted to the anchor · the original probing functions · the gradient-based sampler (MALA) the contribution this thesis built on that foundation · the gradient-free ensemble sampler: the core · neural vs tree on one behavior: a descriptive comparison

Sampling the explanation: a gradient-free ensemble on the same target

Goodman & Weare (2010) · Ataş, Aydın, Kıral & Birbil (2026)

accepted 0 · acceptance –
realized return +2.53% b median generated return – b stretch-move test Z – accept chance –
target  $p_{\pi,\tau}\,\propto\,e^{-G_\pi/\tau}\,p_0$
 single MALA chain: drift = 0, blind
$\min\{1,\ Z^{\,M-1}\,p_{\pi,\tau}(m')/p_{\pi,\tau}(m_k)\}$  ·  exact rule, M = 9
 illustrative geometry target prior anchor target median half A half B stretch line target level b
generated historical anchor running median
$$m' = m_j + \underbrace{Z\,(m_k - m_j)}_{\text{the walker difference stands in for the slope}}$$
$$\alpha=\min\Big\{1,\;\underbrace{\tfrac{p_{\pi,\tau}(m')}{p_{\pi,\tau}(m_k)}}_{\text{target ratio}}\cdot\underbrace{Z^{\,M-1}}_{\text{the correction }c(m',m)}\Big\}$$

Setting: monthly U.S. equities, one rule

Chen & Zimmermann (2022) · Welch & Goyal (2008) · Gu, Kelly & Xiu (2020)

firm characteristics $c$ · OSAP
$C = 140$, rank-normalized monthly
 release 202510 · ≥ 0.30 non-missing
macro state $m$ · Goyal-Welch
 dp dividend-price · ep earnings-price bm book-to-market · ntis net issuance tbl T-bill rate · tms term spread dfy default spread · svar market variance infl inflation
interaction features  $x = (c,\ c \otimes m)$
$F = 140 \times 10 = 1{,}400$
 the economy enters the forecast only through these interactions neural · NN3
$1400 \to 32 \to 16 \to 8 \to 1$ · smooth in $m$
 tree · HistGBR
piecewise constant in $m$: no slope
 training · 192 months validation · 120 test · 107 1990 2006 2016 2024 early stopping · λ, κ all reported numbers

The pipelines are worth explaining

thesis Table 4.2

Pipeline κ Test Sharpe Ann. return Ann. vol Effective N Equal-weight benchmark — 1.056 35.5% 33.7% 188.5 Neural · predict-then-optimize 0.25 1.111 29.1% 26.2% 39.6 Tree · predict-then-optimize 0.10 1.244 38.8% 31.2% 11.0 Neural · predict-and-optimize 0.25 1.130 29.6% 26.2% 30.3 the neural edge is a volatility story 33.7% cut to 26.2%, and the PAO variant edges PTO the tree edge is a return story 38.8% at the lightest robustness, κ = 0.10 Table 4.2 · test split 2016-2024

Three probes, four generation cells

probing functions: Ataş, Aydın, Kıral & Birbil (2026) · anchors: thesis §4.3

Benchmark return · which economies bring each pipeline to +2.99%? anchor · Nov 2017 neural · preconditioned MALA tree · gradient-free ensemble RQ2 Concentration · which economies push the tree to concentrate? anchor · Jan 2019 tree · gradient-free ensemble Divergence · where do two trainings of one network disagree? anchor · Jan 2016 neural PTO vs PAO · MALA

All four generation cells pass the gates

Gelman & Rubin (1992) · Geyer (1992) · thesis Table 4.3

RQ1 the precondition of any answer: the runs are trustworthy
$\widehat R$ · do the chains agree?
 ESS · how many independent draws? generation cell sampler
$\widehat R \le 1.05$
 ESS ≥ 100 acceptance in band verdict benchmark · neural · Nov 2017 preconditioned MALA 1.010 ● 292.1 ● 0.55 in [0.40, 0.70] ● pass benchmark · tree · Nov 2017 gradient-free ensemble 1.024 ● 106.7 ● 0.30 in [0.20, 0.50] ● pass concentration · tree · Jan 2019 gradient-free ensemble 1.025 ● 212.3 ● 0.28 in [0.20, 0.50] ● pass divergence · neural · Jan 2016 preconditioned MALA 1.015 ● 150.6 ● 0.69 in [0.40, 0.70] ● pass

One target, two opposed economies

Ataş, Aydın, Kıral & Birbil (2026) · thesis §4.3, Figures 4.1–4.3

which economies bring each pipeline to b = +2.99%? · S&P 500, Nov 2017 RQ2 neural PTO · preconditioned MALA +2.41% → +2.65% anchor → generated median · the risk-premium economy tree PTO · gradient-free ensemble +2.53% → +2.82% anchor → generated median · the calm expansion grey history, color generated, marks the anchor and generated mean · same target, opposite economies

