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//! Trait definitions extern crate rand; use self::rand::Rng; use data::DataOrSuffStat; use std::fmt::Debug; /// Random variable /// /// Contains the minimal functionality that a random object must have to be /// useful: a function defining the un-normalized density/mass at a point, /// and functions to draw samples from the distribution. pub trait Rv<X> { /// Probability function /// /// # Example /// /// ``` /// use rv::dist::Gaussian; /// use rv::traits::Rv; /// /// let g = Gaussian::standard(); /// assert!(g.f(&0.0_f64) > g.f(&0.1_f64)); /// assert!(g.f(&0.0_f64) > g.f(&-0.1_f64)); /// ``` fn f(&self, x: &X) -> f64 { self.ln_f(x).exp() } /// Probability function /// /// # Example /// /// ``` /// use rv::dist::Gaussian; /// use rv::traits::Rv; /// /// let g = Gaussian::standard(); /// assert!(g.ln_f(&0.0_f64) > g.ln_f(&0.1_f64)); /// assert!(g.ln_f(&0.0_f64) > g.ln_f(&-0.1_f64)); /// ``` fn ln_f(&self, x: &X) -> f64; /// Single draw from the `Rv` /// /// # Example /// /// Flip a coin /// /// ``` /// # extern crate rv; /// extern crate rand; /// /// use rv::dist::Bernoulli; /// use rv::traits::Rv; /// /// let b = Bernoulli::uniform(); /// let mut rng = rand::thread_rng(); /// let x: bool = b.draw(&mut rng); // could be true, could be false. /// ``` fn draw<R: Rng>(&self, rng: &mut R) -> X; /// Multiple draws of the `Rv` /// /// # Example /// /// Flip a lot of coins /// /// ``` /// # extern crate rv; /// extern crate rand; /// /// use rv::dist::Bernoulli; /// use rv::traits::Rv; /// /// let b = Bernoulli::uniform(); /// let mut rng = rand::thread_rng(); /// let xs: Vec<bool> = b.sample(22, &mut rng); /// /// assert_eq!(xs.len(), 22); /// ``` fn sample<R: Rng>(&self, n: usize, mut rng: &mut R) -> Vec<X> { (0..n).map(|_| self.draw(&mut rng)).collect() } } /// Identifies the support of the Rv pub trait Support<X> { /// Returns `true` if `x` is in the support of the `Rv` /// /// # Example /// /// ``` /// use rv::dist::Uniform; /// use rv::traits::Support; /// /// // Create uniform with support on the interval [0, 1] /// let u = Uniform::new(0.0, 1.0).unwrap(); /// /// assert!(u.supports(&0.5_f64)); /// assert!(!u.supports(&-0.1_f64)); /// assert!(!u.supports(&1.1_f64)); /// ``` fn supports(&self, x: &X) -> bool; } /// Is a continuous probability distributions /// /// This trait uses the `Rv<X>` and `Support<X>` implementations to implement /// itself. pub trait ContinuousDistr<X>: Rv<X> + Support<X> { /// The value of the Probability Density Function (PDF) at `x` /// /// # Panics /// /// If `x` is not in the support. /// /// # Example /// /// Compute the Gaussian PDF, f(x) /// /// ``` /// # extern crate rv; /// use rv::dist::Gaussian; /// use rv::traits::ContinuousDistr; /// /// let g = Gaussian::standard(); /// /// let f_mean = g.pdf(&0.0_f64); /// let f_low = g.pdf(&-1.0_f64); /// let f_high = g.pdf(&1.0_f64); /// /// assert!(f_mean > f_low); /// assert!(f_mean > f_high); /// assert!((f_low - f_high).abs() < 1E-12); /// ``` fn pdf(&self, x: &X) -> f64 { self.ln_pdf(x).exp() } /// The value of the log Probability Density Function (PDF) at `x` /// /// # Panics /// /// If `x` is not in the support. /// /// # Example /// /// Compute the natural logarithm of the Gaussian PDF, ln(f(x)) /// /// ``` /// # extern crate rv; /// use rv::dist::Gaussian; /// use rv::traits::ContinuousDistr; /// /// let g = Gaussian::standard(); /// /// let lnf_mean = g.ln_pdf(&0.0_f64); /// let lnf_low = g.ln_pdf(&-1.0_f64); /// let lnf_high = g.ln_pdf(&1.0_f64); /// /// assert!