{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T03:56:01Z","timestamp":1760241361645,"version":"build-2065373602"},"reference-count":40,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2018,1,10]],"date-time":"2018-01-10T00:00:00Z","timestamp":1515542400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>There are several electricity generation technologies based on different sources such as wind, biomass, gas, coal, and so on. The consideration of the uncertainties associated with the future costs of such technologies is crucial for planning purposes. In the literature, the allocation of resources in the available technologies has been solved as a mean-variance optimization problem assuming knowledge of the expected values and the covariance matrix of the costs. However, in practice, they are not exactly known parameters. Consequently, the obtained optimal allocations from the mean-variance optimization are not robust to possible estimation errors of such parameters. Additionally, it is usual to have electricity generation technology specialists participating in the planning processes and, obviously, the consideration of useful prior information based on their previous experience is of utmost importance. The Bayesian models consider not only the uncertainty in the parameters, but also the prior information from the specialists. In this paper, we introduce the classical-equivalent Bayesian mean-variance optimization to solve the electricity generation planning problem using both improper and proper prior distributions for the parameters. In order to illustrate our approach, we present an application comparing the classical-equivalent Bayesian with the naive mean-variance optimal portfolios.<\/jats:p>","DOI":"10.3390\/e20010042","type":"journal-article","created":{"date-parts":[[2018,1,10]],"date-time":"2018-01-10T12:41:10Z","timestamp":1515588070000},"page":"42","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["Classical-Equivalent Bayesian Portfolio Optimization for Electricity Generation Planning"],"prefix":"10.3390","volume":"20","author":[{"given":"Hellinton","family":"Takada","sequence":"first","affiliation":[{"name":"Quantitative Research, Ita\u00fa Asset Management, S\u00e3o Paulo 04538-132, Brazil"},{"name":"Institute of Mathematics and Statistics, University of S\u00e3o Paulo, S\u00e3o Paulo 05508-090, Brazil"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2720-3871","authenticated-orcid":false,"given":"Julio","family":"Stern","sequence":"additional","affiliation":[{"name":"Institute of Mathematics and Statistics, University of S\u00e3o Paulo, S\u00e3o Paulo 05508-090, Brazil"}]},{"given":"Oswaldo","family":"Costa","sequence":"additional","affiliation":[{"name":"Polytechnic School, University of S\u00e3o Paulo, S\u00e3o Paulo 05508-010, Brazil"}]},{"given":"Celma","family":"Ribeiro","sequence":"additional","affiliation":[{"name":"Polytechnic School, University of S\u00e3o Paulo, S\u00e3o Paulo 05508-010, Brazil"}]}],"member":"1968","published-online":{"date-parts":[[2018,1,10]]},"reference":[{"key":"ref_1","unstructured":"International Energy Agency (2016). World Energy Outlook 2016, International Energy Agency."},{"key":"ref_2","unstructured":"United States Energy Information Administration (2016). Annual Energy Outlook."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"693","DOI":"10.1007\/s11027-006-4754-4","article-title":"Portfolio-based electricity generation planning: Policy implications for renewable and energy security","volume":"11","author":"Awerbuch","year":"2006","journal-title":"Mitig. Adapt. Strateg. Glob. Chang."},{"key":"ref_4","unstructured":"Awerbuch, S., and Berger, M. (2003). Applying Portfolio Theory to EU Electricity Planning and Policy-Making, International Energy Agency\/Emerging Energy Technologies."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"246","DOI":"10.1016\/j.iref.2015.02.013","article-title":"Mean-variance portfolio methods for energy policy risk management","volume":"40","author":"Marrero","year":"2015","journal-title":"Int. Rev. Econ. Financ."