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streamline fermi estimation a little

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NunoSempere 2024-01-21 11:44:28 +01:00
parent 9e5d2db23b
commit 1e78617079
3 changed files with 111 additions and 98 deletions
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examples/core/06_dissolving_fermi_paradox/example
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examples/core/06_dissolving_fermi_paradox/example.c
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@ -4,88 +4,23 @@
#include <stdio.h> #include <stdio.h>
#include <stdlib.h> #include <stdlib.h>
#define VERBOSE 0
double sample_loguniform(double a, double b, uint64_t* seed)
{
return exp(sample_uniform(log(a), log(b), seed));
}
int main() int main()
{ {
// Replicate <https://arxiv.org/pdf/1806.02404.pdf>, and in particular the red line in page 11. // Replicate <https://arxiv.org/pdf/1806.02404.pdf>, and in particular the red line in page 11.
// Could also be interesting to just produce and save many samples.
// set randomness seed // set randomness seed
uint64_t* seed = malloc(sizeof(uint64_t)); uint64_t* seed = malloc(sizeof(uint64_t));
*seed = UINT64_MAX / 64; // xorshift can't start with a seed of 0 *seed = 1001; // xorshift can't start with a seed of 0
// Do this naïvely, without worrying that much about numerical precision
double sample_fermi_naive(uint64_t * seed)
{
double rate_of_star_formation = sample_loguniform(1, 100, seed);
double fraction_of_stars_with_planets = sample_loguniform(0.1, 1, seed);
double number_of_habitable_planets_per_star_system = sample_loguniform(0.1, 1, seed);
double rate_of_life_formation_in_habitable_planets = sample_lognormal(1, 50, seed);
double fraction_of_habitable_planets_in_which_any_life_appears = -expm1(-rate_of_life_formation_in_habitable_planets);
// double fraction_of_habitable_planets_in_which_any_life_appears = 1-exp(-rate_of_life_formation_in_habitable_planets);
// but with more precision
double fraction_of_planets_with_life_in_which_intelligent_life_appears = sample_loguniform(0.001, 1, seed);
double fraction_of_intelligent_planets_which_are_detectable_as_such = sample_loguniform(0.01, 1, seed);
double longevity_of_detectable_civilizations = sample_loguniform(100, 10000000000, seed);
if(VERBOSE) printf(" rate_of_star_formation = %lf\n", rate_of_star_formation);
if(VERBOSE) printf(" fraction_of_stars_with_planets = %lf\n", fraction_of_stars_with_planets);
if(VERBOSE) printf(" number_of_habitable_planets_per_star_system = %lf\n", number_of_habitable_planets_per_star_system);
if(VERBOSE) printf(" rate_of_life_formation_in_habitable_planets = %.16lf\n", rate_of_life_formation_in_habitable_planets);
if(VERBOSE) printf(" fraction_of_habitable_planets_in_which_any_life_appears = %lf\n", fraction_of_habitable_planets_in_which_any_life_appears);
if(VERBOSE) printf(" fraction_of_planets_with_life_in_which_intelligent_life_appears = %lf\n", fraction_of_planets_with_life_in_which_intelligent_life_appears);
if(VERBOSE) printf(" fraction_of_intelligent_planets_which_are_detectable_as_such = %lf\n", fraction_of_intelligent_planets_which_are_detectable_as_such);
if(VERBOSE) printf(" longevity_of_detectable_civilizations = %lf\n", longevity_of_detectable_civilizations);
// Expected number of civilizations in the Milky way;
// see footnote 3 (p. 5)
double n = rate_of_star_formation * fraction_of_stars_with_planets * number_of_habitable_planets_per_star_system * fraction_of_habitable_planets_in_which_any_life_appears * fraction_of_planets_with_life_in_which_intelligent_life_appears * fraction_of_intelligent_planets_which_are_detectable_as_such * longevity_of_detectable_civilizations;
return n;
}
double sample_are_we_alone_naive(uint64_t * seed)
{
double n = sample_fermi_naive(seed);
return ((n > 1) ? 1 : 0);
}
double n = 1000000;
double naive_fermi_proportion = 0;
for (int i = 0; i < n; i++) {
double result = sample_are_we_alone_naive(seed);
if(VERBOSE) printf("result: %lf\n", result);
naive_fermi_proportion += result;
}
printf("Naïve %% that we are not alone: %lf\n", naive_fermi_proportion / n);
// Thinking in log space
// Taking care of numerical precision
double sample_fermi_logspace(uint64_t * seed) double sample_fermi_logspace(uint64_t * seed)
{ {
// You can see a simple version of this function in naive.c in this same folder
double log_rate_of_star_formation = sample_uniform(log(1), log(100), seed); double log_rate_of_star_formation = sample_uniform(log(1), log(100), seed);
double log_fraction_of_stars_with_planets = sample_uniform(log(0.1), log(1), seed); double log_fraction_of_stars_with_planets = sample_uniform(log(0.1), log(1), seed);
double log_number_of_habitable_planets_per_star_system = sample_uniform(log(0.1), log(1), seed); double log_number_of_habitable_planets_per_star_system = sample_uniform(log(0.1), log(1), seed);
double log_fraction_of_planets_with_life_in_which_intelligent_life_appears = sample_uniform(log(0.001), log(1), seed);
double log_fraction_of_intelligent_planets_which_are_detectable_as_such = sample_uniform(log(0.01), log(1), seed);
