Latent Variable Estimation in Bayesian Black-Litterman Models
Authors: Thomas Yuan-Lung Lin
Jerry Yao-Chieh Hu
Paul W. Chiou
Peter Lin
Venue: ICML 2025 poster (2025)
Keywords: Black-Litterman Models
Portfolio Optimization
Bayesian Networks
Graphical Models
Hierarchical Models
Latent Variable Models
Uncertainty Quantification
Primary area: applications->time_series
TL;DR: We reformulate the Black-Litterman model to infer investor views as latent variables, directly estimating asset returns from data.

Abstract: We revisit the Bayesian Black–Litterman (BL) portfolio model and remove its reliance on subjective investor views. 
Classical BL requires an investor “view”: a forecast vector $q$ and its uncertainty matrix $\Omega$ that describe how much a chosen portfolio should outperform the market.
Our key idea is to treat $(q,\Omega)$ as latent variables and learn them from market data within a single Bayesian network.
Consequently, 
the resulting posterior estimation admits closed-form expression, enabling fast inference and stable portfolio weights.
Building on these, we propose two mechanisms to capture how features interact with returns: shared-latent parametrization and feature-influenced views; both recover classical BL and Markowitz portfolios as special cases.
Empirically, on 30-year Dow-Jones and 20-year sector-ETF data, we improve Sharpe ratios by 50\% and cut turnover by 55\% relative to Markowitz and the index baselines.
This work turns BL into a fully data-driven, view-free, and coherent Bayesian framework for portfolio optimization.

Latent Variable Estimation in Bayesian Black-Litterman Models
Thomas Y.L. Lin
Jerry Yao-Chieh Hu"
Paul W. Chiou 4 Peter Lin
Abstract
value of investor views and corresponding uncertainties (q, For example. the i-th views "the 2nd asset will out- perform the 1st asset by 9 + 3%" is encoded as {P; [-1, ....,.0::: = 0.09; f2 = 0.03²}. Across decades, this heuristic framework attracts many research (Beach & Orlov, 2007; Palomba, 2008; Duqi et al., 2014; Silva et al., 2017; Deng. 2018; Kara et al., 2019; Kolm & Ritter, 2021) working on this estimation. Among them, one common approach is take asset features and generate (q by external models. However, relying on external estimators in these methods leads to incoherent parameter learning and error propagation across the separate models.
We revisit the Bayesian Black-L.iterman (BL. portfolio model and remove its reliance on sub- jective investor views. Classical BL requires an investor "view": a forecast vector q and its un- certainty matrix 5 that describe how much a chosen portfolio should outperform the market. Our key idea is to treat (g. Q2) latent variables and learn them from market data within single Bayesian network. Consequently, the resulting posterior estimation admits closed-form expres- sion, enabling fast inference and stable portfo- lio weights. Building on these, we propose two mechanisms capture how features interact with returns: shared-latent parametrization and feature- influenced views; both recover classical BL and Markowitz portfolios as special cases. Empiri- cally, on 30-year Dow-Jones and 20-year sector- ETF data, we improve Sharpe ratios by 50% and cut turnover by 55% relative to Markowitz and the index baselines. This work turns BL into a fully data-driven, view-free, and coherent Bayesian framework for portfolio optimization.
In this work, we offer a new approach. Our reformulation recasts the Black- model Bayesian network to integrate the features. Under different scenarios, we identify two potential effects caused by the features and accordingly present this network as specific models. Under the scenarios without subjective investor views, this network treats and Ω as latent - rather than externally estimated parameters, and thereby estimates posterior distribution over both asset returns r and their parameters directly from data, i.e., features. In summary, our approach provides unification of feature integration and parameter inference within single framework, ensuring coherent estimations and mitigating error propagation.
Introduction
Contributions. Our contributions include:
We propose a Bayessian reformulation of the Black- Litterman model for portfolio optimization. Our moti- vation comes from the early works of the model (Black & Litterman, 1992; Lee, 2000; Salomons, 2007; Idzorek, 2007) where human experts are required specify the
Eliminating Subjective Human Input. We introduce Bayesian network formulation of the Black-Litterman model, treating (q. 92) as latent variables. This enables direct estimation from feature data, bypassing heuristic human inputs and potential bias while subsuming the classical Black-Litterman model as a special case.
'qqual contribution 'Department of Physics, National Tai- wan University, Taipei, Taiwan DDepartment of Computer Sci- ence, Northwestern University. Evanston. IL. USA CCntee for Foundation Models and Generative AI, Northwestern Univer- sity, Evanston, II, USA "D' A Amore-McKim School of Business, Northeastern University, Boston, MA, USA 'Gamma Paradigm Capital. New York, NY, USA "Whiting School of Engineer- ing, Johns Hopkins University, Baltimore, MD, USA. Corre- spondence to: Thomas Y.L. Lin <bl <b122020266> ntu. adt.eed..w>. Jerry Cao- Hu <jhu w.worrhwestemm.cdu..... Paul W. Chiou <w.chiou @ northeastem..co Sdu
Unified Feature Integration and Parameter Inference. Unlike prior works, our approach avoids error propaga- tion from external estimators by unifying feature integra- tion and parameter inference into a single framework.
Empirical Outperformance.
Our model achieves a 49.8% mean improvement in Sharpe ratios over the Markowitz model (0.66-0.87 vs. 0.35-0.62) and market indices (S&P 500, DJIA) on 20-year and 30-year datasets, respectively. It achieves a 55.19 1% reduction in turnover rates while showing robustness to hyperparameters.
edu>, Peter Lin <peter.lin Saterliin@juucduus @hu..duu>
Proceedings of the nd International Conference Machine Learning, Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).

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Latent Variable Estimation in Bayesian Black-Litterman Models
Organization. Section 2 includes preliminaries. Section 3 introduces our models and their theoretical analysis. Sec- tion 4 presents empirical studies to backup our work. Ap- pendix A offers a practical guide for our models. We defer conclusions and related works to Section 5 and Appendix B.
and diversified portfolio weights (Black & Litterman, 1992).
The following context elaborates this model in detail.
BlatknLit Black-Litterman (BL) model outputs posterior of the asset returns mean E(r], termed Black- Litterman formula, by Bayes' theorems, taking investor views and market equilibrium price as input. Upon this, the model offers a predictive estimate F on asset returns:
2 Preliminaries
Consider m assets, and let E R be the returns of the m assets. Consider k sets of specified portfolio weights on the m assets, and encode each weight into each row of a portfolio weight matrix P E _xxm Let q E k be the investor views on the h specified portfolio returns and encode the variance of each view into diagonal elements of a diagonal uncertainty matrix S R Larger S2 implies greater uncertainty in (PE(r)): and Sz = 0 implies absolute certainty.
Theorem 2.1
(BL) Formula and Predic- tive Estimation, Theorem I of (Satchell & Scowcroft, 2007)), Let E R' be the vector of asset returns with covariance Σ == Covfr). Let P E RR be the portfolio weight matrix for k specified portfolios, and (q, 12) E E *** represent investor views and their uncertainty. Let II E R" repre- sent the market equilibrium price and > 0 be scaling factor. Assume a prior P PE[ ~ N(g, Q2) and a likelihood 11 Eir] ~ (B(r), 2)), then the posterior mean of r given is
Markowitz (1952) introduces the theory of portfolio opti- mization, suggesting a suitable portfolio weight is the opti- mal off between the mean and variance of the portfolio. To be concrete, we provide a formal definition below.
E(r II] N G, [ - P - G),
where G, = (T 2 + P 2- P. Moreover, the predic- tive distribution 7 = r is
Definition 2.1 (Unconstrained Rijk-Adjusted Mean-Vari- ance Optimization). Let E Ihm be the returns of the m assets, F denote the unobserved (or future) asset returns, and 5 E [0, oo] be risk-adjusted coefficient. The optimization goal is to find w R maximizing the objective function:
F~ N G- ( PT
2 2). G.+))
The Black-Litteemme formula (Theorem 2.1) is a well- known result of the Black-Litterman model. However, the derivation lacks explanations of the assumptions used. Most early works, including the original paper (Black & Litter- man, 1992), provide heuristic derivation, while many (Lee, 2000; Salomons, 2007; Idzorek, 2007) share different under- lying assumptions. This inconsistency leads to confusion for both the analysis of the model and a rigorous inter- pretation with Bayesian statistics. To solve the issues, the Black-Litterman-BBaae Bayes (BLB) model (Kolm & Ritter, 2017) provides reformulation of the Black- model.
8 " Cov
w [
2
max 10
One major challenge of this framework is the reliance estimating F:
Problem (Predictive Estimation of Unobserved Asset Re- turns). Letr E Rn represent the returns the m assets and F denote the unobserved (or future) asset returns. Precisely, given observed data D, the goal is to estimate unobserved asset returns F ~ D).
(BLB). Kolm & Ritter (2017)
In this work, we refer to methods addressing such estima- tion challenge (Problem 1) as portfolio models or simply portfolios. Following the predictive estimation of F, we apply the mean-variance optimization framework to obtain decision vector w, referred to as portfolio weights.
introduce the Litterman-Baaeee model to perform Bayesian inference on 0, treating the market equilibrium as prior and the investor views as likelihood:
Definition 2.2 (BLB Model (0, q, 22), Modified from Def- inition I of (Kolm & Ritter, 2017)). Let r E 1 be the returns of the m assets, parametrized by 8 with r ~ p(r(0). Let y E RR& represent the views on the returns of the k speci- fied portfolio and l E g be the uncertainty matrix. The Minkk-Liteerma taaterrma-BByeesssse (BLB) model is portfolio model composed of three fundamental density functions:
A basic approach, termed the traditional Markowitz model (Markowitz, 1952), involves predicting the expected returns and the covariance matrix of asset returns directly from historical data using the sample mean and sample covari- ance. This method relies on the assumption that historical estimates are accurate representations of future parameters. However, in practice, estimation errors in the expected re- turns and covariance matrix lead to extreme and highly sensitive portfolio weights (Michaud, 1989; DeMiguel et al., 2009). To mitigate this issue, the Black-Litterme model integrates the market equilibrium with investor views by Black- Litterman formula, thereby producing more stable
1. Parametrized Asset Returns: p(r)0), the distribution of asset returns given the parameter.
2. Prior: (0), representing market equilibrium.
3. Likelihood: L(0 (10lqq,3 .2) = p(g. 2266), capturing the rela- tionship between the parameter and the investor views.
2

