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---

### Setting, foundations, and perspective (Sections 2–5 excerpts)
- Forecast target: vector outcome y with
  - Baseline predictive density p0(y) (denoted S0).
  - Alternative scenario predictive densities pj(y) for scenarios Sj (j = 1 : J−1).
  - Statistical reference predictive density p(y).
- Overarching goal: identify “closeness” of each scenario to the reference p(y) and rank scenarios accordingly.
- Bayesian Predictive Synthesis (BPS) and the “mixture BPS” class:
  - Scenario mixture p.d.f.: f(y|α) = Σ_{j=0:J} α_j p_j(y). (eqn. (1))
  - Scenario-set incompleteness handled by adding a backstop p.d.f. SJ; SJ may be a synthetic over-dispersed average of initial model p.d.f.s.
  - Indexing: baseline j = 0, alternatives j = 1 : J−1, backstop j = J.

### Partial scenario information and Entropic Tilting (ET)
- Partial information representation:
  - Scenario constraints as target score m_j = E[s_j(y)|S_j], with s_j(y) a q_j-vector. In practice assume q_j = q and s_j(y) = s(y) for all j.
- ET construction:
  - p_j(y) = k_j exp{τ'_j s_j(y)} p0(y), with k_j^{-1} = ∫ exp{τ'_j s_j(y)} p0(y) dy. (eqn. (2))
  - τ_j solved (e.g., Newton-Raphson) to satisfy moment constraint m_j.
- Monte Carlo representation of baseline:
  - Discrete baseline sample {y_i, w_i^0}_{i=1:n}.
  - ET sample weights u_{ij} ∝ exp{τ'_j s_j(y_i)}; normalized IS weights w_{ij} ∝ u_{ij} w_i^0.
- Effective sample size (ESS):
  - ESS = reciprocal of Σ_i u_{ij}^2 provides interpretable concordance assessment of Sj constraints with the baseline.

### Predictive concordance and Expected Misclassification Rate (EMR)
- Definitions:
  - Hypotheses: Hp: y ∼ p(y); Hf: y ∼ f(y).
  - Expected posterior probability of misclassification when y ∼ p(y):
    - π_{pf} ≡ E[P(Hf|y)|Hp] = ∫ f(y)p(y)/{f(y) + p(y)} dy. (eqn. (3))
  - Symmetry: π_{fp} = π_{pf}. Range: π_{pf} ≤ 0.5 with equality only when f(·) ≡ p(·).
  - 1 − π_{pf} equals population sensitivity and overall accuracy of optimal Bayesian classifier comparing f(·) and p(·).
- Relationship to KL divergences:
  - k(y) = log{p(y)/f(y)}; KL(p∥f) = E[k(y)|Hp].
  - Delta approximation: π_{pf} ≈ 1/[1 + exp{KL(p∥f)}].
  - Lower bound: π_{pf} ≥ 1/[1 + exp{KL(p∥f)}]; refined: π_{pf} ≥ 1/[1 + exp(κ_{pf})] with κ_{pf} = min{KL(p∥f), KL(f∥p)}.
  - Caveat: KL-based expressions require finiteness of KL; π_{pf} is always finite in (0,0.5].

### Scenario synthesis: optimizing mixture probabilities via EMR
- Concordance objective:
  - π_{pf}(α) = ∫ f(y|α) p(y)/{f(y|α) + p(y)} dy. (eqn. (4))
- EMR maximizer:
  - bα = argmax_α π_{pf}(α) yields bπ_{pf} = π_{pf}(bα).
  - Each bα_j quantifies concordance of Sj with the reference relative to others.
  - Low bπ_{pf} signals scenario-set incompleteness; bα_J (backstop weight) informative.
- Regularized optimization:
  - Maximize λ(α) = log{π_{pf}(α)} + ε Σ_{j=0:J} log(α_j),
    subject to α_j > 0 (j = 0:J) and Σ_{j=0:J} α_j = 1. (eqn. (5))
  - ε > 0 is a very small regularization parameter.
- Bayesian foundation and sparsity:
  - EMR π_{pf}(α) is a likelihood for α from a hypothetical z = 1 with Pr(z = 1|y,α) = f(y|α)/{p(y) + f(y|α)}; bα is an MLE.
  - Convexity gives unique MLE bα; simplex solutions typically sparse (some bα_j = 0).
  - Sparsity can be unstable; a minimally informative regularizing prior recommended.
- Prior and penalty:
  - Natural prior α ∼ Dir(a) with a_j = 1 + ε (symmetric) yields posterior log λ(α) in eqn. (5) (up to constant).
  - Default scaling recommended: ε = c/(J+1) with c = 0.005.

### Computational diagnostics and complementary metrics
- Convex EMR maximization with regularizing prior yields unique solution; see Appendix C for convexity and uniqueness proofs.
- ESS from ET and EMR-based metrics provide complementary diagnostics of scenario–baseline and scenario–reference concordance.
- Optional constraints (e.g., α_0 ≥ α_j for j = 1:J) are implementable and preserve convexity.

