This paper incorporates Bayesian estimation and optimization into a portfolio selection framework, particularly for high-dimensional portfolios in which the number of assets is larger than the number of observations. We leverage a constrained \(\ell _1\) minimization approach, called the linear programming optimal (LPO) portfolio, to directly estimate effective parameters appearing in the optimal portfolio. We propose two refinements for the LPO strategy. First, we explore improved Bayesian estimates, instead of sample estimates, of the covariance matrix of asset returns. Second, we introduce Bayesian optimization (BO) to replace traditional grid-search cross-validation (CV) in tuning hyperparameters of the LPO strategy. We further propose modifications in the BO algorithm by (1) taking into account the time-dependent nature of financial problems and (2) extending the commonly used expected improvement acquisition function to include a tunable trade-off with the improvement’s variance. Allowing a general case of noisy observations, we theoretically derive the sublinear convergence rate of BO under the newly proposed EIVar and thus our algorithm has no regret. Our empirical studies confirm that the adjusted BO results in portfolios with higher out-of-sample Sharpe ratio, certainty equivalent, and lower turnover compared to those tuned with CV. This superior performance is achieved with a significant reduction in time elapsed, thus also addressing time-consuming issues of CV. Furthermore, LPO with Bayesian estimates outperforms the original proposal of LPO, as well as the benchmark equally weighted and plugin strategies.


  1. sequential portfolio selection
  2. Bayesian estimation
  3. Bayesian optimization
  4. high dimensionality
  5. sequential regularization
  6. sequential hyperparameter tuning

MSC codes

  1. 91G10
  2. 91G70
  3. 93E20

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Information & Authors


Published In

cover image SIAM Journal on Financial Mathematics
SIAM Journal on Financial Mathematics
Pages: 127 - 157
ISSN (online): 1945-497X


Submitted: 15 June 2021
Accepted: 19 October 2022
Published online: 27 January 2023


  1. sequential portfolio selection
  2. Bayesian estimation
  3. Bayesian optimization
  4. high dimensionality
  5. sequential regularization
  6. sequential hyperparameter tuning

MSC codes

  1. 91G10
  2. 91G70
  3. 93E20



Godeliva Petrina Marisu
School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore.
Corresponding author. School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore.

Funding Information

Funding: This work was funded by the Ministry of Education, Singapore (MOE2017-T2-1-044).

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