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diff --git a/src/lib/math/prob_distr.c b/src/lib/math/prob_distr.c new file mode 100644 index 0000000000..548d256023 --- /dev/null +++ b/src/lib/math/prob_distr.c @@ -0,0 +1,1690 @@ +/* Copyright (c) 2018-2020, The Tor Project, Inc. */ +/* See LICENSE for licensing information */ + +/** + * \file prob_distr.c + * + * \brief + * Implements various probability distributions. + * Almost all code is courtesy of Riastradh. + * + * \details + * Here are some details that might help you understand this file: + * + * - Throughout this file, `eps' means the largest relative error of a + * correctly rounded floating-point operation, which in binary64 + * floating-point arithmetic is 2^-53. Here the relative error of a + * true value x from a computed value y is |x - y|/|x|. This + * definition of epsilon is conventional for numerical analysts when + * writing error analyses. (If your libm doesn't provide correctly + * rounded exp and log, their relative error is usually below 2*2^-53 + * and probably closer to 1.1*2^-53 instead.) + * + * The C constant DBL_EPSILON is actually twice this, and should + * perhaps rather be named ulp(1) -- that is, it is the distance from + * 1 to the next greater floating-point number, which is usually of + * more interest to programmers and hardware engineers. + * + * Since this file is concerned mainly with error bounds rather than + * with low-level bit-hacking of floating-point numbers, we adopt the + * numerical analysts' definition in the comments, though we do use + * DBL_EPSILON in a handful of places where it is convenient to use + * some function of eps = DBL_EPSILON/2 in a case analysis. + * + * - In various functions (e.g. sample_log_logistic()) we jump through hoops so + * that we can use reals closer to 0 than closer to 1, since we achieve much + * greater accuracy for floating point numbers near 0. In particular, we can + * represent differences as small as 10^-300 for numbers near 0, but of no + * less than 10^-16 for numbers near 1. + **/ + +#define PROB_DISTR_PRIVATE + +#include "orconfig.h" + +#include "lib/math/prob_distr.h" + +#include "lib/crypt_ops/crypto_rand.h" +#include "lib/cc/ctassert.h" +#include "lib/log/util_bug.h" + +#include <float.h> +#include <math.h> +#include <stddef.h> + +#ifndef COCCI +/** Declare a function that downcasts from a generic dist struct to the actual + * subtype probablity distribution it represents. */ +#define DECLARE_PROB_DISTR_DOWNCAST_FN(name) \ + static inline \ + const struct name##_t * \ + dist_to_const_##name(const struct dist_t *obj) { \ + tor_assert(obj->ops == &name##_ops); \ + return SUBTYPE_P(obj, struct name ## _t, base); \ + } +DECLARE_PROB_DISTR_DOWNCAST_FN(uniform) +DECLARE_PROB_DISTR_DOWNCAST_FN(geometric) +DECLARE_PROB_DISTR_DOWNCAST_FN(logistic) +DECLARE_PROB_DISTR_DOWNCAST_FN(log_logistic) +DECLARE_PROB_DISTR_DOWNCAST_FN(genpareto) +DECLARE_PROB_DISTR_DOWNCAST_FN(weibull) +#endif /* !defined(COCCI) */ + +/** + * Count number of one bits in 32-bit word. + */ +static unsigned +bitcount32(uint32_t x) +{ + + /* Count two-bit groups. */ + x -= (x >> 1) & UINT32_C(0x55555555); + + /* Count four-bit groups. */ + x = ((x >> 2) & UINT32_C(0x33333333)) + (x & UINT32_C(0x33333333)); + + /* Count eight-bit groups. */ + x = (x + (x >> 4)) & UINT32_C(0x0f0f0f0f); + + /* Sum all eight-bit groups, and extract the sum. */ + return (x * UINT32_C(0x01010101)) >> 24; +} + +/** + * Count leading zeros in 32-bit word. + */ +static unsigned +clz32(uint32_t x) +{ + + /* Round up to a power of two. */ + x |= x >> 1; + x |= x >> 2; + x |= x >> 4; + x |= x >> 8; + x |= x >> 16; + + /* Subtract count of one bits from 32. */ + return (32 - bitcount32(x)); +} + +/* + * Some lemmas that will be used throughout this file to prove various error + * bounds: + * + * Lemma 1. If |d| <= 1/2, then 1/(1 + d) <= 2. + * + * Proof. If 0 <= d <= 1/2, then 1 + d >= 1, so that 1/(1 + d) <= 1. + * If -1/2 <= d <= 0, then 1 + d >= 1/2, so that 1/(1 + d) <= 2. QED. + * + * Lemma 2. If b = a*(1 + d)/(1 + d') for |d'| < 1/2 and nonzero a, b, + * then b = a*(1 + e) for |e| <= 2|d' - d|. + * + * Proof. |a - b|/|a| + * = |a - a*(1 + d)/(1 + d')|/|a| + * = |1 - (1 + d)/(1 + d')| + * = |(1 + d' - 1 - d)/(1 + d')| + * = |(d' - d)/(1 + d')| + * <= 2|d' - d|, by Lemma 1, + * + * QED. + * + * Lemma 3. For |d|, |d'| < 1/4, + * + * |log((1 + d)/(1 + d'))| <= 4|d - d'|. + * + * Proof. Write + * + * log((1 + d)/(1 + d')) + * = log(1 + (1 + d)/(1 + d') - 1) + * = log(1 + (1 + d - 1 - d')/(1 + d') + * = log(1 + (d - d')/(1 + d')). + * + * By Lemma 1, |(d - d')/(1 + d')| < 2|d' - d| < 1, so the Taylor + * series of log(1 + x) converges absolutely for (d - d')/(1 + d'), + * and thus we have + * + * |log(1 + (d - d')/(1 + d'))| + * = |\sum_{n=1}^\infty ((d - d')/(1 + d'))^n/n| + * <= \sum_{n=1}^\infty |(d - d')/(1 + d')|^n/n + * <= \sum_{n=1}^\infty |2(d' - d)|^n/n + * <= \sum_{n=1}^\infty |2(d' - d)|^n + * = 1/(1 - |2(d' - d)|) + * <= 4|d' - d|, + * + * QED. + * + * Lemma 4. If 1/e <= 1 + x <= e, then + * + * log(1 + (1 + d) x) = (1 + d') log(1 + x) + * + * for |d'| < 8|d|. + * + * Proof. Write + * + * log(1 + (1 + d) x) + * = log(1 + x + x*d) + * = log((1 + x) (1 + x + x*d)/(1 + x)) + * = log(1 + x) + log((1 + x + x*d)/(1 + x)) + * = log(1 + x) (1 + log((1 + x + x*d)/(1 + x))/log(1 + x)). + * + * The relative error is bounded by + * + * |log((1 + x + x*d)/(1 + x))/log(1 + x)| + * <= 4|x + x*d - x|/|log(1 + x)|, by Lemma 3, + * = 4|x*d|/|log(1 + x)| + * < 8|d|, + * + * since in this range 0 < 1 - 1/e < x/log(1 + x) <= e - 1 < 2. QED. + */ + +/** + * Compute the logistic function: f(x) = 1/(1 + e^{-x}) = e^x/(1 + e^x). + * Maps a log-odds-space probability in [-infinity, +infinity] into a + * direct-space probability in [0,1]. Inverse of logit. + * + * Ill-conditioned for large x; the identity logistic(-x) = 1 - + * logistic(x) and the function logistichalf(x) = logistic(x) - 1/2 may + * help to rearrange a computation. + * + * This implementation gives relative error bounded by 7 eps. + */ +STATIC double +logistic(double x) +{ + if (x <= log(DBL_EPSILON/2)) { + /* + * If x <= log(DBL_EPSILON/2) = log(eps), then e^x <= eps. In this case + * we will approximate the logistic() function with e^x because the + * relative error is less than eps. Here is a calculation of the + * relative error