### Monte Carlo Importance Sampling (Section 5.3)
- MC integration setup:
  - p(y) and p_j(y) available only on large MC sample y_i ∼ p(y), i = 1:n.
  - Baseline IS weights: w_i0 ∝ p_0(y_i)/p(y_i); normalized discrete distribution {y_i, w_i0}_{i=1:n}.
- IS proposal requirement and efficiency:
  - p(y) should be heavier-tailed than p_0(y).
  - % effective sample size ESS = n^{-1} 100 / Σ_{i=1:n} (w_i0)^2 provides guidance; low ESS suggests increasing n.
- Scenario-specific ET and weights:
  - ET weights u_ij ∝ exp{τ'_j s_j(y_i)} defining p_j(·) relative to baseline.
  - Compound (ET–IS) weights: w_ij ∝ u_ij w_i0.
  - ESS computed on u_ij or w_ij; very low ESS for a scenario indicates discordance with baseline and suggests separate treatment or adaptive IS.
- Computational notes:
  - Adaptive IS can be considered for severe discordance but is more computationally demanding and outside current scope.
  - Severe discordance suggests treating the scenario separately and assessing its full distribution.

### Case study: setup, implementation, and key numerical findings
- Targets and data:
  - Analysis restricts to one-year ahead GDP growth, scalar y.
  - Draws from December 2007 and 2018 Tealbooks and NY Fed Outlook-at-Risk.
- Reference distributions:
  - Reference percentiles from NY Fed Outlook-at-Risk and Tealbook Time-Varying Macroeconomic Risk (five predictive percentiles).
  - Reference distributions fitted by skew-t (Azzalini and Capitanio, 2003) to percentiles by minimizing squared distance.
  - Table 3 skew-t parameters (lc, sc, sk, df):
    - 2007 Reference NY Fed: lc 2.7, sc 2.2, sk −0.5, df 3.4
    - 2007 Baseline: lc 1.3, sc 1.1, sk 0.05, df 50.0
    - 2018 Reference NY Fed: lc 2.5, sc 1.3, sk −0.3, df 3.0
    - 2018 Reference Tealbook: lc 2.1, sc 1.1, sk 0.55, df 50.0
    - 2018 Baseline: lc 1.2, sc 1.9, sk 2.1, df 50.0
- Baseline construction:
  - Tealbook point forecast used as median; 70% CI extremes used as 15th and 85th percentiles.
  - Fit skew-t with df = 50 to those percentiles to represent distribution “close to normal”.
- Monte Carlo sample size:
  - MC sample size used in the case study is 10^6 and resulting MC analysis summaries are stable and robust across reanalyses with such a large sample size.
- Scenario construction and ET:
  - TB Dec. 2007: 6 alternative scenarios (some excluded if identical to baseline).
  - TB Dec. 2018: 4 alternative scenarios.
  - Primary analysis treats scenario point forecasts as medians of p_j(y); ET maps baseline to each p_j(y) constrained to specified median (P50).
  - When only medians available, ET-constructed p_j(y) used to derive implied percentiles to define backstop:
    - P50_B = median_{j=1,...,J} P50_j
    - P15_B = min_{j=1,...,J} P15_j
    - P85_B = max_{j=1,...,J} P85_j
  - ET outcomes: tilted p_j(y) may remain close to baseline for concordant scenarios or show skewness/multimodality when poorly supported.
- Key numerical findings:
  - Concordance measures:
    - 2018 EMR = 0.48
    - 2007 EMR = 0.43
  - Synthesis ESS and scenario-set incompleteness:
    - TB 2007: ESS of f(y|α^∗) is about 71-72%; scenario set incompleteness is about 28-29%.
    - TB 2018: ESS of f(y|α^∗) is about 91%; scenario set incompleteness is much lower.
- Interpretation:
  - TB 2007: mixture synthesis light-tailed relative to reference; scenario set under-represents downside GDP ranges supported by reference; substantial weight on backstop.
  - TB 2018 (NY Fed reference): strong concordance between mixture synthesis and reference; baseline dominates synthesis; alternatives add small value.
  - TB 2018 (Tealbook reference): baseline as precise as reference; baseline dominates; scenarios add little predictive discrimination.
- Model-choice implications:
  - “Normal” times (2018): reference tends to be light-tailed; synthesis close to reference; limited role for backstop.
  - Higher-uncertainty periods (2007): reference should exhibit fatter tails; synthesis of baseline and scenarios may under-represent reference risk unless scenario set includes more extreme cases.

