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This Agreement shall be construed, governed, interpreted and applied in accordance with the internal laws of the Commonwealth of Massachusetts, U.S.A., without regard to conflict of laws principles. */ package org.broadinstitute.gatk.utils; import org.apache.commons.math3.special.Gamma; import org.apache.commons.math3.util.MathArrays; import java.util.Arrays; import java.util.Collections; import java.util.stream.IntStream; /** * The Dirichlet distribution is a distribution on multinomial distributions: if pi is a vector of positive multinomial weights * such that sum_i pi[i] = 1, the Dirichlet pdf is P(pi) = [prod_i Gamma(alpha[i]) / Gamma(sum_i alpha[i])] * prod_i pi[i]^(alpha[i] - 1) * * The vector alpha comprises the sufficient statistics for the Dirichlet distribution. * * Since the Dirichlet is the conjugate prior to the multinomial, if one has a Dirichlet prior with concentration alpha * and observes each category i n_i times (assuming categories are drawn from a multinomial distribution pi) * the posterior is alpha_i -> alpha_i + n_i * * * @author David Benjamin <davidben@broadinstitute.org> */ public class Dirichlet { final double[] alpha; public Dirichlet(final double... alpha) { Utils.nonNull(alpha); Utils.validateArg(alpha.length >= 1, "Dirichlet parameters must have at least one element"); Utils.validateArg(MathUtils.allMatch(alpha, x -> x >= 0), "Dirichlet parameters may not be negative"); Utils.validateArg(MathUtils.allMatch(alpha, Double::isFinite), "Dirichlet parameters must be finite"); this.alpha = alpha.clone(); } /** * Create a symmetric distribution Dir(a/K, a/K, a/K . . .) where K is the number of states and * a is the concentration. */ public static Dirichlet symmetricDirichlet(final int numStates, final double concentration) { Utils.validateArg(numStates > 0, "Must have at leat one state"); Utils.validateArg(concentration > 0, "concentration must be positive"); return new Dirichlet( Collections.nCopies(numStates, concentration / numStates).stream().mapToDouble(x -> x).toArray()); } // in variational Bayes one often needs the effective point estimate of a multinomial distribution with a // Dirichlet prior. This value is not the mode or mean of the Dirichlet but rather the exp of the expected log weights. // note that these effective weights do not add up to 1. This is fine because in any probabilistic model scaling all weights // amounts to an arbitrary normalization constant, but it's important to keep in mind because some classes may expect // normalized weights. In that case the calling code must normalize the weights. public double[] effectiveMultinomialWeights() { final double digammaOfSum = Gamma.digamma(MathUtils.sum(alpha)); return MathUtils.applyToArray(alpha, a -> Math.exp(Gamma.digamma(a) - digammaOfSum)); } public double[] effectiveLog10MultinomialWeights() { final double digammaOfSum = Gamma.digamma(MathUtils.sum(alpha)); return MathUtils.applyToArray(alpha, a -> (Gamma.digamma(a) - digammaOfSum) * MathUtils.LOG10_OF_E); } public double[] meanWeights() { final double sum = MathUtils.sum(alpha); return MathUtils.applyToArray(alpha, x -> x / sum); } public double[] log10MeanWeights() { final double sum = MathUtils.sum(alpha); return MathUtils.applyToArray(alpha, x -> Math.log10(x / sum)); } public int size() { return alpha.length; } }