between the logistic() function and e^x and a proof + * that it's less than eps: + * + * |e^x - e^x/(1 + e^x)|/|e^x/(1 + e^x)| + * <= |1 - 1/(1 + e^x)|*|1 + e^x| + * = |e^x/(1 + e^x)|*|1 + e^x| + * = |e^x| + * <= eps. + */ + return exp(x); /* return e^x */ + } else if (x <= -log(DBL_EPSILON/2)) { + /* + * e^{-x} > 0, so 1 + e^{-x} > 1, and 0 < 1/(1 + + * e^{-x}) < 1; further, since e^{-x} < 1 + e^{-x}, we + * also have 0 < 1/(1 + e^{-x}) < 1. Thus, if exp has + * relative error d0, + has relative error d1, and / + * has relative error d2, then we get + * + * (1 + d2)/[(1 + (1 + d0) e^{-x})(1 + d1)] + * = (1 + d0)/[1 + e^{-x} + d0 e^{-x} + * + d1 + d1 e^{-x} + d0 d1 e^{-x}] + * = (1 + d0)/[(1 + e^{-x}) + * * (1 + d0 e^{-x}/(1 + e^{-x}) + * + d1/(1 + e^{-x}) + * + d0 d1 e^{-x}/(1 + e^{-x}))]. + * = (1 + d0)/[(1 + e^{-x})(1 + d')] + * = [1/(1 + e^{-x})] (1 + d0)/(1 + d') + * + * where + * + * d' = d0 e^{-x}/(1 + e^{-x}) + * + d1/(1 + e^{-x}) + * + d0 d1 e^{-x}/(1 + e^{-x}). + * + * By Lemma 2 this relative error is bounded by + * + * 2|d0 - d'| + * = 2|d0 - d0 e^{-x}/(1 + e^{-x}) + * - d1/(1 + e^{-x}) + * - d0 d1 e^{-x}/(1 + e^{-x})| + * <= 2|d0| + 2|d0 e^{-x}/(1 + e^{-x})| + * + 2|d1/(1 + e^{-x})| + * + 2|d0 d1 e^{-x}/(1 + e^{-x})| + * <= 2|d0| + 2|d0| + 2|d1| + 2|d0 d1| + * <= 4|d0| + 2|d1| + 2|d0 d1| + * <= 6 eps + 2 eps^2. + */ + return 1/(1 + exp(-x)); + } else { + /* + * e^{-x} <= eps, so the relative error of 1 from 1/(1 + * + e^{-x}) is + * + * |1/(1 + e^{-x}) - 1|/|1/(1 + e^{-x})| + * = |e^{-x}/(1 + e^{-x})|/|1/(1 + e^{-x})| + * = |e^{-x}| + * <= eps. + * + * This computation avoids an intermediate overflow + * exception, although the effect on the result is + * harmless. + * + * XXX Should maybe raise inexact here. + */ + return 1; + } +} + +/** + * Compute the logit function: log p/(1 - p). Defined on [0,1]. Maps + * a direct-space probability in [0,1] to a log-odds-space probability + * in [-infinity, +infinity]. Inverse of logistic. + * + * Ill-conditioned near 1/2 and 1; the identity logit(1 - p) = + * -logit(p) and the function logithalf(p0) = logit(1/2 + p0) may help + * to rearrange a computation for p in [1/(1 + e), 1 - 1/(1 + e)]. + * + * This implementation gives relative error bounded by 10 eps. + */ +STATIC double +logit(double p) +{ + + /* logistic(-1) <= p <= logistic(+1) */ + if (1/(1 + exp(1)) <= p && p <= 1/(1 + exp(-1))) { + /* + * For inputs near 1/2, we want to compute log1p(near + * 0) rather than log(near 1), so write this as: + * + * log(p/(1 - p)) = -log((1 - p)/p) + * = -log(1 + (1 - p)/p - 1) + * = -log(1 + (1 - p - p)/p) + * = -log(1 + (1 - 2p)/p). + * + * Since p = 2p/2 <= 1 <= 2*2p = 4p, the floating-point + * evaluation of 1 - 2p is exact; the only error arises + * from division and log1p. First, note that if + * logistic(-1) <= p <= logistic(+1), (1 - 2p)/p lies + * in the bounds of Lemma 4. + * + * If division has relative error d0 and log1p has + * relative error d1, the outcome is + * + * -(1 + d1) log(1 + (1 - 2p) (1 + d0)/p) + * = -(1 + d1) (1 + d') log(1 + (1 - 2p)/p) + * = -(1 + d1 + d' + d1 d') log(1 + (1 - 2p)/p). + * + * where |d'| < 8|d0| by Lemma 4. The relative error + * is then bounded by + * + * |d1 + d' + d1 d'| + * <= |d1| + 8|d0| + 8|d1 d0| + * <= 9 eps + 8 eps^2. + */ + return -log1p((1 - 2*p)/p); + } else { + /* + * For inputs near 0, although 1 - p may be rounded to + * 1, it doesn't matter much because the magnitude of + * the result is so much larger. For inputs near 1, we + * can compute 1 - p exactly, although the precision on + * the input is limited so we won't ever get more than + * about 700 for the output. + * + * If - has relative error d0, / has relative error d1, + * and log has relative error d2, then + * + * (1 + d2) log((1 + d0) p/[(1 - p)(1 + d1)]) + * = (1 + d2) [log(p/(1 - p)) + log((1 + d0)/(1 + d1))] + * = log(p/(1 - p)) + d2 log(p/(1 - p)) + * + (1 + d2) log((1 + d0)/(1 + d1)) + * = log(p/(1 - p))*[1 + d2 + + * + (1 + d2) log((1 + d0)/(1 + d1))/log(p/(1 - p))] + * + * Since 0 <= p < logistic(-1) or logistic(+1) < p <= + * 1, we have |log(p/(1 - p))| > 1. Hence this error + * is bounded by + * + * |d2 + (1 + d2) log((1 + d0)/(1 + d1))/log(p/(1 - p))| + * <= |d2| + |(1 + d2) log((1 + d0)/(1 + d1)) + * / log(p/(1 - p))| + * <= |d2| + |(1 + d2) log((1 + d0)/(1 + d1))| + * <= |d2| + 4|(1 + d2) (d0 - d1)|, by Lemma 3, + * <= |d2| + 4|d0 - d1 + d2 d0 - d1 d0| + * <= |d2| + 4|d0| + 4|d1| + 4|d2 d0| + 4|d1 d0| + * <= 9 eps + 8 eps^2. + */ + return log(p/(1 - p)); + } +} + +/** + * Compute the logit function, translated in input by 1/2: logithalf(p) + * = logit(1/2 + p). Defined on [-1/2, 1/2]. Inverse of logistichalf. + * + * Ill-conditioned near +/-1/2. If |p0| > 1/2 - 1/(1 + e), it may be + * better to compute 1/2 + p0 or -1/2 - p0 and to use logit instead. + * This implementation gives relative error bounded by 34 eps. + */ +STATIC double +logithalf(double p0) +{ + + if (fabs(p0) <= 0.5 - 1/(1 + exp(1))) { + /* + * logit(1/2 + p0) + * = log((1/2 + p0)/(1 - (1/2 + p0))) + * = log((1/2 + p0)/(1/2 - p0)) + * = log(1 + (1/2 + p0)/(1/2 - p0) - 1) + * = log(1 + (1/2 + p0 - (1/2 - p0))/(1/2 - p0)) + * = log(1 + (1/2 + p0 - 1/2 + p0)/(1/2 - p0)) + * = log(1 + 2 p0/(1/2 - p0)) + * + * If the error of subtraction is d0, the error of + * division is d1, and the error of log1p is d2, then + * what we compute is + * + * (1 + d2) log(1 + (1 + d1) 2 p0/[(1 + d0) (1/2 - p0)]) + * = (1 + d2) log(1 + (1 + d') 2 p0/(1/2 - p0)) + * = (1 + d2) (1 + d'') log(1 + 2 p0/(1/2 - p0)) + * = (1 + d2 + d'' + d2 d'') log(1 + 2 p0/(1/2 - p0)), + * + * where |d'| < 2|d0 - d1| <= 4 eps by Lemma 2, and + * |d''| < 8|d'| < 32 eps by Lemma 4 since + * + * 1/e <= 1 + 2*p0/(1/2 - p0) <= e + * + * when |p0| <= 1/2 - 1/(1 + e). Hence the relative + * error is bounded by + * + * |d2 + d'' + d2 d''| + * <= |d2| + |d''| + |d2 d''| + * <= |d1| + 32 |d0| + 32 |d1 d0| + * <= 33 eps + 32 eps^2. + */ + return log1p(2*p0/(0.5 - p0)); + } else { + /* + * We have a choice of computing logit(1/2 + p0) or + * -logit(1 - (1/2 + p0)) = -logit(1/2 - p0). It + * doesn't matter which way we do this: either way, + * since 1/2 p0 <= 1/2 <= 2 p0, the sum and difference + * are computed exactly. So let's do the one that + * skips the final negation. + * + * The result is + * + * (1 + d1) log((1 + d0) (1/2 + p0)/[(1 + d2) (1/2 - p0)]) + * = (1 + d1) (1 + log((1 + d0)/(1 + d2)) + * / log((1/2 + p0)/(1/2 - p0))) + * * log((1/2 + p0)/(1/2 - p0)) + * = (1 + d') log((1/2 + p0)/(1/2 - p0)) + * = (1 + d') logit(1/2 + p0) + * + * where + * + * d' = d1 + log((1 + d0)/(1 + d2))/logit(1/2 + p0) + * + d1 log((1 + d0)/(1 + d2))/logit(1/2 + p0). + * + * For |p| > 1/2 - 1/(1 + e), logit(1/2 + p0) > 1. + * Provided |d0|, |d2| < 1/4, by Lemma 3 we have + * + * |log((1 + d0)/(1 + d2))| <= 4|d0 - d2|. + * + * Hence the relative error is bounded by + * + * |d'| <= |d1| + 4|d0 - d2| + 4|d1| |d0 - d2| + * <= |d1| + 4|d0| + 4|d2| + 4|d1 d0| + 4|d1 d2| + * <= 9 eps + 8 eps^2. + */ + return log((0.5 + p0)/(0.5 - p0)); + } +} + +/* + * The following random_uniform_01 is tailored for IEEE 754 binary64 + * floating-point or smaller. It can be adapted to larger + * floating-point formats like i387 80-bit or IEEE 754 binary128, but + * it may require sampling more bits. + */ +CTASSERT(FLT_RADIX == 2); +CTASSERT(-DBL_MIN_EXP <= 1021); +CTASSERT(DBL_MANT_DIG <= 53); + +/** + * Draw a floating-point number in [0, 1] with uniform distribution. + * + * Note that the probability of returning 0 is less than 2^-1074, so + * callers need not check for it. However, callers that cannot handle + * rounding to 1 must deal with that, because it occurs with + * probability 2^-54, which is small but nonnegligible. + */ +STATIC double +random_uniform_01(void) +{ + uint32_t z, x, hi, lo; + double s; + + /* + * Draw an exponent, geometrically distributed, but give up if + * we get a run of more than 1088 zeros, which really means the + * system is broken. + */ + z = 0; + while ((x = crypto_fast_rng_get_u32(get_thread_fast_rng())) == 0) { + if (z >= 1088) + /* Your bit sampler is broken. Go home. */ + return 0; + z += 32; + } + z += clz32(x); + + /* + * Pick 32-bit halves of an odd normalized significand. + * Picking it odd breaks ties in the subsequent rounding, which + * occur only with measure zero in the uniform distribution on + * [0, 1]. + */ + hi = crypto_fast_rng_get_u32(get_thread_fast_rng()) | UINT32_C(0x80000000); + lo = crypto_fast_rng_get_u32(get_thread_fast_rng()) | UINT32_C(0x00000001); + + /* Round to nearest scaled significand in [2^63, 2^64]. */ + s = hi*(double)4294967296 + lo; + + /* Rescale into [1/2, 1] and apply exponent in one swell foop. */ + return s * ldexp(1, -(64 + z)); +} + +/*******************************************************************/ + +/* Functions for specific probability distributions start here: */ + +/* + * Logistic(mu, sigma) distribution, supported on (-infinity,+infinity) + * + * This is the uniform distribution on [0,1] mapped into log-odds + * space, scaled by sigma and translated by mu. + * + * pdf(x) = e^{-(x - mu)/sigma} sigma (1 + e^{-(x - mu)/sigma})^2 + * cdf(x) = 1/(1 + e^{-(x - mu)/sigma}) = logistic((x - mu)/sigma) + * sf(x) = 1 - cdf(x) = 1 - logistic((x - mu)/sigma = logistic(-(x - mu)/sigma) + * icdf(p) = mu + sigma log p/(1 - p) = mu + sigma logit(p) + * isf(p) = mu + sigma log (1 - p)/p = mu - sigma logit(p) + */ + +/** + * Compute the CDF of the Logistic(mu, sigma) distribution: the + * logistic function. Well-conditioned for negative inputs and small + * positive inputs; ill-conditioned for large positive inputs. + */ +STATIC double +cdf_logistic(double x, double mu, double sigma) +{ + return logistic((x - mu)/sigma); +} + +/** + * Compute the SF of the Logistic(mu, sigma) distribution: the logistic + * function reflected over the y axis. Well-conditioned for positive + * inputs and small negative inputs; ill-conditioned for large negative + * inputs. + */ +STATIC double +sf_logistic(double x, double mu, double sigma) +{ + return logistic(-(x - mu)/sigma); +} + +/** + * Compute the inverse of the CDF of the Logistic(mu, sigma) + * distribution: the logit function. Well-conditioned near 0; + * ill-conditioned near 1/2 and 1. + */ +STATIC double +icdf_logistic(double p, double mu, double sigma) +{ + return mu + sigma*logit(p); +} + +/** + * Compute the inverse of the SF of the Logistic(mu, sigma) + * distribution: the -logit function. Well-conditioned near 0; + * ill-conditioned near 1/2 and 1. + */ +STATIC double +isf_logistic(double p, double mu, double sigma) +{ + return mu - sigma*logit(p); +} + +/* + * LogLogistic(alpha, beta) distribution, supported on (0, +infinity). + * + * This is the uniform distribution on [0,1] mapped into odds space, + * scaled by positive alpha and shaped by positive beta. + * + * Equivalent to computing exp of a Logistic(log alpha, 1/beta) sample. + * (Name arises because the pdf has LogLogistic(x; alpha, beta) = + * Logistic(log x; log alpha, 1/beta) and mathematicians got their + * covariance contravariant.) + * + * pdf(x) = (beta/alpha) (x/alpha)^{beta - 1}/(1 + (x/alpha)^beta)^2 + * = (1/e^mu sigma) (x/e^mu)^{1/sigma - 1} / + * (1 + (x/e^mu)^{1/sigma})^2 + * cdf(x) = 1/(1 + (x/alpha)^-beta) = 1/(1 + (x/e^mu)^{-1/sigma}) + * = 1/(1 + (e^{log x}/e^mu)^{-1/sigma}) + * = 1/(1 + (e^{log x - mu})^{-1/sigma}) + * = 1/(1 + e^{-(log x - mu)/sigma}) + * = logistic((log x - mu)/sigma) + * = logistic((log x - log alpha)/(1/beta)) + * sf(x) = 1 - 1/(1 + (x/alpha)^-beta) + * = (x/alpha)^-beta/(1 + (x/alpha)^-beta) + * = 1/((x/alpha)^beta + 1) + * = 1/(1 + (x/alpha)^beta) + * icdf(p) = alpha (p/(1 - p))^{1/beta} + * = alpha e^{logit(p)/beta} + * = e^{mu + sigma logit(p)} + * isf(p) = alpha ((1 - p)/p)^{1/beta} + * = alpha e^{-logit(p)/beta} + * = e^{mu - sigma logit(p)} + */ + +/** + * Compute the CDF of the LogLogistic(alpha, beta) distribution. + * Well-conditioned for all x and alpha, and the condition number + * + * -beta/[1 + (x/alpha)^{-beta}] + * + * grows linearly with beta. + * + * Loosely, the relative error of this implementation is bounded by + * + * 4 eps + 2 eps^2 + O(beta eps), + * + * so don't bother trying this for beta anywhere near as large as + * 1/eps, around which point it levels off at 1. + */ +STATIC double +cdf_log_logistic(double x, double alpha, double beta) +{ + /* + * Let d0 be the error of x/alpha; d1, of pow; d2, of +; and + * d3, of the final quotient. The exponentiation gives + * + * ((1 + d0) x/alpha)^{-beta} + * = (x/alpha)^{-beta} (1 + d0)^{-beta} + * = (x/alpha)^{-beta} (1 + (1 + d0)^{-beta} - 1) + * = (x/alpha)^{-beta} (1 + d') + * + * where d' = (1 + d0)^{-beta} - 1. If y = (x/alpha)^{-beta}, + * the denominator is + * + * (1 + d2) (1 + (1 + d1) (1 + d') y) + * = (1 + d2) (1 + y + (d1 + d' + d1 d') y) + * = 1 + y + (1 + d2) (d1 + d' + d1 d') y + * = (1 + y) (1 + (1 + d2) (d1 + d' + d1 d') y/(1 + y)) + * = (1 + y) (1 + d''), + * + * where d'' = (1 + d2) (d1 + d' + d1 d') y/(1 + y). The + * final result is + * + * (1 + d3) / [(1 + d2) (1 + d'') (1 + y)] + * = (1 + d''') / (1 + y) + * + * for |d'''| <= 2|d3 - d''| by Lemma 2 as long as |d''| < 1/2 + * (which may not be the case for very large beta). This + * relative error is therefore bounded by + * + * |d'''| + * <= 2|d3 - d''| + * <= 2|d3| + 2|(1 + d2) (d1 + d' + d1 d') y/(1 + y)| + * <= 2|d3| + 2|(1 + d2) (d1 + d' + d1 d')| + * = 2|d3| + 2|d1 + d' + d1 d' + d2 d1 + d2 d' + d2 d1 d'| + * <= 2|d3| + 2|d1| + 2|d'| + 2|d1 d'| + 2|d2 d1| + 2|d2 d'| + * + 2|d2 d1 d'| + * <= 4 eps + 2 eps^2 + (2 + 2 eps + 2 eps^2) |d'|. + * + * Roughly, |d'| = |(1 + d0)^{-beta} - 1| grows like beta eps, + * until it levels off at 1. + */ + return 1/(1 + pow(x/alpha, -beta)); +} + +/** + * Compute the SF of the LogLogistic(alpha, beta) distribution. + * Well-conditioned for all x and alpha, and the condition number + * + * beta/[1 + (x/alpha)^beta] + * + * grows linearly with beta. + * + * Loosely, the relative error of this implementation is bounded by + * + * 4 eps + 2 eps^2 + O(beta eps) + * + * so don't bother trying this for beta anywhere near as large as + * 1/eps, beyond which point it grows unbounded. + */ +STATIC double +sf_log_logistic(double x, double alpha, double beta) +{ + /* + * The error analysis here is essentially the same as in + * cdf_log_logistic, except that rather than levelling off at + * 1, |(1 + d0)^beta - 1| grows unbounded. + */ + return 1/(1 + pow(x/alpha, beta)); +} + +/** + * Compute the inverse of the CDF of the LogLogistic(alpha, beta) + * distribution. Ill-conditioned for p near 1 and beta near 0 with + * condition number 1/[beta (1 - p)]. + */ +STATIC double +icdf_log_logistic(double p, double alpha, double beta) +{ + return alpha*pow(p/(1 - p), 1/beta); +} + +/** + * Compute the inverse of the SF of the LogLogistic(alpha, beta) + * distribution. Ill-conditioned for p near 1 and for large beta, with + * condition number -1/[beta (1 - p)]. + */ +STATIC double +isf_log_logistic(double p, double alpha, double beta) +{ + return alpha*pow((1 - p)/p, 1/beta); +} + +/* + * Weibull(lambda, k) distribution, supported on (0, +infinity). + * + * pdf(x) = (k/lambda) (x/lambda)^{k - 1} e^{-(x/lambda)^k} + * cdf(x) = 1 - e^{-(x/lambda)^k} + * icdf(p) = lambda * (-log (1 - p))^{1/k} + * sf(x) = e^{-(x/lambda)^k} + * isf(p) = lambda * (-log p)^{1/k} + */ + +/** + * Compute the CDF of the Weibull(lambda, k) distribution. + * Well-conditioned for small x and k, and for large lambda -- + * condition number + * + * -k (x/lambda)^k exp(-(x/lambda)^k)/[exp(-(x/lambda)^k) - 1] + * + * grows linearly with k, x^k, and lambda^{-k}. + */ +STATIC double +cdf_weibull(double x, double lambda, double k) +{ + return -expm1(-pow(x/lambda, k)); +} + +/** + * Compute the SF of the Weibull(lambda, k) distribution. + * Well-conditioned for small x and k, and for large lambda -- + * condition number + * + * -k (x/lambda)^k + * + * grows linearly with k, x^k, and lambda^{-k}. + */ +STATIC double +sf_weibull(double x, double lambda, double k) +{ + return exp(-pow(x/lambda, k)); +} + +/** + * Compute the inverse of the CDF of the Weibull(lambda, k) + * distribution. Ill-conditioned for p near 1, and for k near 0; + * condition number is + * + * (p/(1 - p))/(k log(1 - p)). + */ +STATIC double +icdf_weibull(double p, double lambda, double k) +{ + return lambda*pow(-log1p(-p), 1/k); +} + +/** + * Compute the inverse of the SF of the Weibull(lambda, k) + * distribution. Ill-conditioned for p near 0, and for k near 0; + * condition number is + * + * 1/(k log(p)). + */ +STATIC double +isf_weibull(double p, double lambda, double k) +{ + return lambda*pow(-log(p), 1/k); +} + +/* + * GeneralizedPareto(mu, sigma, xi), supported on (mu, +infinity) for + * nonnegative xi, or (mu, mu - sigma/xi) for negative xi. + * + * Samples: + * = mu - sigma log U, if xi = 0; + * = mu + sigma (U^{-xi} - 1)/xi = mu + sigma*expm1(-xi log U)/xi, if xi =/= 0, + * where U is uniform on (0,1]. + * = mu + sigma (e^{xi X} - 1)/xi, + * where X has standard exponential distribution. + * + * pdf(x) = sigma^{-1} (1 + xi (x - mu)/sigma)^{-(1 + 1/xi)} + * cdf(x) = 1 - (1 + xi (x - mu)/sigma)^{-1/xi} + * = 1 - e^{-log(1 + xi (x - mu)/sigma)/xi} + * --> 1 - e^{-(x - mu)/sigma} as xi --> 0 + * sf(x) = (1 + xi (x - mu)/sigma)^{-1/xi} + * --> e^{-(x - mu)/sigma} as xi --> 0 + * icdf(p) = mu + sigma*(p^{-xi} - 1)/xi + * = mu + sigma*expm1(-xi log p)/xi + * --> mu + sigma*log p as xi --> 0 + * isf(p) = mu + sigma*((1 - p)^{xi} - 1)/xi + * = mu + sigma*expm1(-xi log1p(-p))/xi + * --> mu + sigma*log1p(-p) as xi --> 0 + */ + +/** + * Compute the CDF of the GeneralizedPareto(mu, sigma, xi) + * distribution. Well-conditioned everywhere. For standard + * distribution (mu=0, sigma=1), condition number + * + * (x/(1 + x xi)) / ((1 + x xi)^{1/xi} - 1) + * + * is bounded by 1, attained only at x = 0. + */ +STATIC double +cdf_genpareto(double x, double mu, double sigma, double xi) +{ + double x_0 = (x - mu)/sigma; + + /* + * log(1 + xi x_0)/xi + * = (-1/xi) \sum_{n=1}^infinity (-xi x_0)^n/n + * = (-1/xi) (-xi x_0 + \sum_{n=2}^infinity (-xi x_0)^n/n) + * = x_0 - (1/xi) \sum_{n=2}^infinity (-xi x_0)^n/n + * = x_0 - x_0 \sum_{n=2}^infinity (-xi x_0)^{n-1}/n + * = x_0 (1 - d), + * + * where d = \sum_{n=2}^infinity (-xi x_0)^{n-1}/n. If |xi| < + * eps/4|x_0|, then + * + * |d| <= \sum_{n=2}^infinity (eps/4)^{n-1}/n + * <= \sum_{n=2}^infinity (eps/4)^{n-1} + * = \sum_{n=1}^infinity (eps/4)^n + * = (eps/4) \sum_{n=0}^infinity (eps/4)^n + * = (eps/4)/(1 - eps/4) + * < eps/2 + * + * for any 0 < eps < 2. Thus, the relative error of x_0 from + * log(1 + xi x_0)/xi is bounded by eps. + */ + if (fabs(xi) < 1e-17/x_0) + return -expm1(-x_0); + else + return -expm1(-log1p(xi*x_0)/xi); +} + +/** + * Compute the SF of the GeneralizedPareto(mu, sigma, xi) distribution. + * For standard distribution (mu=0, sigma=1), ill-conditioned for xi + * near 0; condition number + * + * -x (1 + x xi)^{(-1 - xi)/xi}/(1 + x xi)^{-1/xi} + * = -x (1 + x xi)^{-1/xi - 1}/(1 + x xi)^{-1/xi} + * = -(x/(1 + x xi)) (1 + x xi)^{-1/xi}/(1 + x xi)^{-1/xi} + * = -x/(1 + x xi) + * + * is