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Latent Variable Estimation in Bayesian Black-Litterman Models
Appendix C.I details modelings of the prior and likelihood.
To address this issue, in this work, propose a Bayesian reformulation of the Black-LLtterman model without the need for subjective (g. ??) from humans.
To recap, the Black-Litterman-Bayy model is a portfolio model aiming to address the estimation challenge of asset re- turns (Problem 1). It solves the problem by using Bayesian inference to obtain the posterior of the model parameter p(Olg, S2), and subsequently produce the predictive estima- tion of unobserved asset returns V. Following the estimation, we apply the mean-variance optimization framework (Defi- nition 2.1) to obtain the portfolio weights WBLB-
3 Method
In this work, recast the Black-Littermal model as Bayesian networks for principled estimation of both investor views and asset returns, eliminating the need for subjective inputs. This Bayesian formulation serves as conceptual baseline for subsequent portfolio model specifications.
The intuition of the model is to simulate the dynamics of how the market adapts to new information after observing it. Here, the market equilibrium represents the initial state, and the investor views approximates the unobserved information. When there is no investor views, the model remains in the initial state of market equilibrium.
In Section 3.1, we introduce the Bayesian Black-Litterman Bayes
network, which underpins the model (Kolm & Ritter, 2017). Building this, Section 3.2 extends the network to incorporate external features, yield- ing the feature-integrated Black- network.
Here we show the Black-Litee Litee-nnteemann formula (Theorem 2. is the posterior estimation on & of the Mlack-Litterrma-- Bayes model under Assumptions C.1 and C.2:
We then examine two scenarios:
In Section 3.3, where investor views are observed, we illustrate the corresponding network (Figure 2) and define the Mixed-effect Black-Litterma (M-BL) model (Defini- tion 3.3).
Lemma 2.1 (Estimations by BLB Model), Let the market capitalization weight on m assets be Weap E Rm and 5 E [0, oo| be a risk-adjusted coefficient. Let P E Rxxm be the portfolio weight matrix for specified portfolios. Given a BLB model (0,r, q, S2) (Definition 2.2), assume
Section 3.4, where no subjective views are given, we present two alternative probabilistic graphical models (Figure 3) and define the AhareedLatenn-Puraantttaill Black-Littermas (SLP-BL) and Feature Influecce- Views Black Litte-nn (FIV-BL) models (Definitions 3.4 and 3.6).
((0, E),
(2.1)
N(0,, Σo);
(2.2)
P0 - E~ (0.
(2.3)
3.1 Bayesian Black-Litterman Network
where Σ, No mx are given intrinsic and prior covari-
ance. The posterior mean is p(0/ R) N (0; G (Ep 10
We introduce a Black-Litterman network (0, r, r. S2) with the two causal relationships. First, the asset returns r are realizations of the process governed by its parameter 0. Sec- ond, the investor views g are formed based on with an associated error term ~ (N Q). We visualize this con- ceptual network in Figure 1.
pT
(2.4)
where G := Enj + PTΩ- 1 P. The predictive estimation of unobserved asset returns := rlg, 52 is
F~ (T E l'Wicap + p- p"st='g],= +
(2.5)
Proof. See Appendix E.1 for detailed proof.
9.S
Lemma 2.1 provides a solution to Problem 1. With the predictive estimation of asset returns 7, the mean-variance optimization framework (Definition 2.1) determines the port- folio weights WBLB- However, it relies on the subjective investor views and corresponding uncertainty (2)
Figure 1: Black-Litterman network (0, r, g, 22).
3.2 Bayesian Feature-Integrated Black-Litterran Network
'Besides providing the Bayesian inference formulation of the Black-Litterman model (Definition 2.2), the original work (Kolm Ritter, 2017) and its follow-up work (Kolm & Ritter, 2021) consider external data, specifically factors in Arbitrage Pricing Theory (APT) model. Yet, these works focus on how their Black- Litterman-Baye (BLB) approach applies the APT model and do not address the issues of subjective investor views.
Building upon the Black-Litterman Network, we introduce a feature-integrated Black-Litttermn network as Black- Litterman network that integrates features and their ef- fects. Specifically, features exert two causal effects:
Efffect
Features F are extracted from the parameter
3

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Latent Variable Estimation in Bayesian Black-Litterman Models
Effect 2: Features F influence the formation of views q-
To capture Effect 1, we define a 0 F relationship:
We quantitatively specify Effect and 2 in Sections 3.3 and 3.4.
Definition 3.1 (0 <> F Linear Model). Given features F E Rmxddm let IR be the returns of the m assets, parametrized by 0, with r ~ p(rl0). Define regression intercept vector a² R regression coefficient vector 3 E &Rdm random error cf E "R" and error matrix S2 Rmxx such that:
3.3 General Scenario: Features with Observed Views
In this section, we discuss the general scenario where views are observed (Problem 2) by specifying the two causal ef- feets introduced in Section 3.2. Incorporating these effects into the network, we showcase it in Figure 2 and define the Mixed-effect Black-Litterman (M-BL.) model (Defini- tion 3.3) based on it. Then, we estimate posterior distribu- tion over both asset returns r and their parameter 0 (Corol- lary 3.1.1 and Theorem 3.1).
0 a= + F3 +e,, e ~ N(0,
To capture Effect 2, we define a ++ F ++ 9 relationship:
Definition 3.2 (q ++ F + 0 Linear Model). Given features F' E RRmxdm and portfolio weight matrix for k specified portfolios P E R let r E R" be the returns of the m assets, parametrized by 0, with r ~ p(r[0) and g E R& be the views on returns of the k portfolios. Define regression intercept vector a E I"" regression coefficient vector 3 ddm scale constant Y E R. random error R", and uncertainty matrix Ω E R Rmxm such that:
Consider the following problem:
Problem 2 (Feature-and- Views Hybrid Predictive Estima- tion). Let E Ihm represent the returns of the m assets and F denote the unobserved (or future) asset returns, Let q E R represent the views on returns of the k portfolios. Let Sa E Idd represent the features on the i-th asset and F E R.xxdm be block-diagonal matrix defined as
q + P(a + F3 + 70), e~ ((,
(3.1)
Remark 3.1 (Rationale). (3.1) extends the classical noisy views model + = P0 (Black & Litterman, 1992) to incorporate the features F, where the LHS remains the k-dimensional noisy views and the RHS generalizes the classical P0 term by introducing a feature-driven term a + F3, and a scaled parameter 78. It recovers the classical model when 0, B = and y = 1.
F diag(f] fT......TT...
= (g. Q, F, where (2, are estimated from observations of asset returns, views, and features ((11, (t kr) and 2 is the homoscedastic error matrix corresponding to the observations (= the goal is to estimate unobserved asset returns 7 ~ DDr
Given D=
We aim to solve Problem 2 by Litterman network, incorporating a mix of two causal effects of features in Section 3.2. Specifically,
Black-
By mixing two effects of the features characterized by the two linear models (Definition 3. and 3.2), we showcase the feature-integrated Black- network as the following Mixed-effect Black-Litterman (M-BL) model:
Effect 1: Features/ F' are extracted from the parameter 9. Consequently, the features F, along with their error term ef ~ (O, DF),, share the common parameter 0 with asset returns r and investor views q.
Definition 3.3 (Mixed-effect Black-Litterma
(M-BL)
Model (0, r. S2, F, (2F).. Let r E T be the returns of the m assets, parametrized by 6, with r ~ p(r(0). Let q RR represent the views on the returns of the l specified portfolio and Ω E R**k be the uncertainty matrix. Let F E R.xxdm be the features of the m assets and error ma- trix Ω E mR The M-BL model is a portfolio model composed of four fundamental density functions:
Effect 2: Features influence the formation of views q- Consequently, the features F and the parameter 0 jointly determine the views g with an uncertain
(0,
We visualize the network in Figure 2 under this general scenario.
Parametrized Asset Returns: p(r(0), the distribution of asset returns given the parameter.
0
ΩF
2. Prior: (0), representing market equilibrium.
3. Likelihood of Features: L[O (20 = P(F (0), the
0 F relationship (Definition 3.1).
4. Observation Likelihood: L(B, F Flg. = p(g, 2(0,
the q +> F t 0 relationship (Definition 3.2).
9.9
We show the posterior estimation on 9 of the M-BL model:
Theorem 3.1 (Parameter Estimation of the M-BL Model). Given M-BL model (0,r, g. Ω. F,SPP) (Definition 3.3) and regression parameters (af ,3 a, B, 7) E R x R4mm
Figure 2: Crature-IInteraaie BL. network with features and observed views (0, r, S2 P.
4

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Latent Variable Estimation in Bayesian Black-Litterman Models
m. Rdmm R. assume
The q ++ F 0 linear model (Definition 3.2) reduces the classical noisy views model (Black & Litterman, 1992): (a, 3, 7) = 0, so the residual (q Pa
0 ~ No:: En),
(3.2)
0 = at + F3 + + e N(0,99))
(3.3)
PF +
g - P(a+ + FB + -0), c~ N()
(3.4)
Define GM 2g- +(?2) 2 PP The posterior
The M-BL. posterior mean satisfies:
mean is
p PT - Pa - PFB) lim M0 = (G
p(0 02 F,5)) (0; uol / (
(GM)-1),
9-50
q-
dominant torm
where
+ -00000 00 + ta + FBB)
F, = (GM)-1 E 0u +
+ F3P)
bounded
+yPTsT (4 Pa PF3)]
(G Pr +
=r" (a.s.)
Proof.
See Appendix E.2 for detailed proof.
where the last step follows GM = Eg + (92 + PTΩ-IP - PT (GM)-11(9(-)) - 0. As re-
The posterior estimation of parameter 0 enables a predictive estimation on P:
sult, Corollary 3.1.1 becomes
74 8,.- as M2 + 0, q + Pr
Corollary 3.1.1 (Predictive Estimation by the M-BL Model), Define G.M = Σg ( + PT Assume N (0, Σ). Then, under Theorem 3.1, M-BL model gives the predictive estimation of unobserved asset returns = r q, G. F, S as
where 8,, denotes the Dirac measure at r*
Corollary 3.1.1 estimates asset returns under the general scenario where views are observed, i.e., solves Problem 2. It generalizes the classical ering its form when features are uninformative, and ap- proaches the true returns with accurate and precise views (Remarks 3.2 and 3.3). With the predictive estimation of asset returns T, the mean-variane variance optimization framework (Definition 2.1) determines the portfolio weights WM-BI.
model, recov-
F~ NO0; E+ (GM)))),
where
F, (GM)-1 [En 17/00 Gu + +S2 ((22))-((a)))) (a f FF*)
+gpTs - Pa PF3)]
Remark 3.2 (Classical Black-L.tttemm Recovery). M-BI. model recovers classical Black-Litterman model when:
3.4 Scenario: Features with Latent Views
In the previous section, solve Problem 2 with features and observed views. However, investor using the Black- Litterman model may not be an expert at quantifying the views. this section, we discuss the scenario without the views (Problem 3) by specifying two effects the features introduced in Section 3.2 and treating q and f2 as latent variables. Incorporating the effects into the network, showcase two graphical forms in Figure 2 and define the Lairedd-Laten-Paaaaetiiiaae
1. Features becomes uninformative, ie., uncertainty ap- proaches infinity: Q + 20, equivalently (125)-1 -1 0, the error matrix (S2F)- disappears from G
2. The q ++ F ++ 0 linear model (Definition 3.2) reduces to the classical noisy views model (Black & Litterman, 1992): (a, (,,) = (0, 0, 1), so the residual (q - Pa PF3) + 9-
Black-Litterman (SLP-BL) -P-B)) and Feature- Influenced Views Black-Liiternn models (Definitions 3.4 and 3.6) based on them. Then, we estimate posterior distribution over both asset returns r and their parameter 9 (Corollary 3.1.1 and Theorem 3.1).
(FIV-BL)
Under these conditions, we have
GM + 251 + 1 PTp--p=G:
and thus, we recover (2.5):
Consider the following problem:
rlg, 2 12 (( E 0u + PTΩ pTT--11 [0)
Problem 3 (Fecatre-Intterrated Integrated Predictive Estimation). Let E T represent the returns of the m assets and F denote the unobserved (or future) asset returns. Let ft E IRd rep- resent the features of the i-th asset and F e 12 be a block-diagonal matrix defined as
Remark 3.3 (Ground- Truth Limit). Corollary 3.1.1 accu- rately and precisely predict ground-truth asset returns with:
Perfect information: Q - 0, equivalently Q - oo.
== diag(f) f. .II (T.,,.
2. Accurate views: q -> Pr where r is true asset returns.
5