### Scenario synthesis based on P15, P50 and P85 (Section 6.5)
- Methodological summary:
  - Scenario p.d.f.s constructed as tilted versions of the baseline to match scenario-specific P15, P50, and P85.
  - In reported analysis P15_j and P85_j computed assuming distances of tail percentiles from the median agree with baseline:
    - P15_j = P50_j − (P50_0 − P15_0)
    - P85_j = P50_j + (P85_0 − P50_0)
  - p_j(y) ∝ w_j(y) p(y) with w_j(y) ∝ w_0(y) exp{τ′_j s_j(y)}, w_0(y) ∝ p_0(y)/p(y); normalized p_j(y) = c_j w_j(y) p(y).
  - Monte Carlo discrete implementation: w^i_j = w_j(y_i) for j = 0:J.
  - BPS-justified mixture: f(y|α) = w(y|α) p(y) where w(y|α) = Σ_{j=0:J} α_j c_j w_j(y); holds for any α.
- TB 2007 (NY Fed reference) observations:
  - P15 and P85 under three-percentile tilt similar to median-only tilt when scenarios close to baseline; diverge as percentiles depart (example S2).
  - Optimal weights and EMR results under three-percentile tilt similar to median-only specification.
  - Alternative specifications of P15 and P85 can lead to different results; methodology flexible to accommodate them.
  - Method addresses location shifts, scale perturbations, and skewness perturbations across scenarios.
- TB 2007: Table 6 reported rows (sequence of reported values per row)
  - 0  Baseline 0.11.32.5100.062.60.410.300.26
  - 1  Greater housing correction −0.21.02.292.357.20.390.000.01
  - 2  Credit crunch −1.5−0.30.920.031.60.320.100.11
  - 3  Stronger domestic demand 0.51.72.990.165.90.420.000.07
  - 4  Better export performance 0.71.93.179.366.10.420.300.26
  - 5  Greater cost pressure 0.01.22.499.461.20.400.000.01
  - 6  Market-based Fed Funds rate 0.41.62.896.065.00.420.000.03
  - 7  backstop −1.51.43.127.762.40.430.300.26
  - f(y|bα) −0.31.42.873.00.44
  - f(y|α∗) −0.31.42.872.70.44
  - (Details as in Table 4.)
- Policy and practical implications:
  - Provides concrete approach to evaluate baseline and judgmental scenarios against formal statistical density forecasts of risk.
  - Relevant for monetary policy decisions reliant on baseline plus alternative risk scenarios.
  - Applicable in IMF GaR-based global financial stability assessment, IMF World Economic Outlook scenario-based risk assessment, portfolio allocation, supervisory stress testing, commercial forecasting.
  - Supports scenario information specified in more percentiles and approximates fully-specified scenario p.d.f.s as more percentiles provided.
- Extensions and technical considerations:
  - ET extendable to point forecasts treated as subjective modes or means and to uncertainty in scenario information (percentiles with ± uncertainties).
  - Methodology extends to multivariate y (multiple indicators, multiple future periods); main scalability issues concern importance sampling and ET in higher dimensions.

### Key theoretical results, prior calibration, optimization properties, and computational flow (Appendices A–D)
- Appendix A — Relating KL to EMR:
  - π_pf = E_k[π(k)] with π(k) = 1/{1 + exp(k)} and k = log{p(y)/f(y)}; E[k(y)|H_p] = KL(p∥f) ≥ 0.
  - Conjectured bound: π_pf ≥ 1/[1 + exp{KL(p∥f)}] has empirical support under specific sufficient conditions (finite positive mean of k, unimodality, Pr[k(y) > 0] > 0.5, tail decay conditions).
  - Exact result under normal scale-mixture assumption: if g(k) continuous, symmetric, unimodal with mode and finite mean m > 0, then π_pf ≥ π(m).
  - First-order approximation: 1/{1 + exp(k)} ≈ (2 − k)/4 at k = 0 is an exact lower bound for k ≥ 0 and exact upper bound for k ≤ 0; absolute error < 0.68% on |k| ≤ 0.5.
  - Illustrative normal-shift example: p = N(0,1), f = N(a,1), KL = a^2/2; Monte Carlo n = 10^6 shows for a ≤ 1: ESS ≥ 40%, π_pf ≥ 0.40; π_pf roughly linear in ESS up to 0.5.
- Appendix B — Prior and regularization parameter specification:
  - Prior α ∼ Dir(1(1 + ε)); choose ε small via imaginary fractional observation x (e.g., x = 0.01).
  - Constraint α_j^* ≥ p/(J + 1) implies ε ≥ c x/(J + 1) with c = p/(1 − p).
  - Example recommendation: x = 0.01 and c = 0.5 imply ε ≥ 0.005/(J + 1); recommended default.
- Appendix C — Optimization of scenario mixtures:
  - Derivatives: h(α) ≡ δπ_pf(α)/δα = ∫_y p(y) h(y|α) dy with h(y|α) = p(y)^2/{p(y) + f(y|α)}^2.
  - Hessian H(α) = −2 E[p(y) p(y)′ a(y|α)] with a(y|α) = p(y)/{p(y) + f(y|α)}^3; H(α) strictly negative definite when y has sufficient support.
  - Maximizing π_pf(α) over simplex is convex with unique maximizer b_α; adding Dirichlet penalty preserves convexity and yields unique posterior mode α^* for ε > 0.
  - KL-based optimization tends to produce sparser mixtures; EMR is more conservative and numerically robust.
- Appendix D — Computational flow (stepwise recipe):
  1. Generate large sample y_i, (i = 1:n), from p(y) for MC evaluation.
  2. Compute baseline IS weights w_i0 ∝ p_0(y_i)/p(y_i), normalized.
  3. Evaluate scenario p_j(y):
     - (a) If fully specified, compute normalized IS weights w_ij on reference sample.
     - (b) If partially specified:
       i. Compute scenario distributions using random sample x_i from baseline p_0(x) to evaluate tilting integrals.
       ii. Compute tilting weights u_ij on reference sample y_i.
       iii. Compute ET-IS weights w_ij ∝ w_i0 u_ij, normalized, for each S_j.
  4. Compute synthesis weights by optimizing eqn. (5) over α; optional constraint α_0 ≥ max_{j=1:J} α_j.
     - At each α during optimization evaluate EMR in eqn. (4) by MC average over i = 1:n of w_if(α)/{w_if(α) + w_ip} with w_if(α) = Σ_{j=0:J} α_j w_ij and w_ip = 1/n.