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"2808","DOI":"10.1016\/j.rser.2010.06.007","article-title":"Electricity generation cost in isolated system: The complementarities of natural gas and renewables in the canary islands","volume":"14","author":"Marrero","year":"2010","journal-title":"Renew. Sustain. Energy Rev."},{"key":"ref_7","unstructured":"Markowitz, H. (1959). Portfolio Selection: Efficient Diversification of Investments, John Wiley & Sons, Inc."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"7","DOI":"10.3905\/jfi.1991.408013","article-title":"Asset allocation: Combining investor views with market equilibrium","volume":"1","author":"Black","year":"1991","journal-title":"J. Fixed Income"},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Fabozzi, F.J., Kolm, P.N., Pachamanova, D.A., and Focardi, S.M. (2007). Robust Portfolio Optimization and Management, John Wiley & Sons, Inc.","DOI":"10.3905\/jpm.2007.684751"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"103","DOI":"10.1007\/s10957-013-0329-1","article-title":"Recent developments in robust portfolios with a worst-case approach","volume":"161","author":"Kim","year":"2014","journal-title":"J. Optim. Theory Appl."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"464","DOI":"10.1137\/080734510","article-title":"Theory and applications of robust optimization","volume":"53","author":"Bertsimas","year":"2011","journal-title":"SIAM Rev."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"32","DOI":"10.1016\/j.rser.2016.04.030","article-title":"Bayesian networks in renewable energy systems: A bibliographical survey","volume":"62","author":"Borunda","year":"2016","journal-title":"Renew. Sustain. Energy Rev."},{"key":"ref_13","unstructured":"Charniak, E., and McDermott, D. (1989). Introduction to Artificial Intelligence, Addison-Wesley."},{"key":"ref_14","first-page":"50","article-title":"Bayesian networks without tears","volume":"4","author":"Charniak","year":"1991","journal-title":"Artif. Intell. Mag."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"250","DOI":"10.1162\/POSC_a_00098","article-title":"The vernacular architecture of household energy models","volume":"21","author":"Shipworth","year":"2013","journal-title":"Perspect. Sci."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"47","DOI":"10.1016\/j.segan.2016.05.003","article-title":"Bayesian framework for power network planning under uncertainty","volume":"7","author":"Lawson","year":"2016","journal-title":"Sustain. Energy Grids Netw."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"1290","DOI":"10.1016\/j.ress.2005.11.025","article-title":"Bayesian analysis of computer code outputs: A tutorial","volume":"91","year":"2006","journal-title":"Reliab. Eng. Syst. Saf."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"425","DOI":"10.1111\/1467-9868.00294","article-title":"Bayesian calibration of computer models","volume":"63","author":"Kennedy","year":"2001","journal-title":"J. R. Stat. Soc. Ser. B"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"158","DOI":"10.1016\/j.eneco.2017.03.021","article-title":"Robust portfolio optimization for electricity planning: An application based on the Brazilian electricity mix","volume":"64","author":"Costa","year":"2017","journal-title":"Energy Econ."},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Meucci, A. (2005). Risk and Asset Allocation, Springer-Verlag.","DOI":"10.1007\/978-3-540-27904-4"},{"key":"ref_21","unstructured":"Rachev, S.T., Hsu, J.S.J., Bagasheva, B.S., and Fabozzi, F.J. (2008). Bayesian Methods in Finance, John Wiley & Sons, Inc."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"515","DOI":"10.1111\/j.1540-6261.1974.tb03064.x","article-title":"Portfolio analysis under uncertain means, variances, and covariances","volume":"29","author":"Barry","year":"1974","journal-title":"J. Financ."},{"key":"ref_23","unstructured":"Brown, S. (1976). Optimal Portfolio Choice under Uncertainty: A Bayesian Approach. [Ph.D. Thesis, University of Chicago]."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"215","DOI":"10.1016\/0304-405X(76)90004-0","article-title":"The effect of estimation risk on optimal portfolio choice","volume":"3","author":"Klein","year":"1976","journal-title":"J. Financ. Econ."