# Empirical Prior This tutorial shows how to use the [`EmpiricalPrior`](https://skfolio.org/generated/skfolio.prior.EmpiricalPrior.html.md#skfolio.prior.EmpiricalPrior) estimator in the [`MeanRisk`](https://skfolio.org/generated/skfolio.optimization.MeanRisk.html.md#skfolio.optimization.MeanRisk) optimization. A [Prior Estimator](https://skfolio.org/user_guide/prior.html.md#prior) in `skfolio` fits a `ReturnDistribution` containing your pre-optimization inputs ($\mu$, $\Sigma$, returns, sample weight, Cholesky decomposition). The term “prior” is used in a general optimization sense, not confined to Bayesian priors. It denotes any **a priori** assumption or estimation method for the return distribution before optimization, unifying **Frequentist**, **Bayesian** and **Information-theoretic** approaches into a single cohesive framework: 1. Frequentist: : * [`EmpiricalPrior`](https://skfolio.org/generated/skfolio.prior.EmpiricalPrior.html.md#skfolio.prior.EmpiricalPrior) * [`TimeSeriesFactorModel`](https://skfolio.org/generated/skfolio.prior.TimeSeriesFactorModel.html.md#skfolio.prior.TimeSeriesFactorModel) * [`SyntheticData`](https://skfolio.org/generated/skfolio.prior.SyntheticData.html.md#skfolio.prior.SyntheticData) 2. Bayesian: : * [`BlackLitterman`](https://skfolio.org/generated/skfolio.prior.BlackLitterman.html.md#skfolio.prior.BlackLitterman) 3. Information-theoretic: : * [`EntropyPooling`](https://skfolio.org/generated/skfolio.prior.EntropyPooling.html.md#skfolio.prior.EntropyPooling) * [`OpinionPooling`](https://skfolio.org/generated/skfolio.prior.OpinionPooling.html.md#skfolio.prior.OpinionPooling) In skfolio’s API, all such methods share the same interface and adhere to scikit-learn’s estimator API: the `fit` method accepts `X` (the asset returns) and stores the resulting [`ReturnDistribution`](https://skfolio.org/generated/skfolio.prior.ReturnDistribution.html.md#skfolio.prior.ReturnDistribution) in its `return_distribution_` attribute. The [`ReturnDistribution`](https://skfolio.org/generated/skfolio.prior.ReturnDistribution.html.md#skfolio.prior.ReturnDistribution) is a dataclass containing: > * `mu`: Estimated expected returns of shape (n_assets,) > * `covariance`: Estimated covariance matrix of shape (n_assets, n_assets) > * `returns`: (Estimated) asset returns of shape (n_observations, n_assets) > * `sample_weight` : Sample weight for each observation of shape (n_observations,) (optional) > * `cholesky` : Lower-triangular Cholesky factor of the covariance (optional) The `EmpiricalPrior` estimator estimates the `ReturnDistribution` by fitting its `mu_estimator` and `covariance_estimator` independently. In this tutorial we will build a Maximum Sharpe Ratio portfolio using the `EmpiricalPrior` estimator with James-Stein shrinkage for the estimation of expected returns and Denoising for the estimation of the covariance matrix. ## Data We load the S&P 500 [dataset](https://skfolio.org/user_guide/datasets.html.md#datasets) composed of the daily prices of 20 assets from the SPX Index composition starting from 1990-01-02 up to 2022-12-28: ```Python from plotly.io import show from sklearn.model_selection import train_test_split from skfolio import Population, RiskMeasure from skfolio.datasets import load_sp500_dataset from skfolio.moments import DenoiseCovariance, ShrunkMu from skfolio.optimization import MeanRisk, ObjectiveFunction from skfolio.preprocessing import prices_to_returns from skfolio.prior import EmpiricalPrior prices = load_sp500_dataset() X = prices_to_returns(prices) X_train, X_test = train_test_split(X, test_size=0.33, shuffle=False) ``` ## Model We create a Maximum Sharpe Ratio model with shrinkage for the estimation of the expected returns and denoising for the estimation of the covariance matrix: ```Python model = MeanRisk( risk_measure=RiskMeasure.VARIANCE, objective_function=ObjectiveFunction.MAXIMIZE_RATIO, prior_estimator=EmpiricalPrior( mu_estimator=ShrunkMu(), covariance_estimator=DenoiseCovariance() ), portfolio_params=dict(name="Max Sharpe - ShrunkMu & DenoiseCovariance"), ) model.fit(X_train) model.weights_ ``` ```none array([5.29957573e-02, 5.62258516e-07, 2.19431587e-07, 5.78329525e-02, 1.05704853e-01, 6.42635886e-07, 1.25145591e-02, 1.64813030e-01, 3.95272189e-07, 8.40687639e-02, 1.20296728e-06, 1.33374541e-06, 6.51482298e-02, 7.44911613e-02, 7.02327401e-06, 1.27177410e-01, 3.87852705e-02, 6.81199403e-02, 4.34872838e-02, 1.04849409e-01]) ``` ## Benchmark For comparison, we also create a Maximum Sharpe Ratio model using the default moments estimators: ```Python bench = MeanRisk( risk_measure=RiskMeasure.VARIANCE, objective_function=ObjectiveFunction.MAXIMIZE_RATIO, portfolio_params=dict(name="Max Sharpe"), ) bench.fit(X_train) bench.weights_ ``` ```none array([9.43631399e-02, 1.13184579e-06, 5.04970598e-07, 1.20834667e-01, 3.18126275e-02, 8.57806907e-07, 7.11596802e-04, 1.24104939e-01, 9.49223801e-07, 2.77547553e-02, 1.23409042e-06, 1.37593860e-06, 1.16299875e-01, 5.73516411e-02, 9.58498590e-06, 1.09493919e-01, 8.64761638e-02, 1.83992252e-01, 1.32350165e-02, 3.35537683e-02]) ``` ## Prediction We predict both models on the test set: ```Python pred_model = model.predict(X_test) pred_bench = bench.predict(X_test) population = Population([pred_model, pred_bench]) fig = population.plot_cumulative_returns() show(fig) ``` [plotly figure stripped from llms output] **Total running time of the script:** (0 minutes 0.730 seconds)