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Latent Variable Estimation in Bayesian Black-Litterman Models
Given D == (F, (F where F estimated from observa- tions of asset returns and features {(1,, F=) and Ω is the homoscedastic enror matrix corresponding to the obser- vations {-i) the goal is to estimate unobserved asset returns F ~ D)r
F, 12
9.92
We aim to solve Problem 3 by feature-integrated Black- Litterman network. We approach this by considering the two causal effects in Section 3.2 and treating the views and uncertainty matrix (q. 52) as latent parameters. Specifically,
Contiguration
Shared Latent Parametrization.
F
Effect 1: Features F' are extracted from the parameter 0. Consequently, the features F, along with their enror term ex (0, ( share the common parameter 0 with asset returns and investor views q.
q
Ω
Configuration 2: Feature-Influenced Views.
Effect 2: Features F influence the formation of views q. In this scenario, the features F, along with their error term ef (0, (t"), are related the latent views q through separate equation from the parameter 8.
Figure 3: Feature-Integrater Black-Liteerman network with fea- tures and latent Views (0, r. Q. F, .2
CONFIGURATION 1: SHARED LATENT PARAMETRIZATION
However, in this section, we do not mix the two effects.
To capture Effect 1, we follow the 0 <> relationship (Definition 3.1). By incorporating Effect 1, we showease
Remark 3.4 (Rationale of Differentiating Effect I and 2). We differentiate the two effects because, the scenario without investor views, the previous M-BL model (Defi- nition 3.3) estimates the parameter directly by (F, Ω),, meaning Effect dominates over Effect when both are present. If, in the general scenario where views observed, Effeect 2 is more significant than Effect 1, then, when views are latent, the ignorance of Effect leads to biased estima- tion. This matches the intuition: if we select the features not directly related to the asset (Effect 1) but highly influence the investor views (Effect 2), such as macroeconomic indi- cators like interest rates or CPI, then using these features to estimate asset returns directly is biased. To avoid this bias, differentiate the two effects with two modeling strate- gies. One handles the case where Effect dominates, and the other handles the case where Effect 2 is more significant.
network as Shared- the reature-integraate Black-Littermant (SLP-BL) model: Patant-Parameetrization
Definition 3.4 (SLP-BL model r0, q, Q, F,2 (...)) Let R" be the returns of the assets, parametrized by 0, with ~ p(r(0). Let g E RR represent the views on the returns of the k specified portfolio and Ω E R be the uncertainty matrix. Let F E mm be the features of the m assets and error matrix Ω2 E gm The SLP-BL model a portfolio model composed of four fundamental density functions: 1. Parametrized Asset Returns: p(r(0), the distribution of asset returns given the parameter.
Prior: (0). representing market equilibrium.
3. Likelihood of Views:
L(0 i2) == p(g, 3210), the rela- tionship between the parameter and the views.
We showcase the feature-integrated Black-Litterman net- work as two configurations: one incorporating Effect 1 and another incorporating Effectt 2. Intuitively, the first better captures generic features while the second more effectively handles the ren-aseet-related features.
Likelihood Features: L(
(2 = ()) the 0 F relationship (Definition 3.1).
We show the posterior estimation of the SLP-BL model:
Theorem 3.2 (Parameter Estimation of the SLP-BL Model),
r.9, Given SLP-BL model (0, (f,r,e F.r,g, Ω F,S2F (Definition 3.4) and regression parameters (a ,,8F) E R x Rdm assume
This implies that, in practice, an investor takes generic fea- tures of assets (e.g. indicators derived from the time series of each asset, as shown our experiment), configuration should be used. If an investor takes features not specific to individual assets (e.g. interest rates), configuration 2 should be used. The two configurations are not contradicting, one can take both types of features and incorporate them correspondingly.
0 ~ N (6. En).
(3.5)
0 0= oF FBF + +E. F N(0.2"
(3.6)
The Define G = vo' + +(Z+)) -(2 The posterior mean is
((0 N (0: (G [X 10
+(-²) - =F (GF)-1).
We visualize two configurations of the network in Figure 3 and define one model for each configuration accordingly.
Proof: See Appendix E.3 for a detailed proof.
6

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Latent Variable Estimation in Bayesian Black-Litterman Models
The posterior estimation of parameter 0 enables a predictive estimation on P:
Remark 3.6 allows estimations of regression parameters and the error matrix (a, B, 22) based on the observations {(11, [i))
See Appendix A for details.
Corollary 3.2.1 (Predictive Estimation by the SLP-BL
Model). Define G == Σg (I² Assume r N (0, Σ). Then, under Theorem 3.2, SLP-BL model gives the predictive estimation of unobserved asset returns F == r| r| as
We now introduce the final piece in this configuration: Re- garding the uncertainty matrix Q. a simplified assumption is that when an investor forms views based on features, the error matrix Ω2 captures all the information about this un- certainty. This would suggest omitting S from our model due the replacement with S2. Yet, in the general case, P remains necessary as it represents the intrinsic uncertainty of the views, regardless of the (F, )² Additionally, re- taining Ω allows our model to remain applicable when both (g, Q)) and F are observed. To treat 52 as a latent parameter, prior π (52) must be specified to enable Bayesian inference.
F~J N (GF I 90 +(22)- +F E (G+))))
Remark 3.5 (Features Replace Views). Corollary 3.1.1 is equialvent to classical Black-Litterman model if:
1. Features recover investor views: a" FF + P-19
2. Error matrix recovers uncertainty; (9"))-1 Pp
By incorporating Effect of the features characterized by the 9 ++ F linear models (Definition 3.5) and a given prior (I2), we showcase the network as the following Feature-Influenced--iii Views Black- Litterman (FIV-BL) model:
Mlack-Litterma
Under these conditions, we have GF + -Ip -
G, and thus recover (2.5):
rlg, Ω N ( |X 6u + pT pT 91
Definition 3.6 (FIV-BL model (0,r; q, 22 F,D2)). Let E RT be the returns of the m assets, parametrized by 0, with p(r(0). Let q E R& represent the views on the returns of the k specified portfolio and Ω E R be the uncertainty matrix. Let F E m be the features of the m assets and error matrix Ω2" E m The FIV-BL model is a portfolio model composed of five fundamental density functions:
Corollary 3.2.1 estimates asset returns without the views, i.e, solves Problem 3. With the predictive estimation of asset returns 7, the mcan-variance optimization framework (Definition 2.1) outputs the portfolio weights WSLP-BAL See Appendix A for the selection of {E, Eo, Oo, of ,3F,, DP..
CONFIGURATION 2:
INFUUREEEINUENNCE VIEWS
1. Parametrized Asset Returns: p(r(0), the distribution of asset returns given the parameter.
To capture Effect 2, we define a ++ F relationship by multivariate linear model with local dependency:
2. Prior: π (0). representing market equilibrium.
Definition 3.5 (g + F Linear Model). Given features F R and portfolio weight matrix for specified portfolios e R let q R* be the views returns of the k portfolios. Define regression intercept vector E R., regression coefficient vector 3 E Rdd,, random error ex E R"",, and сгтоr matrix Ω. E R such that:
Likelihood of Views: L( q (2) == p(g. 220)), the rela-
tionship between the parameter and the views.
4. Views given Features: F, the g ++ F relation- ship (Definition 3.5).
5. Prior on Uncertainty Matrix: T()), representing intrin- sic uncertainty of the views.
q= P(a + 6 e N0,,2")
The FIV-BL model marginalizing out latent parameters (g, ()) to estimate the posterior of
Furthermore, define $1, S2,..., Bm E Rd as m partitions of the vector such that:
Theorem 3.3 (Parameter Estimation of the FIV-BL Model),
B B 8 331
Let P E R be the portfolio weight matrix for k spec- ified portfolios. Given a FIV-BL model (0, r, g, Ω F, ( (Definition 3.6), regression parameters (a,, 3) E RT x ddd and a prior π (S2), assume
This captures the relationship without the loss of generality:
Remark 3.6 (Noisy Implied Asset Returns). Based the intuition that the views 9 are formed on the returns of the k specified portfolios, define the noisy implied asset returns
0 ~ N (fo, Eo) P0 q cG, 6~ NO,,
(3.7)
p == a=
(3.8)
q = P(a + FB+e"), e ~ NO,
(3.9)
such that: g = F. Then, the dependency between each element of this implied asset returns and d features becomes local:
TThis concept is similar to the existence of the intrinsic covari- ance Σ regardless of the prior parameter (9o, Eo) of in Lem- mas 2.1 and C.
of = aa Fr.B+ - = =a + +3 + E i E [m].
7

---

Latent Variable Estimation in Bayesian Black-Litterman Models
where On E m and En I are given prior mean and covariance, and (e, ) mutually independent. Define
0/22, F, (Np',E)) G- where [n'= | (- + PT Pla la F3)
G= E PT P. The posterior mean distribution is:
(eps"PT)1 PG-1
((01F,5)))= / N (0: Kojo F,p), (())))())))
((2)dS2,
Assume Σ' have an Inverse- -iishrt prior:
(3.10)
T(Σ') = (W": V ,2),, (V. E R" (3.11)
where
Then the marginal posterior of 0 given (F,S²) follows a multivariate-f distribution:
Poja = G-1 (Ig 8. + P EER.F G G- (PP
p(0 F.S2.) ~ to (0: v' -
Proof. See Appendix E,4 for detailed proof.
Proof. See Appendix E.5 for a detailed proof.
Remark 3.7 (Posterior Collapse Under Perfect Views). If we omit the intrinsic uncertainty matrix by Ω + 0, or equivalently Q - 0, we have
Since l-distribution lacks conjugate prior, we omit intrinsic covariance 2 in estimating unobserved asset returns i:
Corollary 3.3.2 (Approximated Predictive Estimation by the FIV-BL Model). Assume r ~ N(0, E). Under Theo- rem 3.3, considering 0|52, F, 2 N (n', E), assume: (W")
G G P P,
MOIR - G (P P(a
Cop. + G + G - (PΩ2 PT)S-1PG-! PG
Thus, the posterior collapses to 0 012, F, F,St ~ N(a F3,
p' is a constant G (Ep 0. + P' P(a FB)). Then, as Σ + FIV-BL model gives the predictive estima- tion := F| as
effectively recovering the 0 ++ F' relationship 0 x++ F3+ c" (Definition 3.1) except losing the prior information on 0.
For P:
Furthermore, reintroducing this prior leads to Theorem 3.2.
V' m
The integral (3.10) is a form of Infinite Gaussian Mixture model (IGMM) (Rasmussen, 1999). In general. there is no further closed-form solution for it unless Ω2 is restricted to special conjugate family effectively collapses to point mass (i.e.. Ω is known and fixed). In non-conjugate settings, the expression remains a continuous mixture of Gaussian distributions, and must be evaluated or approximated nu- merically (e.g. via Monte Carlo or approximation methods (Newman & Barkema, 1999; Kruschke, 2010; Wainwright et al., 2008; Blei et al., 2017)).
Corollary 3.3.2 also estimates asset returns without the views, i.., solves Problem 3. With the predictive es- timation of asset returns F, the mean-variance optimiza- tion framework (Definition 2.1) determines the portfolio weights -rIV-BL- See Appendix for the selection of {E, Eo, 0o P,a, 3,92, W ",,'
4 Proof-of-Concepe Experiments
Depart from the classic Black-Litternen model that relies on subjective investor views, our model estimates the poste- rior distribution over asset returns directly from the feature data. To demonstrate this concept, we consider the setting without subjective investor views. Specifically, integrating asset-specific features, discussed in Re- mark 3.4, and choose the SLP-BL model (Definition 3.4) accordingly. We show the model works under this setting and consistently outperforms the benchmarks.
Since there is no trivial conjugate prior (s)) for the likeli- hood 0|5,, F here we offer an approximation method. We first substitute Ω with Enj. Then, we approxi- mate the mean of the likelihood Mon as a constant. Finally, we assign a conjugate prior to E as an Inverse-Wishart (IW) distribution. This allows to obtain tractable joint distribution p(0, P) - specifically Normal-Inverse -ishhst (NIW) distribution. As result, the posterior mean distribution p(0) F, 52F) follows a student-t distribution after marginalizing out E,
focus
Dataset SPDR Sector ETFs. We collect adjusted daily closing prices and volume for 11 Sector ETFs (Table 4) from April 13, 2004 to February 22, 2024 (20 years). To avoid selection bias, the portfolio selection list is updated in sync with the introduction of new sectors.
Corollary 3.3.1 (Conjugate Prior). Consider a FIV-BL model (0,r, g, Q, F, f (Definition 3.6) with constants (P, E, 0g En, a, B, R,) E g x mm R. x pmxm x Rmm x Rdm x Ri
Dataset I:: Dow Jones Index. We collect adjusted daily closing prices and volume for 41 stocks (Table 5) that have been part of the Dow Jones index from January 5, 1994 to
Define G = Σg' + P Assume
8