*Source: wpiea2025105-print-pdf — supplied IMF content unit excerpts and appendices.*

### Section 2 discusses foundations and overviews methodology.  Section 3 addresses partial sce-

### Section 2–5 Excerpts: Foundations, Partial Scenario Information, Predictive Concordance, and Scenario Synthesis

### Setting, foundations and perspective
- Forecasting target is a vector outcome y (e.g., path of several macroeconomic indicators) with:
  - Baseline predictive density p0(y) (denoted S0).
  - Alternative scenario predictive densities pj(y) for scenarios Sj (j = 1 : J−1).
  - A statistical reference predictive density p(y).
- Overarching goal: identify “closeness” of each scenario to the reference p(y) and rank scenarios accordingly.
- Bayesian Predictive Synthesis (BPS) and the “mixture BPS” class are used: scenario mixture p.d.f.
  - f(y|α) = Σ_{j=0:J} α_j p_j(y). (eqn. (1))
- BPS theory addresses scenario-set incompleteness by adding a backstop p.d.f. (labeled SJ) to the scenario set; SJ may be a synthetic over-dispersed average of initial model p.d.f.s.
- Indexing convention: baseline j = 0, alternatives j = 1 : J−1, backstop j = J.

### Partial scenario information and entropic tilting (ET)
- Typical data: scenarios Sj are partially specified (means, medians, percentiles); these are treated as constraints on pj(y).
- Represent partial information as target score m_j = E[s_j(y)|S_j], where s_j(y) is a q_j-vector of scenario scores. In practice assume q_j = q and s_j(y) = s(y) for all j.
- ET chooses pj(y) to minimize KL divergence from baseline p0(y) subject to the moment constraint m_j:
  - p_j(y) = k_j exp{τ'_j s_j(y)} p0(y), with k_j^{-1} = ∫ exp{τ'_j s_j(y)} p0(y) dy. (eqn. (2))
  - τ_j is the unique tilting vector solved (e.g., by Newton-Raphson) to satisfy the expectation constraint.
- Monte Carlo (MC) representation of baseline: discrete distribution {y_i, w_i^0}_{i=1:n}. ET weights on sample values y_i are u_{ij} ∝ exp{τ'_j s_j(y_i)} and normalized IS weights w_{ij} ∝ u_{ij} w_i^0.
- Effective sample size (ESS) = reciprocal of Σ_i u_{ij}^2 provides an interpretable concordance assessment of Sj constraints with the baseline.

### Predictive concordance and expected misclassification rate (EMR)
- Compare two densities p(y) and f(y). If a draw y is equally likely from p or f, define hypotheses Hp: y ∼ p(y) and Hf: y ∼ f(y).
- Expected posterior probability of misclassification when y ∼ p(y):
  - π_{pf} ≡ E[P(Hf|y)|Hp] = ∫ f(y)p(y)/{f(y) + p(y)} dy. (eqn. (3))
  - By symmetry π_{fp} = π_{pf}.
- Interpretation:
  - EMR π_{pf} is the expected misclassification rate; larger π_{pf} means f(·) is harder to distinguish from p(·) (higher concordance).
  - π_{pf} ≤ 0.5 with equality only when f(·) ≡ p(·).
  - 1 − π_{pf} equals population sensitivity and overall accuracy of the optimal Bayesian classifier comparing f(·) and p(·).
- Relationship to Kullback-Leibler (KL) divergences:
  - Let k(y) = log{p(y)/f(y)} and KL(p∥f) = E[k(y)|Hp]. Delta approximation: π_{pf} ≈ 1/[1 + exp{KL(p∥f)}].
  - Lower bound: π_{pf} ≥ 1/[1 + exp{KL(p∥f)}]; further refined to π_{pf} ≥ 1/[1 + exp(κ_{pf})] with κ_{pf} = min{KL(p∥f), KL(f∥p)}.
  - Caveat: KL-based expressions require finiteness of KL; π_{pf} is always finite in (0,0.5].