(lnf_mean > lnf_low); /// assert!(lnf_mean > lnf_high); /// assert!((lnf_low - lnf_high).abs() < 1E-12); /// ``` fn ln_pdf(&self, x: &X) -> f64 { if !self.supports(&x) { panic!("x not in support"); } self.ln_f(x) } } /// Has a cumulative distribution function (CDF) pub trait Cdf<X>: Rv<X> { /// The value of the Cumulative Density Function at `x` /// /// # Example /// /// The proportion of probability in (-∞, μ) in N(μ, σ) is 50% /// /// ``` /// # extern crate rv; /// use rv::dist::Gaussian; /// use rv::traits::Cdf; /// /// let g = Gaussian::new(1.0, 1.5).unwrap(); /// /// assert!((g.cdf(&1.0_f64) - 0.5).abs() < 1E-12); /// ``` fn cdf(&self, x: &X) -> f64; /// Survival function, `1 - CDF(x)` fn sf(&self, x: &X) -> f64 { 1.0 - self.cdf(x) } } /// Has an inverse-CDF / quantile function pub trait InverseCdf<X>: Rv<X> + Support<X> { /// The value of the `x` at the given probability in the CDF /// /// # Example /// /// The CDF identity: p = CDF(x) => x = CDF<sup>-1</sup>(p) /// /// ``` /// # extern crate rv; /// use rv::dist::Gaussian; /// use rv::traits::Cdf; /// use rv::traits::InverseCdf; /// /// let g = Gaussian::standard(); /// /// let x: f64 = 1.2; /// let p: f64 = g.cdf(&x); /// let y: f64 = g.invcdf(p); /// /// // x and y should be about the same /// assert!((x - y).abs() < 1E-12); /// ``` fn invcdf(&self, p: f64) -> X; /// Alias for `invcdf` fn quantile(&self, p: f64) -> X { self.invcdf(p) } /// Interval containing `p` proportion for the probability /// /// # Example /// /// Confidence interval /// /// ``` /// # extern crate rv; /// use rv::dist::Gaussian; /// use rv::traits::InverseCdf; /// /// let g = Gaussian::new(100.0, 15.0).unwrap(); /// let ci: (f64, f64) = g.interval(0.68268949213708585); // one stddev /// assert!( (ci.0 - 85.0).abs() < 1E-12); /// assert!( (ci.1 - 115.0).abs() < 1E-12); /// ``` fn interval(&self, p: f64) -> (X, X) { let pt = (1.0 - p) / 2.0; (self.quantile(pt), self.quantile(p + pt)) } } /// Is a discrete probability distribution pub trait DiscreteDistr<X>: Rv<X> + Support<X> { /// Probability mass function (PMF) at `x` /// /// # Panics /// /// If `x` is not supported /// /// # Example /// /// The probability of a fair coin coming up heads in 0.5 /// /// ``` /// use rv::dist::Bernoulli; /// use rv::traits::DiscreteDistr; /// /// // Fair coin (p = 0.5) /// let b = Bernoulli::uniform(); /// /// assert!( (b.pmf(&true) - 0.5).abs() < 1E-12); /// ``` fn pmf(&self, x: &X) -> f64 { self.ln_pmf(x).exp() } /// Natural logarithm of the probability mass function (PMF) /// /// # Panics /// /// If `x` is not supported /// /// # Example /// /// The probability of a fair coin coming up heads in 0.5 /// /// ``` /// use rv::dist::Bernoulli; /// use rv::traits::DiscreteDistr; /// /// // Fair coin (p = 0.5) /// let b = Bernoulli::uniform(); /// /// assert!( (b.ln_pmf(&true) - 0.5_f64.ln()).abs() < 1E-12); /// ``` fn ln_pmf(&self, x: &X) -> f64 { if !self.supports(&x) { panic!