bounded by 1/xi. + */ +STATIC double +sf_genpareto(double x, double mu, double sigma, double xi) +{ + double x_0 = (x - mu)/sigma; + + if (fabs(xi) < 1e-17/x_0) + return exp(-x_0); + else + return exp(-log1p(xi*x_0)/xi); +} + +/** + * Compute the inverse of the CDF of the GeneralizedPareto(mu, sigma, + * xi) distribution. Ill-conditioned for p near 1; condition number is + * + * xi (p/(1 - p))/(1 - (1 - p)^xi) + */ +STATIC double +icdf_genpareto(double p, double mu, double sigma, double xi) +{ + /* + * To compute f(xi) = (U^{-xi} - 1)/xi = (e^{-xi log U} - 1)/xi + * for xi near zero (note f(xi) --> -log U as xi --> 0), write + * the absolutely convergent Taylor expansion + * + * f(xi) = (1/xi)*(-xi log U + \sum_{n=2}^infinity (-xi log U)^n/n! + * = -log U + (1/xi)*\sum_{n=2}^infinity (-xi log U)^n/n! + * = -log U + \sum_{n=2}^infinity xi^{n-1} (-log U)^n/n! + * = -log U - log U \sum_{n=2}^infinity (-xi log U)^{n-1}/n! + * = -log U (1 + \sum_{n=2}^infinity (-xi log U)^{n-1}/n!). + * + * Let d = \sum_{n=2}^infinity (-xi log U)^{n-1}/n!. What do we + * lose if we discard it and use -log U as an approximation to + * f(xi)? If |xi| < eps/-4log U, then + * + * |d| <= \sum_{n=2}^infinity |xi log U|^{n-1}/n! + * <= \sum_{n=2}^infinity (eps/4)^{n-1}/n! + * <= \sum_{n=1}^infinity (eps/4)^n + * = (eps/4) \sum_{n=0}^infinity (eps/4)^n + * = (eps/4)/(1 - eps/4) + * < eps/2, + * + * for any 0 < eps < 2. Hence, as long as |xi| < eps/-2log U, + * f(xi) = -log U (1 + d) for |d| <= eps/2. |d| is the + * relative error of f(xi) from -log U; from this bound, the + * relative error of -log U from f(xi) is at most (eps/2)/(1 - + * eps/2) = eps/2 + (eps/2)^2 + (eps/2)^3 + ... < eps for 0 < + * eps < 1. Since -log U < 1000 for all U in (0, 1] in + * binary64 floating-point, we can safely cut xi off at 1e-20 < + * eps/4000 and attain <1ulp error from series truncation. + */ + if (fabs(xi) <= 1e-20) + return mu - sigma*log1p(-p); + else + return mu + sigma*expm1(-xi*log1p(-p))/xi; +} + +/** + * Compute the inverse of the SF of the GeneralizedPareto(mu, sigma, + * xi) distribution. Ill-conditioned for p near 1; conditon number is + * + * -xi/(1 - p^{-xi}) + */ +STATIC double +isf_genpareto(double p, double mu, double sigma, double xi) +{ + if (fabs(xi) <= 1e-20) + return mu - sigma*log(p); + else + return mu + sigma*expm1(-xi*log(p))/xi; +} + +/*******************************************************************/ + +/** + * Deterministic samplers, parametrized by uniform integer and (0,1] + * samples. No guarantees are made about _which_ mapping from the + * integer and (0,1] samples these use; all that is guaranteed is the + * distribution of the outputs conditioned on a uniform distribution on + * the inputs. The automatic tests in test_prob_distr.c double-check + * the particular mappings we use. + * + * Beware: Unlike random_uniform_01(), these are not guaranteed to be + * supported on all possible outputs. See Ilya Mironov, `On the + * Significance of the Least Significant Bits for Differential + * Privacy', for an example of what can go wrong if you try to use + * these to conceal information from an adversary but you expose the + * specific full-precision floating-point values. + * + * Note: None of these samplers use rejection sampling; they are all + * essentially inverse-CDF transforms with tweaks. If you were to add, + * say, a Gamma sampler with the Marsaglia-Tsang method, you would have + * to parametrize it by a potentially infinite stream of uniform (and + * perhaps normal) samples rather than a fixed number, which doesn't + * make for quite as nice automatic testing as for these. + */ + +/** + * Deterministically sample from the interval [a, b], indexed by a + * uniform random floating-point number p0 in (0, 1]. + * + * Note that even if p0 is nonzero, the result may be equal to a, if + * ulp(a)/2 is nonnegligible, e.g. if a = 1. For maximum resolution, + * arrange |a| <= |b|. + */ +STATIC double +sample_uniform_interval(double p0, double a, double b) +{ + /* + * XXX Prove that the distribution is, in fact, uniform on + * [a,b], particularly around p0 = 1, or at least has very + * small deviation from uniform, quantified appropriately + * (e.g., like in Monahan 1984, or by KL divergence). It + * almost certainly does but it would be nice to quantify the + * error. + */ + if ((a <= 0 && 0 <= b) || (b <= 0 && 0 <= a)) { + /* + * When ab < 0, (1 - t) a + t b is monotonic, since for + * a <= b it is a sum of nondecreasing functions of t, + * and for b <= a, of nonincreasing functions of t. + * Further, clearly at 0 and 1 it attains a and b, + * respectively. Hence it is bounded within [a, b]. + */ + return (1 - p0)*a + p0*b; + } else { + /* + * a + (b - a) t is monotonic -- it is obviously a + * nondecreasing function of t for a <= b. Further, it + * attains a at 0, and while it may overshoot b at 1, + * we have a + * + * Theorem. If 0 <= t < 1, then the floating-point + * evaluation of a + (b - a) t is bounded in [a, b]. + * + * Lemma 1. If 0 <= t < 1 is a floating-point number, + * then for any normal floating-point number x except + * the smallest in magnitude, |round(x*t)| < |x|. + * + * Proof. WLOG, assume x >= 0. Since the rounding + * function and t |---> x*t are nondecreasing, their + * composition t |---> round(x*t) is also + * nondecreasing, so it suffices to consider the + * largest floating-point number below 1, in particular + * t = 1 - ulp(1)/2. + * + * Case I: If x is a power of two, then the next + * floating-point number below x is x - ulp(x)/2 = x - + * x*ulp(1)/2 = x*(1 - ulp(1)/2) = x*t, so, since x*t + * is a floating-point number, multiplication is exact, + * and thus round(x*t) = x*t < x. + * + * Case II: If x is not a power of two, then the + * greatest lower bound of real numbers rounded to x is + * x - ulp(x)/2 = x - ulp(T(x))/2 = x - T(x)*ulp(1)/2, + * where T(X) is the largest power of two below x. + * Anything below this bound is rounded to a + * floating-point number smaller than x, and x*t = x*(1 + * - ulp(1)/2) = x - x*ulp(1)/2 < x - T(x)*ulp(1)/2 + * since T(x) < x, so round(x*t) < x*t < x. QED. + * + * Lemma 2. If x and y are subnormal, then round(x + + * y) = x + y. + * + * Proof. It is a matter of adding the significands, + * since if we treat subnormals as having an implicit + * zero bit before the `binary' point, their exponents + * are all the