### Scenario synthesis: optimizing mixture probabilities via EMR
- For scenario mixture f(y|α) = Σ_{j=0:J} α_j p_j(y), define concordance objective:
  - π_{pf}(α) = ∫ f(y|α) p(y)/{f(y|α) + p(y)} dy. (eqn. (4))
- EMR-maximizer bα = argmax_α π_{pf}(α) yields bπ_{pf} = π_{pf}(bα):
  - Each bα_j quantifies how concordant scenario Sj is with the reference relative to other scenarios.
  - Low bπ_{pf} indicates the scenario set (including baseline) is discordant with the reference (scenario-set incompleteness); bα_J (backstop weight) is informative.
- Regularized optimization: choose α* to maximize
  - λ(α) = log{π_{pf}(α)} + ε Σ_{j=0:J} log(α_j),
    subject to α_j > 0 (j = 0:J) and Σ_{j=0:J} α_j = 1. (eqn. (5))
  - ε > 0 is a very small regularization parameter.
- Bayesian foundation:
  - EMR π_{pf}(α) is a likelihood function for α based on a hypothetical observation z = 1 where Pr(z = 1|y,α) = f(y|α)/{p(y) + f(y|α)}. Thus bα is an MLE.
  - Convexity of the optimization implies a unique MLE bα; solutions are typically sparse (some bα_j = 0) due to simplex optimization properties.
  - Sparsity can be unstable and sensitive to small changes in inputs; addressing this requires a minimally informative regularizing prior.
- Prior and penalty:
  - Natural prior: α ∼ Dir(a) with p.d.f. ∝ Π_{j=0:J} α_j^{a_j−1}. With a_j = 1 + ε (symmetric), precision a = Σ a_j, prior mean and joint mode are 1/(J+1) for small ε.
  - Under α ∼ Dir(1(1 + ε)), the log posterior is λ(α) in eqn. (5) (up to constant); prior is zero at simplex boundaries, producing a unimodal posterior and posterior mode α* that replaces boundary zeros with small positive values.
  - Recommended default scaling: ε = c/(J+1), where c = 0.005. Scaling with J is important; small modifications of c have minimal impact.
- Additional constraints (optional):
  - Example: require α_0 ≥ α_j for j = 1:J to enforce baseline as the modal scenario. Such constraints modify the Dirichlet prior by an indicator and do not affect convexity; trivial to implement.
- Computational and theoretical points:
  - EMR maximization with the regularizing prior yields a convex optimization problem with a unique solution; see Appendix C (convexity and uniqueness).
  - ESS from ET and EMR-based metrics provide complementary diagnostics of scenario–baseline and scenario–reference concordance.

*Source: wpiea2025105-print-pdf — Section excerpts (canonical URL: https://www.imf.org/-/media/files/publications/wp/2025/english/wpiea2025105-print-pdf.pdf)*

### 5.3   Monte Carlo Importance Sampling

### 5.3   Monte Carlo Importance Sampling

### Monte Carlo importance sampling framework
- Analysis evaluates p.d.f.s p(y) and each p_j(y) and performs the integration in eqn. (4) by Monte Carlo (MC) integration when forecasts are available as MC samples.
- The analysis is implemented when the p.d.f. p(y) and the p_j(y) are available only on a (large) sample of MC draws from the reference p(y), a reference random sample y_i, (i = 1:n), drawn from p(y).
- Importance sample (IS) for the baseline p_0(y) is defined with normalized IS weights w_i0 ∝ p_0(y_i)/p(y_i).
- The discrete distribution {y_i, w_i0}_{i=1:n} defines the MC approximation to the baseline for downstream expectation evaluation.
- IS proposal requirement: p(y) should be a relevant importance sampling proposal and in particular should be heavier-tailed than p_0(y).
- Efficiency monitoring: % effective sample size ESS = n^{-1} 100 / sum_{i=1:n} (w_i0)^2 provides guidance; low ESS suggests increasing sample size.
- Scenario-specific ET parameters are evaluated using the IS sample; ET weights on the sample are u_ij ∝ exp{τ'_j s_j(y_i)} defining p_j(·) relative to the baseline.
- Scenario-specific ESS measures using the u_ij weights are relevant; very low ESS for a scenario indicates discordance with the baseline and suggests separate treatment or adaptive IS.
- Compound (ET–IS) weights relate the scenario S_j to the reference: w_ij ∝ u_ij w_i0.
- ESS can be evaluated on the compound weights w_ij to provide a direct overall assessment of each p_j(·) relative to the reference p(·).
- Note: cases exist where a scenario is more concordant with the reference than the baseline.

### Computational and methodological notes
- Adaptive IS methods can be considered in cases of severe discordance between scenario and baseline, but these are more computationally demanding and outside current scope.
- Encountering severe discordance suggests considering the scenario separately and directly assessing its full distribution.

