How Entropy-Pooling Algorithms Reweight Financial Scenarios: KL Divergence, Moment Views, Convex Constraints, Lagrange Multipliers, Weight Concentration and Infeasible Views

Reader question: If we already have a set of simulated or historical financial scenarios, how can we impose a new view—such as a changed expected loss, spread level or stress probability—while altering the original scenario distribution as little as possible?

Entropy pooling answers this as a constrained optimisation problem. Start with scenario outcomes and prior probabilities. Express the new information as mathematical constraints. Then choose posterior scenario probabilities that satisfy those constraints while remaining as close as possible to the prior distribution in relative-entropy, or Kullback–Leibler, terms.

This article owns the scenario-reweighting problem: scenario matrix + prior probabilities + equality/inequality views → posterior probabilities, constraint residuals, information-distance diagnostics, concentration diagnostics and evidence about whether the requested views are feasible or dangerously distortive.

It does not own Black–Litterman equilibrium-return blending, interbank-network reconstruction, portfolio advice or a forecast of what markets will do. Entropy pooling changes weights on a supplied scenario set; it cannot create a state of the world that was never represented in that set.

This is public mathematical and computational education, not investment advice.

1. The central problem: change beliefs without rewriting the scenario engine

Suppose a Monte Carlo engine has already produced S scenarios x₁,…,x_S with prior probabilities p₁,…,p_S. The baseline might use equal probabilities, or probabilities from another model.

Now a stress analyst wants the reweighted distribution to satisfy a view such as:

E_q[g(X)] = m.

There are usually many probability vectors q that satisfy the view. Entropy pooling chooses the one that is minimally different from p according to relative entropy.

2. The optimisation problem

A standard discrete formulation is:

minimise D_KL(q || p) = Σ_s q_s log(q_s/p_s)

subject to:

q_s ≥ 0,   Σ_sq_s = 1,

plus view constraints such as:

Aq = b

and

Gq ≤ h.

The objective penalises unnecessary movement away from the prior probabilities. The constraints encode the new information.

3. Why KL divergence is directional

Kullback–Leibler divergence is not an ordinary Euclidean distance. In general:

D_KL(q||p) ≠ D_KL(p||q).

Entropy pooling normally treats p as the reference distribution and searches for a posterior q close to that reference while satisfying the views.

This direction has an important support consequence: if p_s=0, assigning q_s>0 creates an infinite or undefined relative-entropy penalty in the standard formulation. A scenario excluded from the prior support cannot simply be resurrected by reweighting.

4. A two-scenario worked example

Suppose the scenario engine contains only two outcomes for a quantity R:

• down scenario: R = −5%, prior probability 0.5;
• up scenario: R = +5%, prior probability 0.5.

A new view requires:

E_q[R] = +2%.

Let q_U be the up-scenario probability. Then:

0.05q_U − 0.05(1−q_U) = 0.02.

Solving gives q_U=0.70 and q_D=0.30. With only two states and the normalisation constraint, the view uniquely determines the posterior weights.

The example is intentionally small: it shows that entropy pooling changes probabilities, not the scenario outcomes themselves.

5. With many scenarios, the optimisation chooses among many feasible answers

If there are thousands of scenarios, one expectation constraint usually leaves many q vectors feasible. The KL objective selects the least-informative deformation relative to p.

Under expectation constraints, the dual solution often has an exponential-tilting form such as:

q_s ∝ p_s exp(−λᵀg(x_s)),

where the sign depends on the convention used to write the Lagrangian and λ is chosen so the constraints hold.

This dual representation is computationally useful because the number of dual variables can depend on the number of views rather than the number of scenarios.

6. What can be expressed as a view?

Views can be represented through functions of the scenarios. Examples include:

• an expected value equals a target;
• a variance or second moment equals a target;
• one expected quantity exceeds another;
• the probability of a region is at least a specified amount;
• a scenario-weighted loss measure lies within bounds;
• several user or stress views hold simultaneously.

Nonlinear-looking economic statements can sometimes still become linear constraints in q because the scenario function g(x_s) is computed first and only the probabilities are optimised.

7. Inputs

Typical inputs include:

• S scenario vectors x_s;
• prior probabilities p_s;
• view functions g_k(x);
• equality targets and inequality bounds;
• solver tolerances;
• optional confidence or soft-constraint design;
• scenario provenance and generation date;
• support checks and zero-probability handling;
• versioned definitions for each view.