---

Latent Variable Estimation in Bayesian Black-Litterman Models
February 22, 2024 (30 years). To avoid selection bias, the
Table 2: Performance on Dow Jones Index Dataset.
portfolio selection list is updated in syne with the index.
Cumulative
CAGR
Shapee Ratio 1
Max
Volatility
Retum (8)
(%) 1
Drawdown ( (E)
(S/ann.)
Backtest Task. We backtest our SLP-BL model for each dataset period. On each monthly rebalance day, the model outputs a portfolio weight WSLP-BL that maximizes the
EQW
4,606
9.22
0.75
58.90
19.54
DIAA
932.51
5.49
0.52
53.78
17.97
MV (50d)
774.08
5.09
0.45
63.35
20.83
BL (500)
3,980.23
8.86
0.78
42.42
17.84
Sharpe ratio (Definition F.I), ), a standardized mean-variance optimization framework (Definition 2.1). In the model, the prior set as traditional Markowitz model and the features selected based nine generic indicators (Table 3) de- rived from asset-specific data. We follow Appendix A for the choice of {Σ, Co. 8o, a", 3", 2²" except that, to avoid the issues of mismatch scale, we use price and indicators data to derive the regression parameters.
MV (80d)
1,081.47
5.82
0.51
53.98
20.26
BL BL.(80))
4,603.82
9.22
0.84
39.95
16.81
MV (100d)
1,529.60
6.60
0.55
56.06
20.59
BL (100d)
4,557.03
9.19
0.85
39.92
16.36
MV (120d)
1,577.61
6.67
0.57
46.73
20.07
(120d)
4,819.83
9.33
0.87
39.81
16.42
MV (150d)
2,208 84
7.45
0.62
41.02
20.03
BL (150d)
3,405.78
8.49
0.80
40.42
16.51
(Log
Benchmark Indioes
Benchmarks and Evaluation. The benchmarks our port- folio model are (i) market index (e.g. S&P500 and DJIA) (ii) -qual-weighted portfolio model (iii) traditional Markowitz model. We evaluate the models with the follow-
630.94
EGW: Exuaa Weiggied
5Pmh 5aP5500
MXX0536)
Camulativv Raturn (K)
ing metrics: Cumulative Return, Compound Annual Growth Rate (CAGR), Sharpe Ratio, Maximum Drawdown, and Volatility. We present the results for five pairs of traditional Markowitz model and our Black-Litterman model with vary- ing rolling window lengths of historical returns: 50 days, 80 days, 100 days, 120 days, and 150 days.
171
48068
(100e)
(120a))
RL (150d)
0.44
Results. The SLP-BL model consistently outperforms both traditional Markowitz model and market indices across two datasets (Tables 1, 2 and 6 and Figures 4, 5, and 9). This attributed to the more stable portfolio weights based on Bayesian framework, as shown by Figures and 7.
.0 34
Data
Figure 4: Cumulative Return on SPDR Sectors ETFs Dataset.
Cumulative Returns (Log Scale) of Models and Benchmark Ind ices
225015
za'nt Equax Whightdd DUlA: Dow Jones
Table 1: Performance on SPDR Sector ETFs Dataset.
3211.2%
5P%: 5505500
NW [5ce)
Cumulative
CAGR
Sharpe
Max
Volatility (R/ann.) |
MV (804
Return (%) 1
(8)
Ratio
Drawdown ( (
MMV (2006)
MY (1206)
BOW
450.74
44.90
1116.2%
NWV 21506)
S&P500
545.77
0.59
55.19
19.03
Oumulative Rahm
(50/)
(560)
MV (50L)
134.12
3,00
0.35
53.11
15.99
609.9%
BL (50d)
541.99
6.67
0.66
46.56
16.24
343.2%
MV (804)
291.21
4.85
0.50
38.58
16.33
(80d)
609.66
0.69
46.78
16.12
171.8%
MV (100J)
411.83
5.84
0.57
36.37
16.91
66.4%
BL (100d)
602.75
7.01
0.70
46.05
15.91
MV (1203)
412.87
5.84
0.57
36.10
17.12
(120d)
587.50
15:74
1.00
20000
2004
2030
2012
2010
2000
2004
MV (150d)
249.11
4.44
0.45
47.49
17.37
BI. (150d)
556.13
6.75
0.68
44.54
15.91
Figure 5: Cumulltive Return on Dow Jones Index Dataset.
Discussion and Conclusion
of latent views (Problem 3), we differentiate the feature ef- fects to handle distinct features (Remark 3.4). Accordingly, we present two models: the first provides closed-form asset return estimation (Theorem 3.2, Corollary 3.2.1). while the second results in mixture model that requires numerical methods (Theorem 3.3, Corollary 3.3.1, Corollary 3.3.2).
We propose a Bayesian reformulation the Black- Litterman model for portfolio optimization without the need for subjective investor views. Our key contribution is unified Bayesian network that integrates features and infers parameters. In the case of observed views (Problem 2), the network estimates asset returns based on a mix of two fea- ture effects (Theorem 3.1, Corollary 3.1.1), generalizing the classical Black-Litterman model and recovering ground- truth estimation with perfect views (Remark 3.3). In the case
Numerically, our model works without investor views and demonstrates consistent, hyperparameter-ronn robbuut improve- ments over the Markowitz model and market indices across long-term, real-world datasets (Section 4).

---

Latent Variable Estimation in Bayesian Black-Litterman Models
Impact Statement
Chen, S. D. and Lim, A. E. A generalized black-litterman
model. Operations Research, 68(2):381 2020.
This work improves portfolio optimization by reducing sub- jective human inputs. It enhances transparency and pro- motes data-driven mariii.n-maaing.... The framework ben- efits both institutional and individual investors with more reliable and fair strategies. However, data-driven models may amplify biases, so careful evaluation is needed for fair outcomes. Overall, this work advances financial modeling and emphasizes ethical implementation.
Cheung, W. The augmented black model: A ranking-free approach to factor-based portfolio construc- tion and beyond. Quantitative Finance, 13(2):0013316 2013.
Christensen, and Li, Predicting financial stress events: A signal extraction approach. Journal of Financial Sta- bility, 14:54-65, 2014.
Acknowledgments
DeMiguel, V.. Garlappi. L., and Uppal, R. Optimal versus naive diversification: How inefficient is the 1/n portfolio strategy? The Review of Financial Studies, 22(5): 1953, 2009.
TL would like to thank Gamma Paradigm Research and NTU ABC Lab for support. JH would like to thank Han Liu, Mimi Gallagher, Sara Sanchez, Dino Feng and Andrew Chen for enlightening discussions on related topics, and the Red Maple Family for support. The authors would like to thank the anonymous reviewers and program chairs for con- structive comments. JH supported by the Northwestern University. The content is solely the responsibility of the authors and does necessarily represent the official views of the funding agencies.
Deng, Q. A generalized vecm/var-dcc/adec framework and its application in the black-litterman model: Illustrated with china portfolio. China Finance Review Interna- tional, 8(4):453 467, 2018.
Duqi, A., Franci, L., and Torluccio, G. The littermn model: the definition of views based on volatility fore- casts. Applied Financial Economics, 24(19); 2014.
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12

---

Latent Variable Estimation in Bayesian Black-Litterman Models
Appendix
A Hyperparameter Selections
14
B Related Work
15
B.1 Bayesian Portfolio Optimization
15
B.2 Data-Driven Black-Litterman Model
15
C Supplementary Theoretical Backgrounds
16
C.1 Prior and Likelihood of the Black-Litterman Litteeettmaa--aeessss Bayes model
16
D Axillary Lemmas
16
D.I Integral of the Product of Two Gaussian Distributions
16
D.2 Sufficient Statistic for Ω in Corollary 3.3.1
D.3 Integral of Normal-Inverse- Wishhar (NTW) distribution
18
D.4
Regression Estimators
19
Proofs Main Text
20
E.1
Proof of Lemma 2.1
20
E.2
Proof of Theorem 3.1
21
E.3 Proof of Theorem 3.2
22
E,4 Proof of Theorem 3.3
22
E.5
Proof of Corollary 3.3.1
23
F Experimental Details
24
F.I
Sharpe Ratio Maximization
24
F2
Table of Indicators
24
F3
Tables of Datasets
25
F.4
Figures of Asset Allocation
26
F.5
Turnover Rate Analysis
27
13

---

Latent Variable Estimation in Bayesian Black-Litterman Models
A
Hyperparameter Selections
Here we provide a practical guide to derive hyperparameter set {Σ, Ev, 80: ,3 ,2 for Black-Litterman model (SLP-BI model, Definition 3.4) and { Co, 0o: P, P,,,8 Ω W, v', Ωo} for reature-Influenced Views Black-Litee Litte-nnten model (FIV-BL model, Definition 3.6)
Among the entries of {E, Eu. B. practitioner first approximate E using sample variance given fril=1 and let Σg T be a matrix proportional to the covariance matrix, following (Salomons, 2007). Numerous papers (Black & Litterman, 1992; Lee, 2000; Ellison, 2004; Idzorek, 2007; Salomons, 2007) address the choice of the scaling factor mostly suggesting a constant in (0, 1]. Given the approximated Σ and Eg. one obtains 0g by Lemma C.I.
In this work, we take P P= as a m x m identity matrix because the features F are asset-specific (Problems and 3). Given the historical observations {(ri, Fi) in (Problems 2 and 3), define:
T カ= F=
F
Fi
F.
The following context suggests how Fi)EE enables estimating (5F,a' R 3). We estimate the erorr matrix O² based on kernel density estimation every feature. Specifically, consider rule-of-thumb bandwidth parameter (Silverman, 2018):
4
h h=
12
2m
dm +
where m is the number of assets, the number of features for each asset, and n the sample size.
Recall that fi E Idd represent the features on the i-th asset and be part of the features E Rmmxdm;
F == diag(fi f..,,,,
we scale glement-wise variances of fi, or Var(ft;) for j-th feature of the i-th asset by h to construct the diagonal matrix
H diag(h Var(Ji,1), h. Var(Jm.a))-
Then, for each asset return r;, we compute the ordinary least squares (OLS) coefficients B, and intercept 0; by predictors f. These coefficients are aggregated into a block-diagonal matrix B E gmxdm The covariance estimate is π BHB
For {aF 333), we conduct maximum likelihood estimation based on the 0 - F model (Definition 3.1) and r ~ N(0, E).
By Lemma D.4, we obtain
aF = 8
F (Ω -
Σ
ET TÔa -
For {ā, 3). we conduct maximum likelihood estimation in the q F' model (Definition 3.5), assuming that the observed returns are samples from the noisy implied asset returns yf defined in Remark 3.6. By Lemma D.4, we obtain
à = F F3, 3
F
Σ F (QF)-17
1=1
For (v', p', 90 the conjugate prior parameters (V,') in Y' ~ (W ,/) have distinct roles: V encodes prior knowledge about the covariance shape and scale, acting as a "pseudo-covariie
matrix" with mean E[E W'/lv/ (m/-mm++ (if > m - 1). Larger V implies stronger prior beliet's about higher covariances. The degrees freedom v/ control prior strength, functioning as an effective sample size. Smaller v' allows the data to dominate, while larger v enforces V. For a weakly informative prior, the rule of thumb is to set Y (minimal informative scale matrix) and v/ m + 2, assuming no strong prior correlations. Gelman al. (1995); Hoff (2009); Murphy (2012) discuss details this topic.
Selecting (V, v') allows us to compute the approximated constant Sg. Since E' is deterministic function of (Ω, F, F.2F). Ω is likewise deterministic function of (2', F, ). Therefore, we let On 12 F, 2 where 2 = v /v' - m is the mean of the distribution IW( (w), z/).
OOne may adjust the targetted data to avoid the issues of misaligned scale. For example, price versus momentum indicators.
14