### Case study: setup and implementation
- Data and targets:
  - Variables in original Risk and Uncertainty analyses: GDP growth, inflation, unemployment rate, and others. This application restricts attention to one-year ahead GDP growth, y = y, now scalar.
  - The case study draws from the December 2007 and 2018 Tealbooks and the NY Fed Outlook-at-Risk.
- Reference distributions:
  - Reference percentiles are available from the NY Fed Outlook-at-Risk and the Tealbook Time-Varying Macroeconomic Risk (five predictive percentiles).
  - Reference distributions are constructed by fitting skew-t distributions (Azzalini and Capitanio, 2003) to the given percentiles by minimizing squared distance between reference quantiles and skew-t quantiles.
  - Table 3 skew-t parameters (location lc, scale sc, skewness sk, degrees of freedom df):
    - 2007 Reference NY Fed: lc 2.7, sc 2.2, sk −0.5, df 3.4
    - 2007 Baseline: lc 1.3, sc 1.1, sk 0.05, df 50.0
    - 2018 Reference NY Fed: lc 2.5, sc 1.3, sk −0.3, df 3.0
    - 2018 Reference Tealbook: lc 2.1, sc 1.1, sk 0.55, df 50.0
    - 2018 Baseline: lc 1.2, sc 1.9, sk 2.1, df 50.0
  - Baseline distribution construction:
    - Tealbook provides point forecast and 70% intervals for the baseline.
    - Baseline median taken as point forecast; extremes of 70% CI taken as 15th and 85th percentiles.
    - Fit a skew-t with 50 degrees of freedom to those percentiles to represent a distribution “close to normal” but with modest tail weight.
- Monte Carlo sample size and stability:
  - MC sample size used in the case study is 10^6 and resulting MC analysis summaries are stable and robust across reanalyses with such a large sample size.

### Scenario construction and entropic tilting (ET)
- TB scenarios provide point forecasts only; for Dec. 2007 there are 6 alternative scenarios and for Dec. 2018 there are 4 alternative scenarios (some scenarios excluded where identical to baseline).
- Primary analysis treats scenario point forecasts as medians of p_j(y); ET maps the baseline to each scenario p.d.f. constrained to its specified median (P50 in Table 2).
- When only medians are available, ET is used to construct p_j(y), j = 1,...,J, then implied percentiles are used to define a backstop distribution.
  - Backstop percentiles: P50_B = median_{j=1,...,J} P50_j, P15_B = min_{j=1,...,J} P15_j, P85_B = max_{j=1,...,J} P85_j.
- ET outcomes:
  - Tilted distributions p_j(y) can remain close to the baseline for concordant scenarios (e.g., S1 and S5 in 2007) or can show skewness and multimodality when scenarios are poorly supported by the baseline.
  - Multimodality or pronounced skew indicates hypothesized states poorly supported by the baseline and relates to limits of policy analysis when interventions are large.

### Key numerical findings from the case study
- Concordance measures:
  - EMR (π* j) reported: 2018 EMR = 0.48; 2007 EMR = 0.43.
- Synthesis ESS and scenario set incompleteness:
  - In TB 2007, ESS of f(y|α^∗) is about 71-72%; scenario set incompleteness is about 28-29%.
  - In TB 2018, ESS of f(y|α^∗) is about 91%; scenario set incompleteness is much lower.
- Interpretation:
  - TB 2007: mixture synthesis is light-tailed relative to the reference; scenario set under-represents downside GDP ranges supported by the reference. Substantial weight is placed on the backstop in the synthesis.
  - TB 2018 (NY Fed reference): strong concordance between mixture synthesis and reference; baseline dominates the synthesis and alternative scenarios add small value.
  - TB 2018 (Tealbook reference): baseline is as precise as the reference; baseline dominates and scenarios add little predictive discrimination.
- Model-choice implications:
  - In “normal” times (example 2018), the reference tends to be light-tailed, scenarios are modest deviations, and synthesis is close to reference with limited role for the backstop.
  - In higher-uncertainty periods (example 2007), reference should exhibit fatter tails; synthesis of baseline and scenarios may substantially under-represent reference risk unless scenario set includes more extreme cases. This motivates admitting extreme scenario considerations in high uncertainty.

*Source: wpiea2025105-print-pdf - 5.3   Monte Carlo Importance Sampling (case study material as provided).*

### 6.5   Scenario Synthesis based on P15, P50 and P85

### 6.5   Scenario Synthesis based on P15, P50 and P85

### Methodological summary
- The methodology admits specification of multiple features of scenarios as expectations under implicit scenario distributions and is directly applicable to multiple percentiles of scenario information.
- Scenario p.d.f.s can be constructed as tilted versions of the baseline reference that match scenario-specific P15, P50, and P85.
- In the analysis reported, scenario P15 and P85 are computed assuming distances of tail percentiles from the median agree with the baseline: P15_j = P50_j − (P50_0 − P15_0) and P85_j = P50_j + (P85_0 − P50_0).
- The scenario p.d.f. form used is p_j(y) ∝ w_j(y) p(y) with w_j(y) ∝ w_0(y) exp{τ′_j s_j(y)}, where w_0(y) ∝ p_0(y)/p(y). Normalized p.d.f.s are p_j(y) = c_j w_j(y) p(y).
- The Monte Carlo implementation uses discrete versions over reference samples y_i, i.e., w^i_j = w_j(y_i) for j = 0:J.
- The BPS-justified scenario mixture p.d.f. is f(y|α) = w(y|α) p(y) where w(y|α) = Σ_{j=0:J} α_j c_j w_j(y). This holds for any α, not only the EMR-optimized α.