("x not in support"); } self.ln_f(x) } } /// Defines the distribution mean pub trait Mean<X> { /// Returns `None` if the mean is undefined fn mean(&self) -> Option<X>; } /// Defines the distribution median pub trait Median<X> { /// Returns `None` if the median is undefined fn median(&self) -> Option<X>; } /// Defines the distribution mode pub trait Mode<X> { /// Returns `None` if the mode is undefined or is not a single value fn mode(&self) -> Option<X>; } /// Defines the distribution variance pub trait Variance<X> { /// Returns `None` if the variance is undefined fn variance(&self) -> Option<X>; } /// Defines the distribution entropy pub trait Entropy { /// The entropy, *H(X)* fn entropy(&self) -> f64; } pub trait Skewness { fn skewness(&self) -> Option<f64>; } pub trait Kurtosis { fn kurtosis(&self) -> Option<f64>; } /// KL divergences pub trait KlDivergence { /// The KL divergence, KL(P|Q) between this distribution, P, and another, Q /// /// # Example /// /// ``` /// use rv::dist::Gaussian; /// use rv::traits::KlDivergence; /// /// let g1 = Gaussian::new(1.0, 1.0).unwrap(); /// let g2 = Gaussian::new(-1.0, 2.0).unwrap(); /// /// let kl_self = g1.kl(&g1); /// let kl_other = g1.kl(&g2); /// /// // KL(P|P) = 0 /// assert!( kl_self < 1E-12 ); /// /// // KL(P|Q) > 0 if P ≠ Q /// assert!( kl_self < kl_other ); /// ``` fn kl(&self, other: &Self) -> f64; /// Symmetrised divergence, KL(P|Q) + KL(Q|P) /// /// # Example /// /// ``` /// use rv::dist::Gaussian; /// use rv::traits::KlDivergence; /// /// let g1 = Gaussian::new(1.0, 1.0).unwrap(); /// let g2 = Gaussian::new(-1.0, 2.0).unwrap(); /// /// let kl_12 = g1.kl(&g2); /// let kl_21 = g2.kl(&g1); /// /// let kl_sym = g1.kl_sym(&g2); /// /// assert!( (kl_12 + kl_21 - kl_sym).abs() < 1E-10 ); /// ``` fn kl_sym(&self, other: &Self) -> f64 { self.kl(&other) + other.kl(&self) } } /// The data for this distribution can be summarized by a statistic pub trait HasSuffStat<X> { type Stat: SuffStat<X> + Debug; fn empty_suffstat(&self) -> Self::Stat; } /// Is a [sufficient statistic](https://en.wikipedia.org/wiki/Sufficient_statistic) for a /// distribution. /// /// # Example /// /// ``` /// use rv::data::BernoulliSuffStat; /// use rv::traits::SuffStat; /// /// // Bernoulli sufficient statistics are the number of observations, n, and /// // the number of successes, k. /// let mut stat = BernoulliSuffStat::new(); /// /// assert!(stat.n == 0 && stat.k == 0); /// /// stat.observe(&true); // observe `true` /// assert!(stat.n == 1 && stat.k == 1); /// /// stat.observe(&false); // observe `false` /// assert!(stat.n == 2 && stat.k == 1); /// /// stat.forget_many(&vec![false, true]); // forget `true` and `false` /// assert!(stat.n == 0 && stat.k == 0); /// ``` pub trait SuffStat<X> { /// Returns the number of observations fn n(&self) -> usize; /// Assimilate the datum `x` into the statistic fn observe(&mut self, x: &X); /// Remove the datum `x` from the statistic fn forget(&mut self, x: &X); /// Assimilate several observations fn observe_many(&mut self, xs: &[X]) { xs.iter().for_each(|x| self.observe(x)); } /// Forget several observations fn forget_many(&mut self, xs: &[X]) { xs.iter().for_each(|x| self.forget(x)); } } /// A prior on `Fx` that induces a posterior that is the same form as the prior pub trait ConjugatePrior<X, Fx>: Rv<Fx> where Fx: Rv<X> + HasSuffStat<X>, { type Posterior: Rv<Fx>; /// Computes the posterior distribution from the data fn posterior(&self, x: &DataOrSuffStat<X, Fx>) -> Self::Posterior; /// Log marginal likelihood fn ln_m(&self, x: &DataOrSuffStat<X, Fx>) -> f64; /// Log posterior predictive of y given x fn ln_pp(&self, y: &X, x: &DataOrSuffStat<X, Fx>) -> f64; /// Marginal likelihood of x fn m(&self, x: &DataOrSuffStat<X, Fx>) -> f64 { self.ln_m(x).exp() } /// Posterior Predictive distribution fn pp(&self, y: &X, x: &DataOrSuffStat<X, Fx>) -> f64 { self.ln_pp(&y, x).exp() } }