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Latent Variable Estimation in Bayesian Black-Litterman Models
B Related Work
B.1 Bayesian Portfolio Optimization
Why Bayesian? To address the parameter estimation risk in traditional portfolio optimization shown by (Markowitz, 1952; Kalymon, 1971), Barry (1974); Klein & Bawa (1976); Brown (1976) advocate Bayesian framework upon prior information in portfolio optimization. Foundational works by (Jorion, 1986) and (Black & Litterman, 1992) demonstrate how Bayesian shrinkage improves covariance estimation, reducing overfitting and highly sensitive weight in Markowitz-style allocations (Meucci, 2005; DeMiguel et al., 2009). Subsequent studies on robust Bayesian portfolio optimization include multiple approaches such as uncertainty estimation (Qiu et al., 2015; Yang et al., 2015), alternative prior specifications (c.g., heavy-tailed or non-conjugate priors) (Garlappi et al., 2007; Tu & Zhou, 2010; 2011), 1), advanced sampling methods (Michaud & Michaud, 2007; Huang et al., 2021), and regularized optimization considering transaction costs (OlivaresNNada & DeMiguel, 2018).
Issues Around the Bayesian Framework. While these methods leverage analytical tractability to incorporate historical data or expert views (Ulf & Raimond, 2006), they often rely on restrictive assumptions (e.g., conjugate priors) or subjective expert inputs. Recent advancements, such as Markov chain Monte Carlo (MCMC) methods (Greyserman et al., 2006), relax these constraints, enabling inference in more complex hierarchical or time-series models. Meanwhile, contemporary approaches increasingly emphasize data-driven techniques for deriving expert inputs, including investor views in the Black-Litterman model (Black& & Litterman, 1992).
B.2 Data-Driven Black-Littterman Model
Why Data-Driven? Across decades, the heuristic framework for deriving investor views (g. St) in the Black-Litterman model attracts research working on estimating investor views (Beach & Orlov, 2007; Palomba, 2008; Duqi et al., 2014;
Silva et al., 2017; Deng, 2018; Kara et al., 2019; Kolm & Ritter, 2021; Teplova et al., 2023). Early efforts to estimate (g. Q2) employ historical return data within GARCH frameworks, framing view derivation as a time series prediction task (Beach & Orlov, 2007; Palomba, 2008; Duqi et al., 2014). but financial time-series data often exhibit high noise and insufficient signal-to-noiss ratios for reliable prediction (Gómez& & Maravall Herrero, 1998; Christensen & Li, 2014).
Advancement in Generating Views. Recent advances mitigate the previously mentioned weaknesses in time series forecasting by integrating econometric models and machine learning - e.g., GARCH with neural networks (Bildirici & Ersin, 2009) or LSTM (Kim & Won, 2018), support vector machines (Pérez-Cruz et al., 2003; Kara et al., 2019), grey systems (Huang & Jane, 2009), and feature programming (Reneau et al., 2023). To embrace richer information, recent studies incorporate external data sources such as macroeconomic indicators (Zhou, 2009; Cheung, 2013), factors (Geyer Lucivjanská, 2016; Kolm & Ritter, 2017; 2021).
Neglected Issues from Whole Perspective. However, while these advanced methods achieve high forecast accuracy in isolation, errors can propagate through subsequent optimization pipelines when estimators (q, are naively embedded in the Black-L.iterman framework. Finkel et al. (2006) addresses such issues of error propagation in multi-stage pipelines. Furthermore, in some works, estimating and Ω independently risks misaligned confidence assumptions. Without joint modeling, overconfidence in views (low f2) might amplify errors in q. and thus distort portfolio weights. Guo et al. (2017); Kendall & Gal (2017) discuss such confidence calibration and uncertainty estimation.
Introduction of Bayesian Network. Meanwhile, prior work has explored the use of Bayesian networks for the Black- Litterman model, which serve distinct purposes such as transferring the approach to factor models (Kolm & Ritter, 2017; 2021), addressing multiple expert views (Chen & Lim, 2020), or generalizing to multi-period frameworks (Abdelhakmi & Lim, 2024). Yet, these approaches still rely on human experts to specify the parameters (q. 22).
inputs, error propagation, and incoherent estimation - we propose our Bayesian network reformulation of the Black-Littermal model. This framework unifies historical or external features and latent investor views (q. S2) into a single Bayesian network, enabling inference over parameters (Theorems 3.2 and 3.3) and asset returns (Corollaries 3.2.1 and 3.3.2) directly from data. This eliminates reliance on heuristic inputs disjointed estimators, ensuring coherent estimation and fully data-driven portfolio optimization.
Our Work. To bridge these gaps-
15

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Latent Variable Estimation in Bayesian Black-Litterman Models
Supplementary Theoretical Backgrounds
C.1 Prior and Likelihood of the Black-Litterman-Bareee model
Here we show how model the prior and likelihood in the Black- Boock-Littermann Bayes model (Definition 2.2). To obtain the prior (0), an investor sets market portfolio and lets the prior be the market portfolio estimation. the absence of views g. the BLB model reduces to this market portfolio, producing estimation (exactly the prior) asset returns and outputting market portfolio weight Wmarket" by Definition 2.1. For example, if the investor takes the traditional Markowitz model as the market portfolio, it would produce a normal distribution of the historical asset returns as the prior (0).
commonly used market portfolio is market cetpaliizaaio-wwiigghe portfolio". In this case, the market portfolio weight is the market capitalization weight Weap". :
Assumption C.1 (Market Capitalization Equilibrium Prior). the absence of views g, the BLB model (Definition 2.2) produces an estimation asset returns such that the mean-variame optimization framework (Definition 2.1) has an optimal argument w² Weap- In other words, 7 satisfied:
argmax
W B'
w" Varliw
Weap:
(C.1)
2
where 8 E [0, oo] is a given risk-adjusted coefficient.
With Assumption C.I, we use reverse optimization technique to derive the prior:
Lemma C.1 (Reverse Optimization for Prior, page 139 of (Satchell & Scowcroft, 2007)). Let r E 1 be m asset returns,
parametrized by 0, with ~ p(r(0). Let the market capitalization weight on the m assets be Weap E R and 8 E [0, oo] be a risk-adjusted coefficient. Assume
N (O, Σ), 0 ~ Noo, Eo):
where 2, En mmx are given intrinsic and prior covariance. Then the prior mean is
0o+ 8(Σ + Weap
(C.2)
To obtain the likelihood function T((lg), we assume probabilistic relationship between parameter and views q:
Assumption C.2 (Classical Noisy Views Model, page 35 of (Black & Litterman, 1992)), Let r E R" be the returns of the m assets, parametrized by 0, with ~ p(r(0). Let P E Raxm be the specified portfolio weight matrix. Let E TR represent the views on the returns of the specified portfolio and f2 g be the uncertainty matrix. Assume
P8 = +
c (00
Under Assumptions C.1 and C.2, can derive the Black-Litterman formula (Theorem 2.1) as the posterior estimation 0 of the Black-Litterman Bayes model (Definition 2.2).
D Axillary Lemmas
D.1 Integral of the Product of Two Gaussian Distributions
Lemma D.1 (Integral of the Product of Two Gaussian Distributions, page 266 of (Aroian, 1947)). Let R", and let
N Na;; p1, E1) and ((:;; μ2, Σg) be two multivariate Gaussian distributions with means μ4, /22 E R and positive definite covariance matrices Σj, Eg Raxx, respectively. Then, the integral of their product over IR" is given by:
N(x M1, Na; u2, 22)dx = N p2, Ei + Σ²)
(D.I)
Proof. The product of two Gaussian PDFs is
Ei) Ex) (2mr
(
(D.2)
1/2 Eg 1/2 1/2
*Idzorek (2007) names Implied Equilibrium Return Vector. 'A market sgpitalization-weigheed portfolio performs market capitalizaaiin index (e.g., 500).
'The weight vector proportional each asset's market cap.
16

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Latent Variable Estimation in Bayesian Black-Litterman Models
where Q (x 11) 1
m1) + (x - u2) Eng (x (22)-
Expanding and combining terms:
Q (2i + +2 2x (>i #1 Liz u2) +c
[x b)b
b) A(x - b) A- b
with A= x E Mn Eng U1, and / E in 1/2 E2 p2
Substituting back to (D.2):
N1 (I;
1
1
1
1
--
A(x A-
23
A 20)
1/2 (2T)a1>))
/// 1/2
exp
Integrate 2 E :":
(2T) ///1//152/1/3 exp 1 bT 20
(27)m/21A-114
E2)dr=
(D.3)
Using |A- E Eal E1 +Ea
we have
A 1A-1/1/2
(D.4)
Σ + 1/11/2 Σ1 1/2 1/1
Simplify the exponential term in (D.3) using the identity:
1
1
b A-
c
(M) U2) T + -2) 12)
(D.5)
2
Thus, by (D.4) and (D.5), (D.3) becomes
f |N N,) N 22,
1
( 122 + 3/) m2)
1
(2 Σ₁ Σし 1/2 exp
=N(wi 21 232)
This completes the proof.
D.2 Sufficient Statistic for Ω in Corollary 3.3.1
Lemma D.2. Consider the hierarchical model where 0 0 A, B ~ (N(2,, A, B), Σ(Ω, A, B)),
with Q as parameter matrix, and A and B as fixed matrices. The mean p(22, A, B) and covariance (Q2, A, B) depend on (2, A, B). Then, the conditional distribution p(0|(,, A, B) can be expressed only in terms of Σ(Ω, A, B), A, B if and only if, for all pairs (S2, S2)) such that E(2,, A, B) = Σ(Z,, A, B), we also have μ( A, B) = p(2, A, B). That is, 112 A, B) is determined by Σ(Ω, A, B), A, B
+ p(0 Ωt A, B) p(0 Σ(S, A, B), A, B).
Proof. If
m0,, A. A,
whenever Σ(Q,, A, B) A A, B), then for a given value of E, the pair (H, Σ) does not depend on which Ω generated Σ. Henee specifying Σ(Ω, A, B), A, alone suffices to determine the normal distribution 0. Thus p(0 Ω2 A, B) p(0 Σ(8, A, B), A, B). Conversely, if
p(0 Ω2 A) = r(0 Σ(Ω, A,),
Then any two values Ω2 and ai yielding the same Σ(Ω, A, B) and E(Πg, A, B) must produce the same distribution for 0. Since 0 is normally distributed, its mean must also match, i.e., (L, A, B) = p(22, A, B). This completes the proof.
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Latent Variable Estimation in Bayesian Black-Litterman Models
D.3 Integral of Normal--nneere--isshall (NIW) distribution
Lemma D.3. Given a joint Normal-Inverre-Wisharr (NIW) distribution
p(0,E") NOW 1,; 0 012"~ ~ N(,,E"), E'~ ">II(V
the marginal distribution of 0 is multivariate t-distribution:
T
0~t N~tv tv W m
with degrees of freedom v', location parameter pt, and scale matrix
V
w'm
Proof. To derive the marginal distribution of f, we integrate out E':
p(0) = / p(0) E')p(E'ddE'
(D.6)
The conditional distribution p(0 E') is:
(27)m/2|511/2 exp (-210 -x= -1))
1
-
p(0 )')
(D.7)
The marginal distribution p(E') is:
V 1//2
(v'+ exp (-
p(E') 2 ml (2/12)
Σ'
(D.8)
Substituting (D.7) and (D.8) into (D.6):
p(0)= x / |Σ (v' (2 exp (-2 - (1100-000003-11000--00007 NE'.
Let S = (0 -0- (0))()(0 =²)T + V. then:
p(0) / E (0(m2222 exp (-1 tr (1E))1)) dE'.
(D.9)
Using the matrix integral identity (Muirhead, 2009, Chapter 7.2) or (Gupta & Nagar, 2018, Chapter 1.4)):
/ |ل| (0 exp ( - (11((1-113) dΣ x /0/2
1
we identify = 1/ + and B = S. (D.9) becomes:
p(0) x (8/44/21)
(D.10)
Expand |S| using the matrix determinant lemma:
|S| =
-())-
(D.11)
Substituting (D.11) back to (D.10), we have
p(0) (1+ (1-
-(v+m)/2
where = m 1. This matches the kernel of multivariate t-distribution:
~ ty (r' V = t,
(v:=)) v (4'
This completes the proof.
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Latent Variable Estimation in Bayesian Black-Litterman Models
D.4
Regression Estimators
Lemma D.4 (Estimation of a, 9 under Homoscedastic and Correlated Errors). Consider a multivariate linear regression model:
7= a FB+e, ~'~ N(,,
(D.12)
where 2 is a constant positive definite covariance matrix across observations (i.e. the error term e" is homoscedastic but potentially correlated). Given observations {(r, Fi) the maximum likelihood estimators (MLE) for (a, (3) are:
à ā F3,
B
F (22 -
F '
1=1
where
F= 1
F
F f==- -
Fi, 1=1
F Fi F.
1=1
Proof. The log-likelihood function for the model is:
log B) log g-
11
1
Σ m
a ((3) (n
F.(3) + const.
(D.13)
2
1=1
Differentiate the
log- log-likelinn likelihood (D.13) with respect to and set it to zero:
log L(a, 3) -
1
Σ
"
( ( a 113)
da
da
1=1
( (n- - a -
Summing over all observations:
1
n 1n(" Σ (²") 0.
1=1
1=1
Solving for ã:
- FB.
(D.14)
Differentiate the log-likelihoo likelihood (D.13) with respect to 33 and set it to zero:
i
1 Σ 2 (i
log [(a,8)
F.3)
03
Σ (m = =0
(D.15)
1=1
Using the expression for â (D.14), we have
- - FiB = ri - (F FF- - F13 = (n - (A - F)3 = 5. - Fi.3.
Substituting back to (D.15) with a = ã:
F. - -
Since ΣR Fi and Σ Fi O, we have:
5 -
F ( 0.
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Latent Variable Estimation in Bayesian Black-Litterman Models
Solving for 3:
3 = Σ F. (0) (
Σ
Fi )
1=1
1=1
Since ef is neither uncorrelated nor homoscedastic, remaining ÿt is essential to obtain unbiased estimates of B. This completes the proof.
E Proofs of Main Text
E.1 Proof of Lemma 2.1
We separate the proof into two parts. One for (2.4) and another for (2.5):
Proof of (2.4).
p(0|g, 52) ()
19- P0) sq (1- P0) exp {
exp
(0 6o)
(0- Tx' (0- 8b)
(By (2.2) and (2.3))
2
2
1 [0TPTS -
q 00 8 -200 -0 0 + 00 80) }
exp
22 PM
s2 9
2
exp
[0T(PT
Σg 2 12- 0, sT 1 9 o En 0o]
(E.1)
2
where the third equation is the result of the symmetric property of 2g and Ω To simplify the above expression, we introduce
G= + P
D 1 0o + P 4.
A = E 6. + q 12-
then, we have:
TG 2D + -00000000000---000000-------
(GO - -C0-DD)G-(00-D++4--00
(0- (0---11)(((0---------D0D00
Therefore, (E.1) becomes
p(0lg, Ω2 x exp (
1 (0- (00
This completes the proof.
Proof of (2.5). From (2.1), we have:
p(r = N (r; E):
(E.2)
From Lemma C.I, we have:
0 = 5(E + Eo)weap
(E.3)
From (2.4), we have:
p(0lg) = N (0: (Ig '0 + (0011))6-1))
(E.4)
Thus,
/ p(F10)p(01))d00
F p~p(())))
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Latent Variable Estimation in Bayesian Black-Litterman Models
/ N
d0
(By (E.2) and (B.4)
EN (E 8u -
(From Lemma D.I)
N (F:G** [ Wepp + Pla-to) ;
(By (E.3)
This completes the proof.
E.2 Proof of Theorem 3.1
Proof of Theorem 3.1. The posterior 0 given the data is proportional to the product of three Gaussian densities:
p(0]g, Ω F,S2
6 p(0)F,52
(E.5)
Observation Likelihood Features Likelihood
Prior
where the observation likelihood, features likelihood, and prior distribution are respectively:
P(a FB + +0]] 2 (4 P(a FB + -011
p(q F, exp
(By (3.4))
p(0 exp
2 [0 (a FB3)] (52 [0- -(af 1
(By (3.3))
(0) exp
- (0- 0o) Eg (0 0n)
(By (3.2)
Combining all quadratic forms in the exponent (ignoring constants) of p(0 q, F,,,, ",", we have
1 (0 0)) 22, (0- (0) + (0 (a" +F3 (0h)) -a"""a" + F33))
2
+[q - Pa - PP3 - -PF (g Pa PFB PFB-7P8)
1 0 - 20,
0o
2
+ + - 2(a T (a" + + F3²)
pT 27 Pa Pa- 1--Pp
(g - -a Pa- PF3) ( ( Pa PF
TT -o11 + +(22 (00)--=2pp00--p p
= - 2
GM
20
0u -a (a FBF) + p
s 12 Pa PPF)] + constant
where the second step groups similar terms and the first step expands the quadratic form. Completing the square in the form
(0 ( GM (0 (o
the posterior distribution (E.5) becomes:
p(0 QFF (0:HO)gALPAP.GG
where
GM = Ep + (0 + 1pp
H, = (G
(G¹)- -1 [>0 00 "'a" +FB) 4TpTs11 lg Pa PF3]]
This completes the proof.
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Latent Variable Estimation in Bayesian Black-Litterman Models
E.3 Proof of Theorem 3.2
Proof of Theorem 3.2. From the 0 ++ F model (Definition 3.1),
1(0 = p(F (F) & exp (-11133
|F - -00 -00- ((-= - -(0 - -0- (0-af1l).
Thus, we have
((FF,22 & 110 F, FF )T()
1
FF - (0 - -(0- - - - af)) exp (-2(0 (0 - 00) "00'=='10 (0 06)
(00-(1))
x exp
2
(By (3.6) and (3.5)
1 (0- - a - -F3 ( a= -33 + (0 (0-0)) T 00)1
x exp
2
1 (0((() -1 + - 0o + +22) 1a + F + const]
X exp
2
1 o0 GF - 2(DF) + const) const)
x exp
2
2 (0- - po( GG (0 - ((FF)) - Th -o GF Palp + constl)
1
x exp
(E.6)
where G = (52 + 2T² and DF = 2 00 + (10 ( + F and
(5² 0u + {"
+ FFF)).
PolF
Substituting back to (E.6), the posterior distribution is:
((0 = N ( 0o + (0
+ +F3*)] , (G+)-1).
This completes the proof.
E.4 Proof of Theorem 3.3
Proof of Theorem 3.3. The first two relations (3.7) and (3.8), from (2.4), leads to:
p(0 q, Q) = N (0; G (Eg 0. + p=)2-19) ,) where :== 2ij PP. The third relation (3.9) leads to
(E.7)
p( N(g P(a F3), PSP PT).
(E.8)
Then, the posterior mean distribution
((0 = /ʃ a, of Idgd2,
= / ( p(0, g/2, F. )dg क ((2))d2,
= / (/ p(0)g, ()l(l (12)d22
(F.9)
where the last step follows the conditional independence between @ and (F,2²) given (q, Also, it is clear to see that
9., ~ ff p(0) Dpg(g )dg.
(E.
Both p(0gg, (E.7) and P( (F.8) are Gaussian, so the inside integral of (F.9) is also Gaussian, with
/ p(0 g. 1)p(al 22)222 = N (6: (0: Egp..F.)
By the law of total expectation,
22

