Detail: Dirichlet Estimator

The Dirichlet Estimator fits a Bayes Parametric Model to data using a Dirichlet prior over each row of the conditional probability tables (CPTs). It is a Bayesian-smoothing alternative to plain ML estimation, reducing zero-probability problems in sparse data.

This estimator is available whenever the Parametric Model connected to the Estimator box is a Bayes PM.

Dirichlet Estimator

Dirichlet Estimator

Purpose

  • Estimate CPTs using posterior mean probabilities under a Dirichlet prior.

  • Provide smoothed probability estimates that avoid zero counts.

  • Improve stability when sample sizes are modest or some parent configurations are rare or unobserved.

Inputs and requirements

  • Parametric Model: A Bayes PM specifying nodes, states, and parents.

  • Data: Discrete data with variables matching the Bayes PM.

  • Prior settings (when exposed in the GUI):

    • Often summarized as a strength or equivalent sample size for a uniform Dirichlet prior (e.g., α > 0).

    • Some versions may allow non-uniform priors.

How it works (conceptually)

For each node X with parents Pa(X), and each parent configuration pi:

  1. Start with a Dirichlet prior written as Dir(alpha_1, …, alpha_k) over the k states of X.

  2. Observe empirical counts n_1, …, n_k from the data.

  3. Compute the posterior Dirichlet Dir(alpha_1 + n_1, …, alpha_k + n_k).

  4. Use the posterior mean as the CPT entries:

    P(X = x_i | Pa(X) = pi) = (alpha_i + n_i) / sum_j (alpha_j + n_j).

Output

  • A fitted Bayes model whose CPT rows are posterior-mean probabilities under the specified Dirichlet prior(s).

  • Optionally, a summary of:

    • Log-marginal likelihood or other scores (depending on implementation).

  • The result can be saved as an Instantiated Model.

Tips and common issues

  • Larger prior strengths (equivalent sample sizes) lead to heavier smoothing, shrinking probabilities toward the prior.

  • For very small datasets or models with many parents, Dirichlet smoothing is often preferable to raw ML.

  • If you are using the model for score-based structure learning, keep in mind that different Dirichlet hyperparameters can significantly affect scores.