8. Outputs

Useful outputs include:

• posterior scenario probabilities q;
• constraint residuals;
• KL divergence from the prior;
• largest posterior weights;
• weight concentration measures;
• posterior moments and event probabilities;
• dual variables or shadow-price diagnostics;
• solver status and infeasibility certificates where available;
• sensitivity to small changes in the views.

9. Feasibility comes before optimisation quality

A solver cannot satisfy a view that lies outside the convex possibilities represented by the scenario set.

In the two-scenario example above, an expected return of +8% is impossible because the largest scenario return is only +5%.

Falsifier: solve a feasibility problem before interpreting the entropy optimum. If no nonnegative q summing to one can satisfy the constraints, the correct output is “infeasible,” not a creatively distorted probability vector.

10. Scenario support is the first weak link

Entropy pooling is often described as flexible, but it is constrained by the supplied scenarios. If every scenario assumes liquid markets, reweighting cannot manufacture a true market-closure state. If no scenario contains a particular joint default pattern, no posterior weight can create that pattern.

This is a fundamental boundary:

reweighting changes frequency; scenario generation determines possibility.

11. Weight concentration can destroy effective information

A very strong view may force most posterior probability onto a few scenarios. The optimisation may still be mathematically feasible, but the posterior then depends heavily on a small part of the scenario library.

Useful diagnostics include:

• maximum q_s;
• Herfindahl concentration Σq_s²;
• a reciprocal-concentration diagnostic 1/Σq_s²;
• entropy-based effective scenario count exp[−Σq_slog q_s].

These are diagnostics, not universal regulatory thresholds. Their job is to make concentration visible.

12. Equality views versus inequality views

An equality view such as E[R]=2% pins a quantity to one value. An inequality such as E[R]≥2% permits a feasible region.

Inequality views can be more robust when the information itself is approximate. They also create active-set behaviour: a constraint that is loose may have no effect at the optimum, while a binding constraint can have a nonzero dual multiplier.

Inspecting which constraints bind helps explain why the posterior moved.

13. Soft views are different from hard constraints

If a view is uncertain, forcing it as an exact equality can create false precision. One alternative is to represent confidence through relaxed bounds, penalty terms, hierarchical structures or a mixture of prior and fully-viewed posteriors.

The mathematics must match the semantic claim. “We believe the mean is exactly 2.000%” is a much stronger statement than “we want a stress distribution tilted toward higher values.”

14. Counterexample: mutually inconsistent views

Suppose one constraint requires E[R]≥4% and another requires E[R]≤1% on the same posterior distribution.

No probability vector can satisfy both.

Falsifier: run a dedicated feasibility check and identify the smallest conflicting set of constraints if the solver supports it. Do not label solver failure as “extreme market conditions.”

15. Counterexample: the scenario library omits the tail

Imagine reweighting a historical sample that contains no severe liquidity event. A view that increases the probability of “bad days” can only upweight the worst days that happened to be present. That may still understate the mechanism of a genuinely different crisis state.

Falsifier: compare the scenario support with independently designed stress scenarios. If the stress mechanism is absent, expand the scenario library before pooling.

16. Counterexample: a tiny prior probability becomes dominant

A scenario with p=0.0001 might receive q=0.20 after imposing a strong view. The result is feasible, but it relies on a 2,000-fold probability amplification.

Falsifier: inspect q_s/p_s ratios and concentration. Extreme ratios are a signal to challenge the view, the prior or the scenario support.

17. Counterexample: zero prior support

If p_s=0 for a needed scenario, standard KL reweighting cannot assign it positive posterior mass without leaving the support of p.

Falsifier: audit zeros in p before optimisation. If a required state has zero prior support, regenerate or explicitly augment the scenario set rather than hiding the problem in numerical epsilons.

18. Counterexample: a view is fitted after seeing the desired answer

If analysts repeatedly adjust the view until the posterior produces a preferred portfolio metric, the process has become outcome tuning rather than independent scenario conditioning.

Falsifier: freeze the view specification before evaluating the downstream metric, or evaluate on a separate scenario set.

19. Evidence polarity

Evidence for confidence includes exact or tolerance-level constraint satisfaction, moderate divergence from the prior, stable results under small perturbations, broad effective scenario support, clear provenance for every view, reproducible solver convergence and similar conclusions under reasonable alternative priors.

Evidence against confidence includes infeasible or nearly infeasible constraints, a posterior dominated by a handful of scenarios, enormous q/p ratios, dependence on arbitrary numerical floors, large sensitivity to tiny view changes, a scenario library that omits the mechanism being stressed, or views chosen after inspecting the desired output.