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Latent Variable Estimation in Bayesian Black-Litterman Models
Bl0lg, DlppglpppFFFF 22 )dq
(By (E.10)
G = 8 + P PT N (a:
P.)PP)))))))))) p!) dq
(By (E.7) and (E.8)
G- (=o 80 + + - F())
(By plo. = Blg F.
Using the law of total variance,
F
Varl0 F.
[ Ω2 F, F,52 + Var [ 02 F, []]
(By the total variance)
= E[ 2] + Var [ 22]
(By conditional independence)
/ Varl0lg. 22 p(g Ω dg + Var E q.
(By (E.7))
E Bot P 8 99]
G G p Varl PG
(By p .25) and Var[Mg] MVarr M-
s (PD. P
(By (E.8)
This completes the proof.
E.5 Proof of Corollary 3.3.1
Proof of Corollary 3.3.1. From Lemma D.2, we have
0 FF ~ N EW)
From (3.11), we have
E ~ IW(V v')
We obtain the joint distribution by definition of the Normal-Inversee -iihsaa distribution:
Wishart
p(0, ((, p(0|E' F, ,( )x(")) = N (n',E) E) (W("' v/)
= 1.
(E.11)
Then the posterior mean distribution becomes
p(0 = /
p(0, E'FE,DF)
(I V v/ D')
(By (E.11))
=t (0:41 //'
where the last step follows Lemma D.3. This completes the proof.
23

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Latent Variable Estimation in Bayesian Black-Litterman Models
F
Experimental Details
F.1
Sharpe Ratio Maximization
In our experiment (Section 4), we consider a standardized version the mean-variance optimization framework (Defini- tion 2.1), taking Sharpe ratio its maximization objective:
Definition F.1 (Mean- Variance Optimization on Sharpe ratio), Let E Rm be the returns the m assets and be its prediction. The optimization problem, under the constraint of (1) leverage and (2) long only, is:
w E[7
max
S.t.
wy and wy 0.
VWTCooll Cov
F.2
Table Indicators
Indicator
Description
Hyperparameters
ATR
Measures market volatility based on price range.
Window length (default 14)
ADX
Measures the strength of a trend.
Window length (default 14)
EMA
Weighted moving average prioritizing re- cent prices.
Window length (default 14)
MACD
Difference between short- and long-term
Fast EMA window length (12).
EMAs, indicates momentum shifts.
Slow EMA window length (26).
Signal window length (9)
SMA
Average of prices over specified window length, indicating short-term trends.
Window length (default 20)
RSI
Momentum oscillator identifying over- bought/oversold conditions.
Window length (default 14)
BB (Upper & Lower)
Measures price volatility, expanding during high volatility and contracting during low.
Window length (default 20), Stan- dard deviation multiplier (default 2)
OBV (normalized)
Volume indicator combined with price, nor- malized to range [0, 1].
None (computed from price and vol- ume)
Table 3: Common Indicators Used
24

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Latent Variable Estimation in Bayesian Black-Litterman Models
E33
Tables of Datasets
Table 4: S&P Sector ETF Components
Table DJIA Components
ETF Ticker
Start Date
End Date
Stock Ticker
Start Date
End Date
XLB
2004-04-13 13
2024-02-22
AA
1994-01-08
2013-09-22
XLE
2004-04-13
2024-02-22
AIG
2004-04-08
2008-09-21
XLF
2004-04-13 13
2024-02-22
AAPL
2015-03-19
2024-02-22
XLI
2004-04-13
2024-02-22
AMGN
2020-08-31
2024-02-22
XLK
2004-04-13
2024-02-22
AXP
1994-01-05
2024-02-22
XLP
2004-04-13
2024-02-22
BA
1994-01-05
2024-02-22
XLU
2004-04-13
2024-02-22
BAC
2008-02-19
2013-09-22
XLV
2004-04-13
2024-02-22
C
1999-11-01
2009-06-07
XLY
2004-04-13
2024-02-22
CAT
1994-01-05
2024-02-22
XLRE
2015-10-08
2024-02-22
CSCO
2009-06-08
2024-02-22
XLC
2018-06-19 19
2024-02-22
CVX
2008-02-19
2024-02-22
DD
1994-01-0S
2019-04-01
DIS
1994-01-05
2024-02-22
FL
1994-01-05
1997-03-17
GE
1994-01-05
2018-06-25
GS
2013-09-23
2024-02-22
GT
1994-01-05
1999-11-01
HD
1999-11-01
2024-02-22
HON
1994-01-05
2024-02-22
HPQ
1997-03-17
2013-09-22
IBM
1994-01-05
2024-02-22
INTC
1999-11-01
2024-02-22
IP
1994-01-05
2004-04-08
JNJ
1997-03-17
2024-02-22
JPM
1994-01-05
2024-02-22
KO
1994-01-05
2024-02-22
MCD
1994-01-05
2024-02-22
MMM
1994-01-05
2024-02-22
MO
1994-01-05
2008-02-18
MRK
1994-01-05
2024-02-22
MSFT
1999-11-01
2024-02-22
NKE
2013-09-23
2024-02-22
PFE
2004-04-08
2024-02-22
PG
1994-01-0S
2024-02-22
T
1994-01-08
2015-03-18
TRV
1997-03-17
2024-02-22
UNH
2012-09-24
2024-02-22
VZ
2004-04-08
2024-02-22
WBA
2018-06-26
2024-02-22
WMT
1994-01-05
2024-02-22
XOM
1994-01-05
2024-02-22
25

---

Latent Variable Estimation in Bayesian Black-Litterman Models
F,4
Figures of Asset Allocation
Asset Allocation Over Time
1.0
XLY
XLV
XLU
0.8
XLRE
XLP
Portfolio Weight
XLK
XLI
0.6
XLF
XLE
XLC
0.4
XLB
0.2
0.0
2004
2006
2008
2010
2012
2014
2016
2018
2020
2022
2024
Date
Figure 6: Asset Allocation of traditional MV model (rolling window: 100 days) on SPDR Sectors ETFs Dataset.
Asset Allocation Over Time
1.0
XLY
XLV
XLU
0.8
XLRE
XLP
Portfolio Weight
XLK
XLI
0.6
XLF
XLE
XLC
XLB
0.4
0.2
0.0
2004
2006
2008
2010
2012
2014
2016
2018
2020
2022
2024
Date
Figure
Asset Allocation of SLP-BL model (rolling window: 100 days) on SPDR Sectors ETFs Dataset.
26

---

Latent Variable Estimation in Bayesian Black-Litterman Models
F55
Turnover Rate Analysis
Average Turnover Rate (%) on Dow Jones Index Dataset
Average Turnover Rate (%)
on SPDR Sector ETFs Dataset
MV (50d)
65.48
61.75
BL (50d)
34.85
24.30
MV (80d)
53.94
49.38
BL (80d)
25.62
19.56
MV (100d)
47.79
48.73
BL (100d)
23.41
18.63
MV (120d)
44.30
41.72
BL (120d)
20.89
18.09
MV (150d)
39.70
38.84
BL (150d)
19.17
17.00
Table 6: Average Turnover (%) for SLP-BL Markowitz model on the Dow Jones Index SPDR Sector Datasets.
Portfolio Turnover Rate Over Time
100.0%
Tumover Rate
Average Tumover: 0.4779
80.0%
Turnover Rate
60.0%
40.0%
20.0%
0.0%
1996
2000
2004
2016
2020
2024
Date
Figure 8: Turnover rate traditional MV model (rolling window: days) on Dow Jones Index Dataset.
27