same. There is at most one carry/borrow + * bit, which can always be acommodated either in a + * subnormal, or, at largest, in the implicit one bit + * of a normal. + * + * Lemma 3. Let x and y be floating-point numbers. If + * round(x - y) is subnormal or zero, then it is equal + * to x - y. + * + * Proof. Case I (equal): round(x - y) = 0 iff x = y; + * hence if round(x - y) = 0, then round(x - y) = 0 = x + * - y. + * + * Case II (subnormal/subnormal): If x and y are both + * subnormal, this follows directly from Lemma 2. + * + * Case IIIa (normal/subnormal): If x is normal and y + * is subnormal, then x and y must share sign, or else + * x - y would be larger than x and thus rounded to + * normal. If s is the smallest normal positive + * floating-point number, |x| < 2s since by + * construction 2s - |y| is normal for all subnormal y. + * This means that x and y must have the same exponent, + * so the difference is the difference of significands, + * which is exact. + * + * Case IIIb (subnormal/normal): Same as case IIIa for + * -(y - x). + * + * Case IV (normal/normal): If x and y are both normal, + * then they must share sign, or else x - y would be + * larger than x and thus rounded to normal. Note that + * |y| < 2|x|, for if |y| >= 2|x|, then |x| - |y| <= + * -|x| but -|x| is normal like x. Also, |x|/2 < |y|: + * if |x|/2 is subnormal, it must hold because y is + * normal; if |x|/2 is normal, then |x|/2 >= s, so + * since |x| - |y| < s, + * + * |x|/2 = |x| - |x|/2 <= |x| - s <= |y|; + * + * that is, |x|/2 < |y| < 2|x|, so by the Sterbenz + * lemma, round(x - y) = x - y. QED. + * + * Proof of theorem. WLOG, assume 0 <= a <= b. Since + * round(a + round(round(b - a)*t) is nondecreasing in + * t and attains a at 0, the lower end of the bound is + * trivial; we must show the upper end of the bound + * strictly. It suffices to show this for the largest + * floating-point number below 1, namely 1 - ulp(1)/2. + * + * Case I: round(b - a) is normal. Then it is at most + * the smallest floating-point number above b - a. By + * Lemma 1, round(round(b - a)*t) < round(b - a). + * Since the inequality is strict, and since + * round(round(b - a)*t) is a floating-point number + * below round(b - a), and since there are no + * floating-point numbers between b - a and round(b - + * a), we must have round(round(b - a)*t) < b - a. + * Then since y |---> round(a + y) is nondecreasing, we + * must have + * + * round(a + round(round(b - a)*t)) + * <= round(a + (b - a)) + * = round(b) = b. + * + * Case II: round(b - a) is subnormal. In this case, + * Lemma 1 falls apart -- we are not guaranteed the + * strict inequality. However, by Lemma 3, the + * difference is exact: round(b - a) = b - a. Thus, + * + * round(a + round(round(b - a)*t)) + * <= round(a + round((b - a)*t)) + * <= round(a + (b - a)) + * = round(b) + * = b, + * + * QED. + */ + + /* p0 is restricted to [0,1], but we use >= to silence -Wfloat-equal. */ + if (p0 >= 1) + return b; + return a + (b - a)*p0; + } +} + +/** + * Deterministically sample from the standard logistic distribution, + * indexed by a uniform random 32-bit integer s and uniform random + * floating-point numbers t and p0 in (0, 1]. + */ +STATIC double +sample_logistic(uint32_t s, double t, double p0) +{ + double sign = (s & 1) ? -1 : +1; + double r; + + /* + * We carve up the interval (0, 1) into subregions to compute + * the inverse CDF precisely: + * + * A = (0, 1/(1 + e)] ---> (-infinity, -1] + * B = [1/(1 + e), 1/2] ---> [-1, 0] + * C = [1/2, 1 - 1/(1 + e)] ---> [0, 1] + * D = [1 - 1/(1 + e), 1) ---> [1, +infinity) + * + * Cases D and C are mirror images of cases A and B, + * respectively, so we choose between them by the sign chosen + * by a fair coin toss. We choose between cases A and B by a + * coin toss weighted by + * + * 2/(1 + e) = 1 - [1/2 - 1/(1 + e)]/(1/2): + * + * if it comes up heads, scale p0 into a uniform (0, 1/(1 + e)] + * sample p; if it comes up tails, scale p0 into a uniform (0, + * 1/2 - 1/(1 + e)] sample and compute the inverse CDF of p = + * 1/2 - p0. + */ + if (t <= 2/(1 + exp(1))) { + /* p uniform in (0, 1/(1 + e)], represented by p. */ + p0 /= 1 + exp(1); + r = logit(p0); + } else { + /* + * p uniform in [1/(1 + e), 1/2), actually represented + * by p0 = 1/2 - p uniform in (0, 1/2 - 1/(1 + e)], so + * that p = 1/2 - p. + */ + p0 *= 0.5 - 1/(1 + exp(1)); + r = logithalf(p0); + } + + /* + * We have chosen from the negative half of the standard + * logistic distribution, which is symmetric with the positive + * half. Now use the sign to choose uniformly between them. + */ + return sign*r; +} + +/** + * Deterministically sample from the logistic distribution scaled by + * sigma and translated by mu. + */ +static double +sample_logistic_locscale(uint32_t s, double t, double p0, double mu, + double sigma) +{ + + return mu + sigma*sample_logistic(s, t, p0); +} + +/** + * Deterministically sample from the standard log-logistic + * distribution, indexed by a uniform random 32-bit integer s and a + * uniform random floating-point number p0 in (0, 1]. + */ +STATIC double +sample_log_logistic(uint32_t s, double p0) +{ + + /* + * Carve up the interval (0, 1) into (0, 1/2] and [1/2, 1); the + * condition numbers of the icdf and the isf coincide at 1/2. + */ + p0 *= 0.5; + if ((s & 1) == 0) { + /* p = p0 in (0, 1/2] */ + return p0/(1 - p0); + } else { + /* p = 1 - p0 in [1/2, 1) */ + return (1 - p0)/p0; + } +} + +/** + * Deterministically sample from the log-logistic distribution with + * scale alpha and shape beta. + */ +static double +sample_log_logistic_scaleshape(uint32_t s, double p0, double alpha, + double beta) +{ + double x = sample_log_logistic(s, p0); + + return alpha*pow(x, 1/beta); +} + +/** + * Deterministically sample from the standard exponential distribution, + * indexed by a uniform random 32-bit integer s and a uniform random + * floating-point number p0 in (0, 1]. + */ +static double +sample_exponential(uint32_t s, double p0) +{ + /* + * We would like to evaluate log(p) for p near 0, and log1p(-p) + * for p near 1. Simply carve the interval into (0, 1/2] and + * [1/2, 1) by a fair coin toss. + */ + p0 *= 0.5; + if ((s & 1) == 0) + /* p = p0 in (0, 1/2] */ + return -log(p0); + else + /* p = 1 - p0 in [1/2, 1) */ + return -log1p(-p0); +} + +/** + * Deterministically sample from a Weibull distribution with scale + * lambda and