20. Diagnostics ladder

1. Probability audit: verify p≥0 and Σp=1.
2. Support audit: confirm scenarios contain the states relevant to each view.
3. View audit: translate every verbal view into an explicit equation or inequality.
4. Feasibility audit: test whether any q satisfies all constraints.
5. Solver audit: inspect status, residuals and tolerances.
6. Divergence audit: measure D_KL(q||p).
7. Concentration audit: inspect largest weights and effective scenario count.
8. Ratio audit: examine extreme q/p changes.
9. Sensitivity audit: perturb targets and bounds.
10. Prior audit: repeat with reasonable alternative priors.
11. Stress audit: compare with independently constructed stress scenarios.
12. Downstream audit: verify that conclusions are not driven by one or two posterior scenarios.

21. Lagrange multipliers as diagnostics

Dual variables can be interpreted as sensitivity information: they describe how the optimum objective changes locally when a binding constraint is tightened or relaxed, subject to the usual optimisation assumptions.

A very large multiplier is not automatically “important economically,” but it can signal that the requested posterior is strongly constrained by that view.

Dual diagnostics are especially useful when many views are imposed and the posterior movement is otherwise hard to explain.

22. Entropy pooling versus Black–Litterman

Black–Litterman traditionally starts from an equilibrium-return framework and blends investor views with a prior under a structured probabilistic model. Entropy pooling is more general as a scenario-probability reweighting framework: it can represent non-normal distributions and a wide family of constraints on functions of scenarios.

They can address related “prior plus views” questions, but they are not the same page role. One works through a particular return-view model; the other works directly on scenario probabilities under information constraints.

23. Entropy pooling versus maximum-entropy network reconstruction

The Bukit Timah Tutor estate already contains an article on maximum-entropy interbank reconstruction. That problem starts with missing bilateral exposures and tries to infer a network matrix subject to row/column constraints.

Entropy pooling here does something different: the scenario outcomes already exist, and the unknowns are their probabilities. The common information-theoretic language should not hide the different mathematical objects.

24. Alternatives

Direct stress-scenario construction is preferable when the required state is absent from the baseline support.

Black–Litterman is useful when the problem is specifically equilibrium returns plus structured views.

Bayesian model averaging changes probabilities across models rather than only across scenarios of one supplied library.

Robust optimisation can optimise against an uncertainty set instead of choosing one reweighted posterior.

Importance sampling also changes scenario weights, but its primary computational job is variance reduction or rare-event estimation under an explicitly corrected sampling distribution, not expressing subjective or stress views.

25. Connections to the surrounding Bukit Timah Tutor knowledge estate

How Black–Litterman Portfolio Algorithms Blend Market Equilibrium with Views owns the equilibrium-return view-blending role.

How Maximum-Entropy Interbank Reconstruction Algorithms Infer Missing Bilateral Exposures owns network reconstruction, not scenario reweighting.

How Monte Carlo Pricing Algorithms Value Path-Dependent Derivatives explains one way the scenario library itself may be generated.

How Rockafellar–Uryasev CVaR Optimisation Algorithms Turn Tail Scenarios into Linear Programs owns downstream tail-risk optimisation over scenarios.

The full public lane is indexed at Finance & Banking Algorithms | Applied Mathematics in Real Financial Systems.

26. What would falsify confidence?

Confidence should be withdrawn if the views are infeasible, the posterior depends on states absent from the scenario support, the result is dominated by a few weights, small view changes cause discontinuously different conclusions, numerical floors create artificial support, or independently constructed stress scenarios reveal a mechanism the reweighted library cannot represent.

27. Verification and update triggers

Preserve the scenario matrix, prior weights, every view equation, units, solver version, tolerances, posterior weights, dual variables, concentration metrics and constraint residuals. Re-run the analysis when the scenario engine changes, the prior distribution changes, a view is revised, new risk factors become material, scenario support changes, or downstream decisions become sensitive to a small number of states.

28. Primary and high-quality references

• Attilio Meucci, Fully Flexible Views: Theory and Practice, describing the fully flexible views / entropy-pooling framework and implementation.
• S. Kullback and R. A. Leibler, On Information and Sufficiency, Annals of Mathematical Statistics, 1951.
• Stephen Boyd and Lieven Vandenberghe, Convex Optimization, for convex duality, equality/inequality constraints and numerical optimisation foundations.

Educational boundary: entropy pooling expresses assumptions mathematically; it does not turn a subjective view into a fact. The posterior is only as defensible as the scenario support, prior probabilities, constraints and verification that produced it.