---

Latent Variable Estimation in Bayesian Black-Litterman Models
Portfolio Turnover Rate Over Time
Turnover Rate
Average Turnover: 0.2341
80.0%
Turnover Rate
60.0%
40.0%
20.0%
0.0%
1996
2000
2004
2012
2016
2024
Date
Figure 9; Tumooee rate of SLP-BL model (rolling window: 100 days) on Dow Jones Index Dataset.
28


Reviews:
[{"review_id": "P46iZWm0Ap", "reviewer": "Reviewer_B3nT", "summary": "Paper removes the need for heuristic investor views while maintaining a Bayesian framework. \nIt makes the Black-Litterman model more data-driven, robust, and automated.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thanks for the insightful questions and reviews.\nThe latest revision is readily available in the **[anonymous dropbox folder](https://www.dropbox.com/scl/fo/i13bhu138gjk76cf5r44v/ACog6LpDbYdBQbB87jtJA-Q?rlkey=mpv3b4xgbmr1cohaen6roxoyf&st=zndq6hnx&dl=0).**\nAny changes or modifications made from the submitted version are highlighted in blue in this updated draft.\n\n>**Reviewer's Comment:** The backtests (2004-2024) are not enough (for the equities portfolio), and the small asset universe (38 assets) limits generalizability. As noted earlier, turnover, transaction costs, a longer backtest (1990s onward), and larger datasets (Asness et al. 2013, Fama-French 2015) should be included for robustness.\n\nIn response to the suggestion of **backtesting periods**, we have added an extended version of our experiment on the Dow Jones Index to backtest from 1994 to 2024 (previously 2004-2024). As demonstrated in Table 2 (https://imgur.com/KcE5QxV), the general results remain consistent, with our model continuing to show (even greater) outperformance compared to the traditional Markowitz model. This outperformance is likely attributed to the more stable portfolio weights provided by our Bayesian-based model. We have added an example asset allocation visualization (Figure 7, Appendix G.5, https://imgur.com/CCXyE7T) to demonstrate this.\n\nPer the request for **turnover analysis**, we have added a new Appendix G.6 to compare the turnover rates between our SLP-BL model and the benchmark Markowitz model. The analysis includes Table 6 (https://imgur.com/Azl39Y0) presenting the average turnover rates, as well as visualizations of example turnover rates for both models (Figure 8 and 9, https://imgur.com/nqZ3rkk). Specifically, the average turnover rate for all the SLP-BL models is 24.79 on the DJI dataset and 19.51 on the sectors dataset, while for all the Markowitz models, the rates are 50.24 and 48.08, respectively.\n\nWe would like to note that our experiments aim to prove the concept—specifically, to prove that this machine learning model could be readily implemented and useful (compared to the benchmarks): \n - In the theoretical sections, we provide closed-form solutions for every model in our work. In experiments, we use Thm. 3.2 for the SLP-BL model, showing it is readily implemented.\n - We take the asset sets, including the 41 DJI stocks and the 11 sectors, to represent the samples of at least large-cap equities. These sets are practically used as they provide some benefits compared to a larger set of stocks: ease of trading, smaller impact of slippage (due to higher liquidity), fewer trades per period (thus lower transaction costs), lower managerial costs, and lower computational costs. \n - The monthly rebalance setting is also designed to enhance these benefits (particularly reduce the impact of transaction costs) while maintaining a similar performance compared to weekly or daily rebalance. \n\nThus, in response to the reviewer’s suggestion regarding **asset universes** and **transaction costs**, we choose a smaller — but representative — set of assets for their practical usefulness, including the advantage of lower transaction costs. Our proof-of-concept experiment design also aims to minimize the effect of transaction costs, without explicitly considering them.\n\nLastly, we need to clarify that the backtest period should be carefully chosen to avoid unfair comparison. Specifically, although the DJIA covers a long history, some stock data might be inaccessible. For example, companies like American Can Company, Navistar International Corporation, and USX Corporation were part of the DJIA before May 1991 but have since been delisted or replaced, making their data difficult to retrieve. Omitting these stocks (as many portfolio studies do) could introduce selection bias because delisted stocks, in the aftermath, often underperform. This is particularly concerning when backtesting over earlier years, as more data is unavailable. Notably, we observed better performance in the benchmark equal-weighted portfolio (EQW) but worse performance in other benchmarks - we suspect that the EQW may be benefiting from this selection bias. In terms of fair comparison, we believe our experimental results (20 yrs for sector ETFs; 30 yrs on the Dow Jones Index) are now sufficient for demonstrating the outperformance of our models.", "claims_and_evidence": "Claims are well supported.", "methods_and_evaluation_criteria": "The methods and evaluation criteria are reasonable, but the testing is limited to small asset universes (11-38 assets). To assess robustness, the model should be evaluated on larger datasets like those in Asness et al. (2013) and Fama-French (2015). Additionally, turnover analysis and net-of-transaction-cost performance are missing, which are crucial for real-world applicability. \n\nExpanding the analysis to start at least in the 1990s and cover a much larger universe of assets would help assess the model’s robustness across more market regimes and economic conditions.\n\n- Asness, C. S., Moskowitz, T. J., and Pedersen, L. H. (2013). Value and Momentum Everywhere. The Journal of Finance, 68(3):929–985​\n- Fama, E. F. and French, K. R. (2015). A Five-Factor Asset Pricing Model. Journal of Financial Economics, 116(1):1–22​", "theoretical_claims": "The proofs (Theorems 3.1, 3.2, 3.3) appear correct, following standard Bayesian inference.", "experimental_designs_or_analyses": "The backtests (2004-2024) are not enough (for the equities portfolio), and the small asset universe (38 assets) limits generalizability. As noted earlier, turnover, transaction costs, a longer backtest (1990s onward), and larger datasets (Asness et al. 2013, Fama-French 2015) should be included for robustness.", "relation_to_broader_scientific_literature": "The paper extends Bayesian Black-Litterman (Kolm & Ritter) by making views data-driven and explains its contributions well.", "essential_references_not_discussed": "All the essential referenes are well discussed.", "other_strengths_and_weaknesses": "No", "other_comments_or_suggestions": "No", "questions_for_authors": "No", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "A1ponCnJK0", "reviewer": "Reviewer_f7Sa", "summary": "The paper presents a new formulation for the well-known Black-Litterman model, introducing a Bayesian reinterpretation of the model for portfolio optimization, eliminating the need for subjective investor views and their associated uncertainties. The authors analyse the problem from a theoretical perspective and numerically validate their findings.\n\n--- \n\nPost-rebuttal: The authors provided adequate responses to my concerns. I will maintain my (positive) score.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thanks for the reviews.\nThe latest revision is readily available in the **[anonymous dropbox folder](https://www.dropbox.com/scl/fo/i13bhu138gjk76cf5r44v/ACog6LpDbYdBQbB87jtJA-Q?rlkey=mpv3b4xgbmr1cohaen6roxoyf&st=zndq6hnx&dl=0).**\nAny changes made from the submitted version are highlighted in blue in this updated draft.\n\n>**Reviewer's Comment:** My only comment pertains to the highly specific target and scope of the work, which may make it less suitable for the general audience of ICML.\n\nFrom a practical perspective, our work aims to take different types of data under different scenarios — some data is given, while others are not — to make predictions and decisions. Thus, it addresses machine learning problems using machine learning methods, and should be considered relevant to ICML community.\nSpecifically, in our work,\n1. **Different types of data** include: raw data, feature data extracted from raw data, feature data involving additional information, and heuristic expert knowledge (views). One of the major focuses in our model design is to capture the effects of these data through Bayesian networks with latent variables, a common methodology in machine learning research [1][2][3]. These data are clearly and consistently described in the introduction (page 1), Sec. 3.2, 3.3, and 3.4 (page 3-5).\n2. **Different scenarios** refer to Sec. 3.3, where investor views are observed, and Sec. 3.4, where no subjective views are given. In the scenario of Sec. 3.4, our model is showcased by two configurations for handling different types of data. These scenarios are clearly and consistently described in the paragraph before Sec. 3.1 (page 3) and the paragraph after remark. 3.3 (page 5).\n3. **Predictions** refer to the posterior probability distribution of unobserved asset returns given data estimated by each designed model. The prediction problems are described in problems 1, 2, 3 (page 2, 4, 5). The prediction models are Def 2.2, 3.3, 3.4, and 3.6, and they make predictions in Lemma 2.1, Corollary 3.1.1, 3.2.1, and 3.3.1, correspondingly.\n4. **Decisions** refer to the solution of the portfolio optimization problem (Def. 2.1, page 2) based on the estimations of each designed model, including Lemma 2.1, Thm 3.1, 3.2, and 3.3. The portfolio optimization problem is also a popular topic in machine learning research [4][5][6][7][8].\nIn this perspective, we deem our work to fall under the general scope of machine learning and satisfy the topics of interest (https://icml.cc/Conferences/2025/CallForPapers) of ICML.\n\n[1]. Anandkumar et al. (2013). Learning linear Bayesian networks with latent variables. ICML.\n\n[2]. Xie et al. (2016). Diversity-promoting Bayesian learning of latent variable models. ICML.\n\n[3]. Lorch et al. (2021). DiBS: Differentiable Bayesian structure learning. NeurIPS.\n\n[4]. Agarwal et al. (2006). Portfolio management via the Newton method. ICML.\n\n[5]. Qiu et al. (2015). Robust portfolio optimization. NeurIPS.\n\n[6]. Ito et al. (2018). Online portfolio selection with cardinality constraints: Regret bounds. NeurIPS.\n\n[7]. Tsai et al. (2023). Data-dependent bounds for online portfolio selection. NeurIPS.\n\n[8]. Lin et al. (2024). Globally optimal m-sparse Sharpe ratio portfolios. NeurIPS.\n\n>**Reviewer's Comment:** due to space constraints, the authors often omitted discussions of their results and choices.\n\nThe general conclusion of this work is stated in Appendix A:\n\n*“We propose ...... real-world datasets (Section 4).”*\n\nWe also have more detailed discussions for the results of each problem in Sec. 3:\n - For Problem 2 (Sec 3.3), the discussion includes Remark 3.2 (classical Black-Litterman is a special case), Remark C.1/D.1 (the prediction has a ground truth limit), and a summary at the end of Sec. 3.3 (page 5).\n - For Problem 3 (Sec 3.4), the discussion ofthe  SLP-BL model includes Remark 3.4 (equivalency to classical Black-Litterman) and a summary at the end of \\textbf{Configuration 1} (page 6).\n - For Problem 3 (Sec 3.4), the discussion of the FIV-BL model includes Remark C.2/D.2 (equivalency to SLP-BL model) after Thm. 3.3 and a summary at the end of \\textbf{Configuration 2} (page 8).\n - Due to the complex nature of configuration 2, we also discuss the further assumptions (conjugate prior) after Remark C.2 (page 7).\n\nTo clarify the choice between configurations discussed in Section 3.4, we have added a guiding statement for practitioners at the end of page 5.\n\n*“This implies that, in practice, if an investor takes generic features of assets (e.g. indicators derived from the time series of each asset, as shown in our experiment), configuration 1 should be used. If an investor takes features not specific to individual assets (e.g. interest rates), configuration 2 should be used. The two configurations are not contradicting, so one can take both types of features and incorporate them correspondingly.”*", "claims_and_evidence": "All the claims are supported by evidence.", "methods_and_evaluation_criteria": "Yes, the evaluation criteria is coherent with the scope of the proposed model.", "theoretical_claims": "I didn't carefully check the proofs.", "experimental_designs_or_analyses": "The experimental validation is limited but coherent with the scope of the work.", "relation_to_broader_scientific_literature": "This work is relevant to the specific literature about this portfolio optimization model.", "essential_references_not_discussed": "To the best of my knowledge, relevant literature is presented.", "other_strengths_and_weaknesses": "I think this work is very well-presented, even if, due to space constraints, the authors often omitted discussions of their results and choices.", "other_comments_or_suggestions": "My only comment pertains to the highly specific target and scope of the work, which may make it less suitable for the general audience of ICML.", "questions_for_authors": "No questions.", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "QIwfjQ3xOu", "reviewer": "Reviewer_fEJh", "summary": "The paper extends the classical Black-Litterman model by incorporating asset features. In the traditional model, investor views and their associated uncertainty are assumed to be given. The author proposes leveraging asset features to estimate both investor views and their uncertainty. Two models are introduced: the first assumes that asset features influence both investor views and uncertainty through a shared hyperparameter, while the second allows asset features to directly influence investor views. Numerical experiments demonstrate that the proposed approach outperforms the Markowitz and market-index baselines.", "questions": "", "limitations": "", "rating": 2, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thanks for the reviews.