shape k -- just an exponential with a shape parameter in + * addition to a scale parameter. (Yes, lambda really is the scale, + * _not_ the rate.) + */ +STATIC double +sample_weibull(uint32_t s, double p0, double lambda, double k) +{ + + return lambda*pow(sample_exponential(s, p0), 1/k); +} + +/** + * Deterministically sample from the generalized Pareto distribution + * with shape xi, indexed by a uniform random 32-bit integer s and a + * uniform random floating-point number p0 in (0, 1]. + */ +STATIC double +sample_genpareto(uint32_t s, double p0, double xi) +{ + double x = sample_exponential(s, p0); + + /* + * Write f(xi) = (e^{xi x} - 1)/xi for xi near zero as the + * absolutely convergent Taylor series + * + * f(x) = (1/xi) (xi x + \sum_{n=2}^infinity (xi x)^n/n!) + * = x + (1/xi) \sum_{n=2}^infinity (xi x)^n/n! + * = x + \sum_{n=2}^infinity xi^{n-1} x^n/n! + * = x + x \sum_{n=2}^infinity (xi x)^{n-1}/n! + * = x (1 + \sum_{n=2}^infinity (xi x)^{n-1}/n!). + * + * d = \sum_{n=2}^infinity (xi x)^{n-1}/n! is the relative error + * of f(x) from x. If |xi| < eps/4x, then + * + * |d| <= \sum_{n=2}^infinity |xi x|^{n-1}/n! + * <= \sum_{n=2}^infinity (eps/4)^{n-1}/n! + * <= \sum_{n=1}^infinity (eps/4) + * = (eps/4) \sum_{n=0}^infinity (eps/4)^n + * = (eps/4)/(1 - eps/4) + * < eps/2, + * + * for any 0 < eps < 2. Hence, as long as |xi| < eps/2x, f(xi) + * = x (1 + d) for |d| <= eps/2, so x = f(xi) (1 + d') for |d'| + * <= eps. What bound should we use for x? + * + * - If x is exponentially distributed, x > 200 with + * probability below e^{-200} << 2^{-256}, i.e. never. + * + * - If x is computed by -log(U) for U in (0, 1], x is + * guaranteed to be below 1000 in IEEE 754 binary64 + * floating-point. + * + * We can safely cut xi off at 1e-20 < eps/4000 and attain an + * error bounded by 0.5 ulp for this expression. + */ + return (fabs(xi) < 1e-20 ? x : expm1(xi*x)/xi); +} + +/** + * Deterministically sample from a generalized Pareto distribution with + * shape xi, scaled by sigma and translated by mu. + */ +static double +sample_genpareto_locscale(uint32_t s, double p0, double mu, double sigma, + double xi) +{ + + return mu + sigma*sample_genpareto(s, p0, xi); +} + +/** + * Deterministically sample from the geometric distribution with + * per-trial success probability p. + * + * XXX Quantify the error (KL divergence?) of this + * ceiling-of-exponential sampler from a true geometric distribution, + * which we could get by rejection sampling. Relevant papers: + * + * John F. Monahan, `Accuracy in Random Number Generation', + * Mathematics of Computation 45(172), October 1984, pp. 559--568. +*https://pdfs.semanticscholar.org/aca6/74b96da1df77b2224e8cfc5dd6d61a471632.pdf + * + * Karl Bringmann and Tobias Friedrich, `Exact and Efficient + * Generation of Geometric Random Variates and Random Graphs', in + * Proceedings of the 40th International Colloaquium on Automata, + * Languages, and Programming -- ICALP 2013, Springer LNCS 7965, + * pp.267--278. + * https://doi.org/10.1007/978-3-642-39206-1_23 + * https://people.mpi-inf.mpg.de/~kbringma/paper/2013ICALP-1.pdf + */ +static double +sample_geometric(uint32_t s, double p0, double p) +{ + double x = sample_exponential(s, p0); + + /* This is actually a check against 1, but we do >= so that the compiler + does not raise a -Wfloat-equal */ + if (p >= 1) + return 1; + + return ceil(-x/log1p(-p)); +} + +/*******************************************************************/ + +/** Public API for probability distributions: + * + * These are wrapper functions on top of the various probability distribution + * operations using the generic <b>dist</b> structure. + + * These are the functions that should be used by consumers of this API. + */ + +/** Returns the name of the distribution in <b>dist</b>. */ +const char * +dist_name(const struct dist_t *dist) +{ + return dist->ops->name; +} + +/* Sample a value from <b>dist</b> and return it. */ +double +dist_sample(const struct dist_t *dist) +{ + return dist->ops->sample(dist); +} + +/** Compute the CDF of <b>dist</b> at <b>x</b>. */ +double +dist_cdf(const struct dist_t *dist, double x) +{ + return dist->ops->cdf(dist, x); +} + +/** Compute the SF (Survival function) of <b>dist</b> at <b>x</b>. */ +double +dist_sf(const struct dist_t *dist, double x) +{ + return dist->ops->sf(dist, x); +} + +/** Compute the iCDF (Inverse CDF) of <b>dist</b> at <b>x</b>. */ +double +dist_icdf(const struct dist_t *dist, double p) +{ + return dist->ops->icdf(dist, p); +} + +/** Compute the iSF (Inverse Survival function) of <b>dist</b> at <b>x</b>. */ +double +dist_isf(const struct dist_t *dist, double p) +{ + return dist->ops->isf(dist, p); +} + +/** Functions for uniform distribution */ + +static double +uniform_sample(const struct dist_t *dist) +{ + const struct uniform_t *U = dist_to_const_uniform(dist); + double p0 = random_uniform_01(); + + return sample_uniform_interval(p0, U->a, U->b); +} + +static double +uniform_cdf(const struct dist_t *dist, double x) +{ + const struct uniform_t *U = dist_to_const_uniform(dist); + if (x < U->a) + return 0; + else if (x < U->b) + return (x - U->a)/(U->b - U->a); + else + return 1; +} + +static double +uniform_sf(const struct dist_t *dist, double x) +{ + const struct uniform_t *U = dist_to_const_uniform(dist); + + if (x > U->b) + return 0; + else if (x > U->a) + return (U->b - x)/(U->b - U->a); + else + return 1; +} + +static double +uniform_icdf(const struct dist_t *dist, double p) +{ + const struct uniform_t *U = dist_to_const_uniform(dist); + double w = U->b - U->a; + + return (p < 0.5 ? (U->a + w*p) : (U->b - w*(1 - p))); +} + +static double +uniform_isf(const struct dist_t *dist, double p) +{ + const struct uniform_t *U = dist_to_const_uniform(dist); + double w = U->b - U->a; + + return (p < 0.5 ? (U->b - w*p) : (U->a + w*(1 - p))); +} + +const struct dist_ops_t uniform_ops = { + .name = "uniform", + .sample = uniform_sample, + .cdf = uniform_cdf, + .sf = uniform_sf, + .icdf = uniform_icdf, + .isf = uniform_isf, +}; + +/*******************************************************************/ + +/** Private functions for each probability distribution. */ + +/** Functions for logistic distribution: */ + +static double +logistic_sample(const struct dist_t *dist) +{ + const struct logistic_t *L = dist_to_const_logistic(dist); + uint32_t s = crypto_fast_rng_get_u32(get_thread_fast_rng()); + double