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"293","DOI":"10.2307\/2331043","article-title":"An empirical Bayes approach to efficient portfolio selection","volume":"21","author":"Frost","year":"1986","journal-title":"J. Financ. Quant. Anal."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"338","DOI":"10.1080\/07350015.2000.10524875","article-title":"Bayesian dynamic factor models and portfolio allocation","volume":"18","author":"Aguilar","year":"2000","journal-title":"J. Bus. Econ. Stat."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"25","DOI":"10.1146\/annurev-financial-120209-133947","article-title":"Bayesian portfolio analysis","volume":"2","author":"Avramov","year":"2010","journal-title":"Annu. Rev. Financ. Econ."},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"Sekerke, M. (2015). Bayesian Risk Management: A Guide to Model Risk and Sequential Learning in Financial Markets, John Wiley & Sons, Inc.","DOI":"10.1002\/9781118864784"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"292","DOI":"10.1016\/j.ejor.2016.05.044","article-title":"Bayesian estimation of the global minimum variance portfolio","volume":"256","author":"Bodnar","year":"2017","journal-title":"Eur. J. Oper. Res."},{"key":"ref_30","unstructured":"Bawa, V.S., Brown, S.J., and Klein, R.W. (1979). Estimation Risk and Optimal Portfolio Choice, North-Holland Publishing Company."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"285","DOI":"10.1016\/S0304-405X(97)00020-2","article-title":"Analyzing investments whose histories differ in length","volume":"45","author":"Stambaugh","year":"1997","journal-title":"J. Financ. Econ."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"179","DOI":"10.1111\/0022-1082.00204","article-title":"Portfolio selection and asset pricing models","volume":"55","year":"2000","journal-title":"J. Financ."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"335","DOI":"10.1016\/S0304-405X(00)00044-1","article-title":"Comparing asset pricing models: An investment perspective","volume":"56","author":"Stambaugh","year":"2000","journal-title":"J. Financ. Econ."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"959","DOI":"10.1017\/S0022109010000335","article-title":"Incorporating economic objectives into bayesian priors: Portfolio choice under parameter uncertainty","volume":"45","author":"Tu","year":"2010","journal-title":"J. Financ. Quant. Anal."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"301","DOI":"10.1016\/j.enpol.2013.07.049","article-title":"Efficient power generating portfolio in Brazil: Conciliating cost, emissions and risk","volume":"62","author":"Losekann","year":"2013","journal-title":"Energy Policy"},{"key":"ref_36","doi-asserted-by":"crossref","unstructured":"Davis, M.H.A., and Vinter, R.B. (1985). Stochastic Modelling and Control, Chapman and Hall.","DOI":"10.1007\/978-94-009-4828-0"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"185","DOI":"10.1016\/0022-0531(83)90129-1","article-title":"A characterization of the distributions that imply mean-variance utility functions","volume":"29","author":"Chamberlain","year":"1983","journal-title":"J. Econ. Theory"},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"453","DOI":"10.1098\/rspa.1946.0056","article-title":"An Invariant Form for the Prior Probability in Estimation Problems","volume":"186","author":"Jeffreys","year":"1946","journal-title":"Proc. R. Soc. Lond. Ser. A"},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"119","DOI":"10.1086\/294846","article-title":"Mutual fund performance","volume":"39","author":"Sharpe","year":"1966","journal-title":"J. Bus."},{"key":"ref_40","doi-asserted-by":"crossref","unstructured":"Gamerman, D., and Lopes, H.F. (2006). Markov Chain Monte Carlo: Stochastic Simulation for Bayesian Inference, Chapman and Hall.","DOI":"10.1201\/9781482296426"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/20\/1\/42\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T14:50:44Z","timestamp":1760194244000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/20\/1\/42"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,1,10]]},"references-count":40,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2018,1]]}},"alternative-id":["e20010042"],"URL":"https:\/\/doi.org\/10.3390\/e20010042","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2018,1,10]]}}}