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package breeze.stats
package distributions
/*
Copyright 2009 David Hall, Daniel Ramage
Licensed under the Apache License, Version 2.0 (the "License")
you may not use this file except in compliance with the License.
You may obtain a copy of the License at
http://www.apache.org/licenses/LICENSE-2.0
Unless required by applicable law or agreed to in writing, software
distributed under the License is distributed on an "AS IS" BASIS,
WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
See the License for the specific language governing permissions and
limitations under the License.
*/
import breeze.linalg._
import breeze.numerics.{digamma, lgamma}
import breeze.optimize._
import org.apache.commons.math3.distribution.BetaDistribution
import scala.math._
/**
* The Beta distribution, which is the conjugate prior for the Bernoulli distribution
*
* @author dlwh
* @param a the number of pseudo-observations for true
* @param b the number of pseudo-observations for false
*/
class Beta(a: Double, b: Double)(implicit rand: RandBasis = Rand) extends ContinuousDistr[Double] with Moments[Double, Double] with HasCdf with HasInverseCdf {
require(a > 0.0)
require(b > 0.0)
override def unnormalizedLogPdf(x: Double) = {
require(x >= 0)
require(x <= 1)
(a-1) * log(x) + (b-1) * log(1-x)
}
override def pdf(x: Double): Double = {
require(x >= 0)
require(x <= 1)
x match {
case 0.0 => if (a > 1) { 0 } else if (a == 1) { normalizer } else { Double.PositiveInfinity }
case 1.0 => if (b > 1) { 0 } else if (b == 1) { normalizer } else { Double.PositiveInfinity }
case x => math.exp(logPdf(x))
}
}
override def probability(x: Double, y: Double): Double = {
new BetaDistribution(a, b).probability(x, y)
}
lazy val logNormalizer = lgamma(a) + lgamma(b) - lgamma(a+b)
private val aGamma = new Gamma(a,1)(rand)
private val bGamma = new Gamma(b,1)(rand)
override def draw():Double = {
// from tjhunter, a corrected version of numpy's rk_beta sampling in mtrand/distributions.c
if(a <= .5 && b <= .5) {
while (true) {
val U = rand.uniform.draw()
val V = rand.uniform.draw()
if (U > 0 && V > 0) {
// Performing the computations in the log-domain
// The exponentiation may fail if a or b are really small
// val X = math.pow(U, 1.0 / a)
val logX = math.log(U) / a
// val Y = math.pow(V, 1.0 / b)
val logY= math.log(V) / b
val logSum = softmax(logX, logY)
if (logSum <= 0.0) {
return math.exp(logX - logSum)
}
} else {
throw new RuntimeException("Underflow!")
}
}
throw new RuntimeException("Shouldn't be here.")
} else if(a <= 1 && b <= 1) {
while (true) {
val U = rand.uniform.draw()
val V = rand.uniform.draw()
if (U > 0 && V > 0) {
// Performing the computations in the log-domain
// The exponentiation may fail if a or b are really small
val X = math.pow(U, 1.0 / a)
val Y = math.pow(V, 1.0 / b)
val sum = X + Y
if (sum <= 1.0) {
return X / sum
}
} else {
throw new RuntimeException("Underflow!")
}
}
throw new RuntimeException("Shouldn't be here.")
} else {
val ad = aGamma.draw()
val bd = bGamma.draw()
ad / (ad + bd)
}
}
def mean = a / (a + b)
def variance = (a * b) / ( (a + b) * (a+b) * (a+b+1))
def mode = (a - 1) / (a+b - 2)
def entropy = logNormalizer - (a - 1) * digamma(a) - (b-1) * digamma(b) + (a + b - 2) * digamma(a + b)
// Probability that x < a <= Y
override def cdf(x: Double): Double = {
new BetaDistribution(a, b).cumulativeProbability(x)
}
override def inverseCdf(p: Double): Double = {
new BetaDistribution(a, b).inverseCumulativeProbability(p)
}
}
object Beta extends ExponentialFamily[Beta,Double] with ContinuousDistributionUFuncProvider[Double,Beta] {
type Parameter = (Double,Double)
case class SufficientStatistic(n: Double, meanLog: Double, meanLog1M: Double) extends distributions.SufficientStatistic[SufficientStatistic] {
def *(weight: Double) = SufficientStatistic(n*weight,meanLog, meanLog1M)
def +(t: SufficientStatistic) = {
val delta = t.meanLog - meanLog
val newMeanLog = meanLog + delta * (t.n /(t.n + n))
val logDelta = t.meanLog1M - meanLog1M
val newMeanLog1M = meanLog1M + logDelta * (t.n /(t.n + n))
SufficientStatistic(n+t.n, newMeanLog, newMeanLog1M)
}
}
def emptySufficientStatistic = SufficientStatistic(0,0,0)
def sufficientStatisticFor(t: Double) = SufficientStatistic(1,math.log(t),math.log1p(-t))
def mle(stats: SufficientStatistic): (Double, Double) = {
import breeze.linalg.DenseVector.TupleIsomorphisms._
val lensed = likelihoodFunction(stats).throughLens[DenseVector[Double]]
val startingA = stats.meanLog.abs // MoM would include variance, meh.
val startingB = stats.meanLog1M.abs // MoM would include variance, meh
val result = minimize(lensed,DenseVector(startingA,startingB))
val res@(a,b) = (result(0),result(1))
res
}
def distribution(ab: Parameter) = new Beta(ab._1,ab._2)
def likelihoodFunction(stats: SufficientStatistic):DiffFunction[(Double,Double)] = new DiffFunction[(Double,Double)]{
import stats.n
def calculate(x: (Double, Double)) = {
val (a,b) = x
if(a < 0 || b < 0) (Double.PositiveInfinity,(0.0,0.0))
else {
val obj = n * (lgamma(a) + lgamma(b) - lgamma(a+b) - (a-1)*stats.meanLog - (b-1) *stats.meanLog1M)
val gradA = n * (digamma(a) - digamma(a+b) - stats.meanLog)
val gradB = n * (digamma(b) - digamma(a+b) - stats.meanLog1M)
(obj,(gradA,gradB))
}
}
}
}