Concentrating the tree: a fivefold squeeze

Ataş, Aydın, Kıral & Birbil (2026) · thesis §4.3, Figures 4.4–4.6

which economies push the tree to concentrate? · Jan 2019 36 → 7 effective holdings · anchor → generated median · Herfindahl 0.03 → 0.15, a fivefold squeeze grey history, color generated, marks the anchor and generated mean · tms carries the largest shift

Splitting the trainings: from 0.08 to 0.50

Ataş, Aydın, Kıral & Birbil (2026) · thesis §4.3, Figures 4.7–4.9

where do two trainings of one network disagree? · PTO vs PAO, Jan 2016 0.08 → 0.50 ℓ1 distance · anchor → generated median · a quarter of the book reallocated grey history, color generated, marks the anchor and generated mean · infl carries the dominant shift

The two questions, answered

thesis §5

RQ1 yes: a pipeline with no gradient can be explained at the decision level, on the same footing as a neural network the ensemble reaches each targeted behavior, the generated economies stay plausible, and every cell passed the gates RQ2 no: the two model classes reach the same target through opposite economies so a stress narrative must name the pipeline it reads Plausibility-Constrained Generative Scenario Design for Explaining a Predict-Optimize Portfolio Pipeline · pre-defense · 7 July 2026

Limitations, and where this goes next

thesis §5

Limitations · one tactical, firm-level U.S. equity setting · one anchor per probe: explanations are fingerprints of particular pipelines and dates · the class comparison is descriptive, not causal Future work · many anchors and behaviors: fingerprints become distributions · probe firm characteristics, not just macro · carry the sampler to other non-differentiable pipelines

Thank you

Batuhan Ataş · Tinbergen Institute · University of Amsterdam

the gap a tree pipeline could not be explained at the decision level: no gradient to follow the method a gradient-free ensemble that samples the same plausibility- constrained Gibbs target the result the gradient-boosted tree, explained on the same footing as the neural network, gated the finding the two model classes reach the same target through opposite economies Questions? supervisor: Prof. dr. Ş. İlker Birbil · committee: Prof. dr. J.A.S. Gromicho, dr. E. Karakoyun coauthored framework: Ataş, Aydın, Kıral & Birbil (2026) · arXiv:2606.25808 backup slides follow: figures, sampler settings, and predictive detail
Backup
figures · sampler settings · predictive detail · demo pointers

Backup · Two samplers, one target

MALA: Ataş, Aydın, Kıral & Birbil (2026) · stretch move: Goodman & Weare (2010)

one target $p_{\pi,\tau}$
 two ways to walk it
preconditioned MALA · needs the slope
$$m' = m-\eta\,\Sigma^{*}\nabla\widetilde G_\pi(m)+\sqrt{2\tau\eta}\,\big(\Sigma^{*}\big)^{1/2}\xi$$
$$c(m',m)=\frac{q(m\mid m')}{q(m'\mid m)}\qquad \text{the drifted Gaussian ratio}$$
direction from $\nabla\widetilde G_\pi$, scale from $\Sigma^*$  ·  one walker, guided at every step
stretch move · no gradient anywhere
$$m' = m_j + Z\,(m_k - m_j),\qquad\text{accept: }\ \min\!\big\{1,\ \textcolor{#0F6E56}{Z^{\,M-1}}\,p_{\pi,\tau}(m')/p_{\pi,\tau}(m_k)\big\}$$
$$g(z)\propto z^{-1/2}\ \text{on}\ [1/a,\,a],\qquad g(1/z)=z\,g(z)$$
$c = \textcolor{#0F6E56}{Z^{M-1}}$, the Jacobian of the stretch  ·  one knob: $a$  ·  two halves alternate: a partner always comes from the frozen half
the walker spread is the geometry: what $\Sigma^*$ hand-builds for MALA, the crowd learns from its own shape, and for free

Backup · regime composition of the generated states

thesis Figures 4.2, 4.5, 4.8

benchmark · neural · Nov 2017 benchmark · tree · Nov 2017 concentration · tree · Jan 2019 divergence · neural · Jan 2016

Backup · NFCI history and nearest analogs

thesis Figures 4.3, 4.6, 4.9

benchmark · neural · analogs 2017 benchmark · tree · analogs 2017 concentration · tree · analogs 2006-2007 divergence · neural · analogs 2006-2007