### TB 2007 (NY Fed reference) — scenario synthesis observations
- When scenario p.d.f.s are tilted to match P15, P50, and P85, P15 and P85 values for scenarios are similar to those in earlier specification when scenarios are close to the baseline and diverge as scenario percentiles depart further from the baseline (example: S2).
- Optimal weights and EMR results under the three-percentile tilt are quite similar to those obtained when only median (P50) information is used.
- Alternative, scenario-founded specifications of P15 and P85 (other than the synthetic choices here) can lead to different results; the methodology is flexible to accommodate those alternative specifications with negligible analytic and computational burden.
- The methodology can address multiple aspects of the forecast distribution within any one scenario and across a set of scenarios: location shifts, scale perturbations, and skewness perturbations.

### TB 2007: Table 6 — reported rows (sequence of reported values per row)
- 0  Baseline 0.11.32.5100.062.60.410.300.26
- 1  Greater housing correction −0.21.02.292.357.20.390.000.01
- 2  Credit crunch −1.5−0.30.920.031.60.320.100.11
- 3  Stronger domestic demand 0.51.72.990.165.90.420.000.07
- 4  Better export performance 0.71.93.179.366.10.420.300.26
- 5  Greater cost pressure 0.01.22.499.461.20.400.000.01
- 6  Market-based Fed Funds rate 0.41.62.896.065.00.420.000.03
- 7  backstop −1.51.43.127.762.40.430.300.26
- f(y|bα) −0.31.42.873.00.44
- f(y|α∗) −0.31.42.872.70.44
- (Details as in Table 4.)

### Policy and practical implications
- The framework provides a concrete, straightforward approach to evaluate baseline and judgmental scenario assessments against formal statistical density forecasts of risk.
- The approach is relevant for monetary policy settings where decisions rely on baseline forecasts plus alternative risk scenarios; it supports rigorous integration of judgmental information with model-based forecasts.
- The methodology is applicable in broader institutional contexts: IMF GaR-based global financial stability assessment, IMF World Economic Outlook scenario-based risk assessment, portfolio allocation, supervisory stress testing, commercial revenue and supply chain forecasting.
- The framework supports scenario information specified in terms of more than a few percentiles and approximates fully-specified scenario p.d.f.s as more percentiles are provided — relevant where scenarios come from alternative/competing models.

### Extensions and technical considerations
- The ET framework can be extended to scenario information beyond medians, including point forecasts regarded as subjective modes or means, and to uncertainty in scenario information (percentiles provided with ± uncertainties), though details need development for specific applied contexts.
- Elements of y can include multiple economic indicators and multiple future time periods (e.g., the coming eight quarters); the methodology extends immediately to these higher-dimensional settings.
- Main technical scalability questions concern importance sampling and entropic tilting as dimension of y increases and as the underlying Bayesian decision-analytic score functions grow in dimension; these are known issues to be addressed contextually in applications.

*Italic: Source — content unit "6.5   Scenario Synthesis based on P15, P50 and P85" from the supplied IMF PDF content.*

### References

### References

### Key theoretical results (Appendix A: Relating KL Divergences to EMR)
- Definition and focus:
  - π_pf = E_k[π(k)] with π(k) = 1/{1 + exp(k)} and k = log{p(y)/f(y)}; E[k(y)|H_p] = KL(p∥f) ≥ 0.
- Conjectured bound and sufficient conditions:
  - Bound: π_pf ≥ 1/[1 + exp{KL(p∥f)}] has empirical support in specific examples but is not generally true.
  - The bound is conjectured to hold when the distribution of k(y) has finite, positive mean, is unimodal with Pr[k(y) > 0] > 0.5, and has p.d.f. tail decay on k(y) < 0 no heavier than that on k(y) > 0.
- Exact result under normal scale-mixture assumption:
  - If g(k) is continuous, symmetric and unimodal with mode and finite mean m > 0, then g(k) is a normal scale mixture and π_pf ≥ π(m). The inequality is strict unless k = v = 0.
- Extensions and caveats:
  - Possible extension to scale mixtures of skew-normal distributions (Azzalini and Capitanio, 2013) if g(k) is unimodal with m > 0 and/or Pr(k ≥ 0) > 0.5.
  - No extension to cases where the expectation of k (the KL divergence) does not exist or when k’s distribution is multimodal.
  - Interpretation: where the bound holds, κ_pf, κ_fp ≤ log{(1−π_pf)/π_pf}.
- First-order Taylor approximation and bounds:
  - 1/{1 + exp(k)} ≈ (2 − k)/4 at k = 0.
  - This is an exact lower bound for k ≥ 0 and exact upper bound for k ≤ 0.
  - Accuracy: absolute error < 0.68% on |k| ≤ 0.5 (where 1/{1 + exp(k)} ≥ 0.38).
  - Practical implication: when distribution of k(y) concentrates in |k| ≤ 0.5, π_pf ≈ {2 − κ_pf}/4 and choosing f(·) to maximize π_pf approximates minimizing symmetrized KL.
- Simple illustrative example (normal shift):
  - Setup: p(y) = N(0,1), f(y) = N(a,1) for a ≥ 0; KL(p∥f) = KL(f∥p) = κ_pf = a^2/2.
  - Monte Carlo importance sampling example with n = 10^6.
  - For a ≤ 1: ESS ≥ 40%, π_pf ≥ 0.40; π_pf roughly linear in ESS up to maximum 0.5.
  - Practical takeaway: π_pf ≥ 0.4 is expected unless p(·) and f(·) are substantially discordant; π_pf ≥ 0.40 links to approximation π_pf ≈ 1/{1 + exp(κ_pf)} with κ_pf ≤ 0.4.