t = random_uniform_01(); + double p0 = random_uniform_01(); + + return sample_logistic_locscale(s, t, p0, L->mu, L->sigma); +} + +static double +logistic_cdf(const struct dist_t *dist, double x) +{ + const struct logistic_t *L = dist_to_const_logistic(dist); + return cdf_logistic(x, L->mu, L->sigma); +} + +static double +logistic_sf(const struct dist_t *dist, double x) +{ + const struct logistic_t *L = dist_to_const_logistic(dist); + return sf_logistic(x, L->mu, L->sigma); +} + +static double +logistic_icdf(const struct dist_t *dist, double p) +{ + const struct logistic_t *L = dist_to_const_logistic(dist); + return icdf_logistic(p, L->mu, L->sigma); +} + +static double +logistic_isf(const struct dist_t *dist, double p) +{ + const struct logistic_t *L = dist_to_const_logistic(dist); + return isf_logistic(p, L->mu, L->sigma); +} + +const struct dist_ops_t logistic_ops = { + .name = "logistic", + .sample = logistic_sample, + .cdf = logistic_cdf, + .sf = logistic_sf, + .icdf = logistic_icdf, + .isf = logistic_isf, +}; + +/** Functions for log-logistic distribution: */ + +static double +log_logistic_sample(const struct dist_t *dist) +{ + const struct log_logistic_t *LL = dist_to_const_log_logistic(dist); + uint32_t s = crypto_fast_rng_get_u32(get_thread_fast_rng()); + double p0 = random_uniform_01(); + + return sample_log_logistic_scaleshape(s, p0, LL->alpha, LL->beta); +} + +static double +log_logistic_cdf(const struct dist_t *dist, double x) +{ + const struct log_logistic_t *LL = dist_to_const_log_logistic(dist); + return cdf_log_logistic(x, LL->alpha, LL->beta); +} + +static double +log_logistic_sf(const struct dist_t *dist, double x) +{ + const struct log_logistic_t *LL = dist_to_const_log_logistic(dist); + return sf_log_logistic(x, LL->alpha, LL->beta); +} + +static double +log_logistic_icdf(const struct dist_t *dist, double p) +{ + const struct log_logistic_t *LL = dist_to_const_log_logistic(dist); + return icdf_log_logistic(p, LL->alpha, LL->beta); +} + +static double +log_logistic_isf(const struct dist_t *dist, double p) +{ + const struct log_logistic_t *LL = dist_to_const_log_logistic(dist); + return isf_log_logistic(p, LL->alpha, LL->beta); +} + +const struct dist_ops_t log_logistic_ops = { + .name = "log logistic", + .sample = log_logistic_sample, + .cdf = log_logistic_cdf, + .sf = log_logistic_sf, + .icdf = log_logistic_icdf, + .isf = log_logistic_isf, +}; + +/** Functions for Weibull distribution */ + +static double +weibull_sample(const struct dist_t *dist) +{ + const struct weibull_t *W = dist_to_const_weibull(dist); + uint32_t s = crypto_fast_rng_get_u32(get_thread_fast_rng()); + double p0 = random_uniform_01(); + + return sample_weibull(s, p0, W->lambda, W->k); +} + +static double +weibull_cdf(const struct dist_t *dist, double x) +{ + const struct weibull_t *W = dist_to_const_weibull(dist); + return cdf_weibull(x, W->lambda, W->k); +} + +static double +weibull_sf(const struct dist_t *dist, double x) +{ + const struct weibull_t *W = dist_to_const_weibull(dist); + return sf_weibull(x, W->lambda, W->k); +} + +static double +weibull_icdf(const struct dist_t *dist, double p) +{ + const struct weibull_t *W = dist_to_const_weibull(dist); + return icdf_weibull(p, W->lambda, W->k); +} + +static double +weibull_isf(const struct dist_t *dist, double p) +{ + const struct weibull_t *W = dist_to_const_weibull(dist); + return isf_weibull(p, W->lambda, W->k); +} + +const struct dist_ops_t weibull_ops = { + .name = "Weibull", + .sample = weibull_sample, + .cdf = weibull_cdf, + .sf = weibull_sf, + .icdf = weibull_icdf, + .isf = weibull_isf, +}; + +/** Functions for generalized Pareto distributions */ + +static double +genpareto_sample(const struct dist_t *dist) +{ + const struct genpareto_t *GP = dist_to_const_genpareto(dist); + uint32_t s = crypto_fast_rng_get_u32(get_thread_fast_rng()); + double p0 = random_uniform_01(); + + return sample_genpareto_locscale(s, p0, GP->mu, GP->sigma, GP->xi); +} + +static double +genpareto_cdf(const struct dist_t *dist, double x) +{ + const struct genpareto_t *GP = dist_to_const_genpareto(dist); + return cdf_genpareto(x, GP->mu, GP->sigma, GP->xi); +} + +static double +genpareto_sf(const struct dist_t *dist, double x) +{ + const struct genpareto_t *GP = dist_to_const_genpareto(dist); + return sf_genpareto(x, GP->mu, GP->sigma, GP->xi); +} + +static double +genpareto_icdf(const struct dist_t *dist, double p) +{ + const struct genpareto_t *GP = dist_to_const_genpareto(dist); + return icdf_genpareto(p, GP->mu, GP->sigma, GP->xi); +} + +static double +genpareto_isf(const struct dist_t *dist, double p) +{ + const struct genpareto_t *GP = dist_to_const_genpareto(dist); + return isf_genpareto(p, GP->mu, GP->sigma, GP->xi); +} + +const struct dist_ops_t genpareto_ops = { + .name = "generalized Pareto", + .sample = genpareto_sample, + .cdf = genpareto_cdf, + .sf = genpareto_sf, + .icdf = genpareto_icdf, + .isf = genpareto_isf, +}; + +/** Functions for geometric distribution on number of trials before success */ + +static double +geometric_sample(const struct dist_t *dist) +{ + const struct geometric_t *G = dist_to_const_geometric(dist); + uint32_t s = crypto_fast_rng_get_u32(get_thread_fast_rng()); + double p0 = random_uniform_01(); + + return sample_geometric(s, p0, G->p); +} + +static double +geometric_cdf(const struct dist_t *dist, double x) +{ + const struct geometric_t *G = dist_to_const_geometric(dist); + + if (x < 1) + return 0; + /* 1 - (1 - p)^floor(x) = 1 - e^{floor(x) log(1 - p)} */ + return -expm1(floor(x)*log1p(-G->p)); +} + +static double +geometric_sf(const struct dist_t *dist, double x) +{ + const struct geometric_t *G = dist_to_const_geometric(dist); + + if (x < 1) + return 0; + /* (1 - p)^floor(x) = e^{ceil(x) log(1 - p)} */ + return exp(floor(x)*log1p(-G->p)); +} + +static double +geometric_icdf(const struct dist_t *dist, double p) +{ + const struct geometric_t *G = dist_to_const_geometric(dist); + + return log1p(-p)/log1p(-G->p); +} + +static double +geometric_isf(const struct dist_t *dist, double p) +{ + const struct geometric_t *G = dist_to_const_geometric(dist); + + return log(p)/log1p(-G->p); +} + +const struct dist_ops_t geometric_ops = { + .name = "geometric (1-based)", + .sample = geometric_sample, + .cdf = geometric_cdf, + .sf = geometric_sf, + .icdf = geometric_icdf, + .isf = geometric_isf, +}; |