Backup · sampler settings and diagnostics detail

thesis §4.3, Table 4.3

preconditioned MALA cells · 4 chains × 4,000 steps, first 1,000 discarded · step size tuned to hold acceptance in [0.40, 0.70] · exact analytic gradient through cvxpylayers,   no finite differencing · preconditioner: the prior covariance Σ* benchmark: 1.010 / 292.1 / 0.55 · divergence: 1.015 / 150.6 / 0.69 gradient-free ensemble cells · 40 walkers × 4,000 sweeps, first 1,000 discarded · stretch scale a = 2.0, acceptance held in [0.20, 0.50] · two halves updated in alternation: a partner always   comes from the frozen half · out-of-box proposals rejected outright benchmark: 1.024 / 106.7 / 0.30 · concentration: 1.025 / 212.3 / 0.28 how the diagnostics are formed per coordinate on the nine-dimensional state, judged at the worst coordinate: the largest R̂ and the smallest ESS per cell ensemble chains: the forty post-burn-in walkers grouped as four · read per walker instead, R̂ = 1.040 and 1.043, inside the gate either way R̂: classic unsplit Gelman-Rubin (1992) · ESS: Geyer (1992) initial-positive truncation · acceptance bands are where each sampler mixes best too low and the chain rarely moves; too high and it only inches

Backup · predictive accuracy and the PAO losses

thesis §3.1, §4.2 · Gu, Kelly & Xiu (2020) · Donti et al. (2017)

predictive accuracy, out of sample
test $R^2_{\text{oos}}$ against the zero benchmark:
 neural +0.35%  ·  tree +0.31% in line with the 0.33%–0.40% of GKX’s strongest models sub-1% is the standard of evidence here: a thin forecast can still rank stocks well enough to earn reliable portfolio gains the PAO decision loss
primary: negative mean-variance task utility, $\ell_{\text{dec}}(w,r) = -\big(w^{\top}r - \tfrac{\gamma}{2}\,w^{\top}\Sigma\,w\big)$, evaluated at the optimizer’s output
 gradients flow through the differentiable convex layer (cvxpylayers, implicit differentiation of optimality conditions) alternatives trained: realized-return, robust-utility, batch Sharpe live demo pointers, if asked slide 8, the τ-slider: honest Gibbs tempering on the MALA widget slide 11, the piecewise Gπ demo: a lone MALA chain goes blind, the ensemble keeps moving tree hyperparameters: depth, number of trees B, learning rate ν fixed on the validation window (2006-2015), then frozen the tree stays PTO-only by design: a differentiable surrogate would defeat the purpose of a genuinely non-differentiable class

Backup · References

full bibliography: thesis, references section

Ataş, B., Aydın, N., Kıral, E.M. & Birbil, Ş.İ. (2026). Probing-based explanation of portfolio     decision pipelines. arXiv:2606.25808. Ceria, S. & Stubbs, R.A. (2006). Incorporating estimation errors into portfolio selection:     robust portfolio construction. Journal of Asset Management, 7. Chen, A.Y. & Zimmermann, T. (2022). Open source cross-sectional asset pricing.     Critical Finance Review, 11. Donti, P., Amos, B. & Kolter, J.Z. (2017). Task-based end-to-end model learning in     stochastic optimization. NeurIPS. Gelman, A. & Rubin, D.B. (1992). Inference from iterative simulation using multiple     sequences. Statistical Science, 7. Geyer, C.J. (1992). Practical Markov chain Monte Carlo. Statistical Science, 7. Girolami, M. & Calderhead, B. (2011). Riemann manifold Langevin and Hamiltonian     Monte Carlo methods. JRSS-B, 73. Goodman, J. & Weare, J. (2010). Ensemble samplers with affine invariance.     Comm. App. Math. and Comp. Sci., 5. Gu, S., Kelly, B. & Xiu, D. (2020). Empirical asset pricing via machine learning.     Review of Financial Studies, 33. Hamilton, J.D. (1994). Time Series Analysis. Princeton University Press. Hastings, W.K. (1970). Monte Carlo sampling methods using Markov chains     and their applications. Biometrika, 57. Lundberg, S.M. & Lee, S.-I. (2017). A unified approach to interpreting model     predictions. NeurIPS. Markowitz, H. (1952). Portfolio selection. Journal of Finance, 7. Metropolis, N., Rosenbluth, A.W., Rosenbluth, M.N., Teller, A.H. & Teller, E. (1953).     Equation of state calculations by fast computing machines. J. Chem. Phys., 21. Ribeiro, M.T., Singh, S. & Guestrin, C. (2016). "Why should I trust you?"     Explaining the predictions of any classifier. KDD. Roberts, G.O. & Tweedie, R.L. (1996). Exponential convergence of Langevin     distributions and their discrete approximations. Bernoulli, 2. Welch, I. & Goyal, A. (2008). A comprehensive look at the empirical performance     of equity premium prediction. Review of Financial Studies, 21. per-slide sources appear in each slide’s footer line; the thesis bibliography is the complete record