### Prior and regularization parameter specification (Appendix B)
- Prior form and calibration goal:
  - Prior α ∼ Dir(1(1 + ε)); choose small ε by analogy to Dirichlet multinomial conjugacy and a minimally informative posterior.
- Imaginary fractional observation framework:
  - Treat a minimal-information posterior as Dir(1(1 + ε) + x e) for small x (e.g., x = 0.01 representing 1% of a single multinomial draw).
  - Posterior mode on baseline S_0: α_0^* = (ε + x)/{(J + 1)ε + x}; other scenarios j > 0: α_j^* = ε/{(J + 1)ε + x}.
- Constraint to limit shrinkage:
  - Require α_j^* ≥ p/(J + 1) for some p ∈ (0,1), which implies ε ≥ c x/(J + 1) with c = p/(1 − p).
  - Example recommendation: choices x = 0.01 and c = 0.5 imply ε ≥ 0.005/(J + 1); this value is recommended as a default.

### Optimization of scenario mixtures (Appendix C)
- Derivatives of EMR (eqn. (4)) and properties:
  - h(α) ≡ δπ_pf(α)/δα = ∫_y p(y) h(y|α) dy with h(y|α) = p(y)^2/{p(y) + f(y|α)}^2.
  - H(α) ≡ δ^2 π_pf(α)/δα α′ = −2 ∫_y p(y) p(y)′ H(y|α) dy with H(y|α) = h(y|α)/{p(y) + f(y|α)}.
  - Hessian H(α) = −2 E[p(y) p(y)′ a(y|α)] where a(y|α) = p(y)/{p(y) + f(y|α)}^3; since a(y|α) > 0, H(α) is strictly negative definite (full rank) when y has sufficient support.
- Convexity and uniqueness:
  - Maximizing π_pf(α) over the simplex is a convex optimization with a unique maximizer b_α.
  - Adding the prior penalty maintains convexity and yields a unique posterior mode α^* for any specified ε.
  - Similar convexity and uniqueness apply when minimizing KL(p∥f) or KL(f∥p) (when finite) with respect to α.
- Comparison EMR vs KL:
  - KL-based optimization typically yields sparser mixtures (more zeros among α_i) because KL depends on expectations of the unbounded function log{p(y)/f(y|α)} and is sensitive to tail behavior.
  - EMR is more conservative and numerically robust, as it is based on expectations of the bounded function 1/{1 + p(y)/f(y|α)}; still, boundary shrinkage can occur and modest regularization (the Dirichlet prior) is recommended.

### Computational flow for implementing the mixture optimization (Appendix D)
- Stepwise computational recipe:
  1. Generate a large random sample y_i, (i = 1:n), from reference p(y) to form an importance sample for MC evaluation of baseline p_0(y) and scenario p_j(y).
  2. Evaluate baseline IS weights w_i0 ∝ p_0(y_i)/p(y_i), normalized.
  3. Evaluate scenario p.d.f.s p_j(y) for j > 0:
     - (a) If scenario p.d.f.s are fully specified, compute normalized IS weights w_ij directly on the reference sample.
     - (b) If scenarios are partially specified:
       i. Compute scenario distributions using random sample x_i drawn from baseline p_0(x) to evaluate integrals for tilting parameters.
       ii. Compute implied tilting weights u_ij on reference sample values y_i.
       iii. Compute implied ET-IS weights w_ij ∝ w_i0 u_ij, normalized, for each S_j.
  4. Compute synthesis weights by optimizing eqn. (5) over α; optional constraint α_0 ≥ max_{j=1:J} α_j if baseline must be modal.
     - At each α in optimization iterations, evaluate EMR in eqn. (4) by MC integration as the average over i = 1:n of sampled EMR values w_if(α)/{w_if(α) + w_ip} with w_if(α) = Σ_{j=0:J} α_j w_ij and uniform reference weights w_ip = 1/n for all i.

*Source: References and Appendices A–D from the provided IMF content unit.*

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_Source: https://www.imf.org/-/media/files/publications/wp/2025/english/wpiea2025105-print-pdf.pdf_