\nThe latest revision is readily available in the **[anonymous dropbox folder](https://www.dropbox.com/scl/fo/i13bhu138gjk76cf5r44v/ACog6LpDbYdBQbB87jtJA-Q?rlkey=mpv3b4xgbmr1cohaen6roxoyf&st=zndq6hnx&dl=0).**\nAny changes made from the submitted version are highlighted in blue in this updated draft.\n\n## Features Used in Experiments.\n>**Reviewer's Comment:** What asset features are used in the experiment section?\n\nThe asset features we use in our experiments are listed in Appendix G.3. They include common indicators (e.g., EMA, MACD, RSI, ...) in financial analysis. The idea behind selecting these features is to keep things generic, following configuration 1 (as explained in Remark 3.3). We state the description of our features usage in **Backtest Task. (page 8)** with:  \n*“In the model, the prior is set as traditional Markowitz model and the features are selected based on nine generic indicators (Table 5) derived from asset-specific data.”*\n\n>**Reviewer's Comment:** Since the proposed method requires additional input (asset features) compared to the baseline methods, one could easily construct a portfolio that outperforms the baselines by simply assigning greater weight to the tech sector.\n\nWe would like to clarify that our method does not necessarily require “additional input\":\n - Similar to the features used in our experiment, they can be extracted from existing raw data (e.g., price and volume). In this case, the feature selection does not favor any particular sector and is broadly applicable to various assets. \n - That said, in some cases, if an investor has hindsight regarding a range of assets (rather than individual ones), they could include non-asset-specific features (e.g., the QQQ index representing the tech sector) as part of the model. The incorporation of these features should follow configuration 2 (also as explained in Remark 3.3), alongside the generic asset-specific features.\nIn our experiment with the sectors dataset, our model does not assign greater weight to the tech sector \"XLK\". The weight of \"XLK\" is, in fact, lower than that of \"XLP\" or “XLF” for most of the time. We have added an example asset allocation visualization (Figure 7, Appendix G.5, https://imgur.com/CCXyE7T) to demonstrate this.\n\n## Choice Between Configurations\n>**Reviewer's Comment:** Can the authors provide a discussion on how to choose between the two proposed configurations?\n\nYes, we have added a discussion explicitly on the choice of configurations. Below is the details:\n - In the original paper version, we have discussed the distinctive characteristics of the two configurations in the paragraph after Remark 3.3 (page 5):  \n> *“We showcase the feature-integrated Black-Litterman network as two configurations: one incorporating Effect 1 and another incorporating Effect 2. Intuitively, the first one better captures generic features while the second one more effectively handles the non-asset-related features.”*\n\n - To make this concept more explicit to practice, we have added a statement after the above paragraph:  \n> *“This implies that, in practice, if an investor takes generic features of assets (e.g. indicators derived from the time series of each asset, as shown in our experiment), configuration 1 should be used. If an investor takes features not specific to individual assets (e.g. interest rates), configuration 2 should be used. The two configurations are not contradicting, so one can take both types of features and incorporate them correspondingly.”*\n\n>**Reviewer's Comment:** in practice, practitioners primarily need a formula to compute portfolio weights, such a discussion would be valuable. \n\nSimilar to the usage of the Black-Litterman (BL) Formula (Theorem 2.1), the practitioner formula to compute portfolio weights is provided by Theorem 3.2 and Theorem 3.3 corresponding to each configuration.\n\n>**Reviewer's Comment:** Similarly, what kinds of asset features are most suitable for the proposed method?\n\n\nRegarding what asset features are suitable, we do not limit the choice of data in this paper and only distinguish them for the purpose of choosing which configuration to use. For example, in our experiment, we randomly take 9 indicators of each asset price as generic features (see Appendix G.3 on page 25) to demonstrate the practice of configuration 1. In a features-driven model like ours, feature engineering literature [1][2] suggests taking diversified (uncorrelated) \tdata to reduce multicollinearity so that it can provide robust mean/variance estimation and portfolio optimization decisions. However, detailed methods for feature selection problems in our model are not the focus of our work, thus we retain them for future works.\n\n[1]. Gujarati, Damodar, and Dawn Porter. \"Multicollinearity: What happens if the regressors are correlated.\" Basic econometrics 363 (2003).  \n[2]. Alin, Aylin. \"Multicollinearity.\" Wiley interdisciplinary reviews: computational statistics 2, no. 3 (2010): 370-374.", "claims_and_evidence": "The paper is well written and use abundant figures to demonstrate the proposed graphical models.", "methods_and_evaluation_criteria": "Yes. The metrics used in the paper for porfolio selection are standard.", "theoretical_claims": "I checked some of the proofs and did not find obvious errors.", "experimental_designs_or_analyses": "- **Features Used in Experiments:** What asset features are used in the experiment section? Since the proposed method requires additional input (asset features) compared to the baseline methods, one could easily construct a portfolio that outperforms the baselines by simply assigning greater weight to the tech sector.  \n\n- **Choice Between Configurations:** Can the authors provide a discussion on how to choose between the two proposed configurations? Since, in practice, practitioners primarily need a formula to compute portfolio weights, such a discussion would be valuable. Similarly, what kinds of asset features are most suitable for the proposed method?", "relation_to_broader_scientific_literature": "The paper extends Black-Litterman model and is related to that literature.", "essential_references_not_discussed": "The paper cites sufficient papers in the literature.", "other_strengths_and_weaknesses": "Strengths\n\n The paper proposes a Bayesian framework for incorporating asset features, and the resulting formulas have closed-form expressions. This is advantageous and practical, as it allows for easy computation.", "other_comments_or_suggestions": "See Experimental Designs Or Analyses.", "questions_for_authors": "See Experimental Designs Or Analyses.", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "N1Pb76iZxn", "reviewer": "Reviewer_KJCs", "summary": "The paper proposes Bayesian models for Portfolio management with good theoretical results and empirical validation. From what I can understand the contribution of the work is computational efficiency of portfolio management. Could the code be provided to do this?\n\nGiven no code, I see no academic value for this work.\n\nThe empirical validation is far too small to allow for any reasonable examination of the academic value of this work.\n\n## update after rebuttal\nI have raised my score due to code release.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": ">**Reviewer's Comment:** Could the code be provided to do this? Given no code, I see no academic value for this work.\n\nWe provide code for reproducibility and the latest paper revision in this **[anonymous dropbox folder](https://www.dropbox.com/scl/fo/i13bhu138gjk76cf5r44v/ACog6LpDbYdBQbB87jtJA-Q?rlkey=mpv3b4xgbmr1cohaen6roxoyf&st=zndq6hnx&dl=0).**\n\n**On Code Availability.** We originally withheld the code due to industrial IP auditing procedures. However, per the reviewer's request, we have now completed the necessary internal reviews and are able to share an anonymized version of the code for reproducibility purposes.\n\n>**Reviewer's Comment:** The empirical validation is far too small to allow for any reasonable examination of the academic value of this work.\n\nWe have extended the backtest period of the experiment on the DJI dataset from 20 years to 30 years, as shown by Table 2, Section 4 and Figure 5, Appendix G.4. We have also added a turnover rate analysis (Appendix G.6) to further demonstrate our model outperformance.\n\nWe note that our experiments aim to prove the concept—specifically, to prove that this machine learning model could be readily implemented and useful (compared to the benchmarks). The two datasets we choose are practically useful as they provide some benefits compared to a larger set of stocks: ease of trading, smaller impact of slippage (due to higher liquidity), fewer trades per period (thus lower transaction costs), lower managerial costs, and lower computational costs.\n\nWe need to clarify that the backtest period should be carefully chosen to avoid unfair comparison. Specifically, although the DJIA covers a long history, some stock data might be inaccessible. Omitting them (as many portfolio studies do) could introduce selection bias because delisted stocks often underperform. Thus, we believe our experimental results now (20 yrs for sector ETFs; 30 yrs on the Dow Jones Index) are sufficient for showing the outperformance of our models.\n\n>**Reviewer's Comment:** There is no justification given as to why Bayesian comes up in portfolio management.\n\nAs we mention in \\textbf{Why Bayesian? } in the related works (Appendix C), Bayesian framework has long been advocated and used in portfolio management. We state:\n\n*“To address the parameter estimation risk in traditional portfolio optimization shown by (Markowitz, 1952; Kalymon, 1971), Barry (1974); Klein & Bawa (1976); Brown (1976) advocate Bayesian framework upon prior information in portfolio optimization.”*\n\nMoreover, it is proved that the Bayesian framework has several benefits, as we state:\n\n*“Foundational works by (Jorion, 1986) and (Black & Litterman, 1992) demonstrate how Bayesian shrinkage improves covariance estimation, reducing overfitting and highly sensitive weight in Markowitz-style allocations (Meucci, 2005; DeMiguel et al., 2009).”*\n\nWe hope this background information clarifies the use of Bayesian.\n\n>**Reviewer's Comment:** Please answer how this is relevant to the ICML community.\n\nFrom an essential and practical perspective, our work aims to take different types of data under different scenarios — some data is given, while others are not — to make predictions and decisions. Thus, it addresses machine learning problems using machine learning methods, and should be considered relevant to general ICML community.\nSpecifically,\n1. **Different types of data** include: raw data, feature data extracted from raw data, feature data involving additional information, and heuristic expert knowledge (views). One of the major focuses in our model design is to capture the effects of these data through Bayesian networks with latent variables, a common methodology in machine learning research. These data are described in the introduction (page 1), Sec. 3.2, 3.3, and 3.4 (page 3-5).\n2. **Different scenarios** refer to Sec. 3.3, where investor views are observed, and Sec. 3.4, where no subjective views are given. In the scenario of Sec. 3.4, our model is showcased by two configurations for handling different types of data. These scenarios are described in the paragraph before Sec. 3.1 (page 3) and the paragraph after remark. 3.3 (page 5).\n3. **Predictions** refer to the posterior probability distribution of unobserved asset returns given data estimated by each designed model. The prediction problems are described in problems 1, 2, 3 (page 2, 4, 5). The prediction models are Def 2.2, 3.3, 3.4, and 3.6, and they make predictions in Lemma 2.1, Corollary 3.1.1, 3.2.1, and 3.3.1, correspondingly.\n4. **Decisions** refer to the solution of the portfolio optimization problem (Def. 2.1, page 2) based on the estimations of each designed model, including Lemma 2.1, Thm 3.1, 3.2, and 3.3. The portfolio optimization problem is also a popular topic in machine learning research.\nIn this perspective, we deem our work to fall under the scope of machine learning and satisfy the topics of interest (https://icml.cc/Conferences/2025/CallForPapers) of ICML.", "claims_and_evidence": "The claims and evidence are sufficient.", "methods_and_evaluation_criteria": "The methods and evaluation are sufficient.", "theoretical_claims": "There is no justification given as to why Bayesian comes up in portfolio management.", "experimental_designs_or_analyses": "The experimental design seems sufficient for the paper.", "relation_to_broader_scientific_literature": "N/A", "essential_references_not_discussed": "N/A", "other_strengths_and_weaknesses": "N/A", "other_comments_or_suggestions": "I'm not sure how or why this work is relevant or useful to the ICML community or the machine learning community.", "questions_for_authors": "Please answer how this is relevant to the ICML community.", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Meta-review: The authors propose what they describe as a Bayesian reformulation of the Black-Litterman model for portfolio optimization. 

The reviewers acknowledged the relevance of the setting but initially raised several concerns regarding limited theoretical novelty, depth of the empirical validation, and inconsistencies between the paper's title and its actual scope. After the discussion, most of the concerns were addressed to a sufficient degree to move the needle to a positive score for all but one reviewer. Although this is a borderline submission, I believe it passes the threshold of acceptance, provided that the authors commit to incorporating all the points that were raised in the reviewing process in their revised version.
Decision: Accept (poster)