double log_longevity_of_detectable_civilizations = sample_uniform(log(100), log(10000000000), seed);
if(VERBOSE) printf(" log_rate_of_star_formation = %lf\n", log_rate_of_star_formation);
if(VERBOSE) printf(" log_fraction_of_stars_with_planets = %lf\n", log_fraction_of_stars_with_planets);
if(VERBOSE) printf(" log_number_of_habitable_planets_per_star_system = %lf\n", log_number_of_habitable_planets_per_star_system);
if(VERBOSE) printf(" log_fraction_of_planets_with_life_in_which_intelligent_life_appears = %lf\n", log_fraction_of_planets_with_life_in_which_intelligent_life_appears);
if(VERBOSE) printf(" log_fraction_of_intelligent_planets_which_are_detectable_as_such = %lf\n", log_fraction_of_intelligent_planets_which_are_detectable_as_such);
if(VERBOSE) printf(" log_longevity_of_detectable_civilizations = %lf\n", log_longevity_of_detectable_civilizations);
double log_n1 = log_rate_of_star_formation + log_fraction_of_stars_with_planets + log_number_of_habitable_planets_per_star_system + log_fraction_of_planets_with_life_in_which_intelligent_life_appears + log_fraction_of_intelligent_planets_which_are_detectable_as_such + log_longevity_of_detectable_civilizations;
if(VERBOSE) printf("first part of calculation: %lf\n", log_n1);
double log_rate_of_life_formation_in_habitable_planets = sample_normal(1, 50, seed);
double log_fraction_of_habitable_planets_in_which_any_life_appears;
/* /*
Consider: Consider:
a = underlying normal a = underlying normal
@ -93,15 +28,12 @@ int main()
c = 1 - exp(-b) = fraction_of_habitable_planets_in_which_any_life_appears c = 1 - exp(-b) = fraction_of_habitable_planets_in_which_any_life_appears
d = log(c) d = log(c)
Now, is there some way we can get d more efficiently/precisely? Looking at the Taylor expansion for c = 1 - exp(-b), it's
b - b^2/2 + b^3/6 - x^b/24, etc.
Turns out there is!
Looking at the Taylor expansion for c = 1 - exp(-b), it's b - b^2/2 + b^3/6 - x^b/24, etc.
<https://www.wolframalpha.com/input?i=1-exp%28-x%29> <https://www.wolframalpha.com/input?i=1-exp%28-x%29>
When b ~ 0 (as is often the case), this is close to b. When b ~ 0 (as is often the case), this is close to b.
But now, if b ~ 0 But now, if b ~ 0, c ~ b
c ~ b
and d = log(c) ~ log(b) = log(exp(a)) = a and d = log(c) ~ log(b) = log(exp(a)) = a
Now, we could play around with estimating errors, Now, we could play around with estimating errors,
@ -109,46 +41,11 @@ int main()
we could compute this as e.g., a < (nlog(10) + log(2))/2 we could compute this as e.g., a < (nlog(10) + log(2))/2
so for example if we want ten digits of precision, that's a < -11 so for example if we want ten digits of precision, that's a < -11
But more empirically, the two numbers do become really close around 11 or so, and at 38 that calculation results in a -inf (so probably an overflow.) Empirically, the two numbers as calculated in C do become really close around 11 or so,
and at 38 that calculation results in a -inf (so probably a floating point error or similar.)
So we should be using that formula for somewhere between -38 << a < -11 So we should be using that formula for somewhere between -38 << a < -11
I chose -16 for the sake of it after playing with:
<https://www.wolframalpha.com/input?i=log%281-exp%28-exp%28-16%29%29%29>
*/
double log_rate_of_life_formation_in_habitable_planets = sample_normal(1, 50, seed);
if(VERBOSE) printf("log_rate_of_life_formation_in_habitable_planets: %lf\n", log_rate_of_life_formation_in_habitable_planets);
double log_fraction_of_habitable_planets_in_which_any_life_appears; I chose -16 as a happy medium after playing around with
if (log_rate_of_life_formation_in_habitable_planets < -16) {
log_fraction_of_habitable_planets_in_which_any_life_appears = log_rate_of_life_formation_in_habitable_planets;
} else {
double rate_of_life_formation_in_habitable_planets = exp(log_rate_of_life_formation_in_habitable_planets);
double fraction_of_habitable_planets_in_which_any_life_appears = -expm1(-rate_of_life_formation_in_habitable_planets);
log_fraction_of_habitable_planets_in_which_any_life_appears = log(fraction_of_habitable_planets_in_which_any_life_appears);
}
if(VERBOSE) printf(" log_fraction_of_habitable_planets_in_which_any_life_appears: %lf\n", log_fraction_of_habitable_planets_in_which_any_life_appears);
double log_n = log_n1 + log_fraction_of_habitable_planets_in_which_any_life_appears;
return log_n;
}
double sample_are_we_alone_logspace(uint64_t * seed)
{
double n = sample_fermi_logspace(seed);
return ((n > 0) ? 1 : 0);
}
double logspace_fermi_proportion = 0;
for (int i = 0; i < n; i++) {
double result = sample_are_we_alone_logspace(seed);
if(VERBOSE) printf("result: %lf\n", result);
logspace_fermi_proportion += result;
}
printf("Using more accurate logspace computations, %% that we are not alone: %lf\n", logspace_fermi_proportion / n);
free(seed);
/*
double invert(double x){ double invert(double x){
return log(1-exp(-exp(-x))); return log(1-exp(-exp(-x)));
} }
@ -156,5 +53,42 @@ int main()
double j = i; double j = i;
printf("for %lf, log(1-exp(-exp(-x))) is calculated as... %lf\n", j, invert(j)); printf("for %lf, log(1-exp(-exp(-x))) is calculated as... %lf\n", j, invert(j));
} }
and <https://www.wolframalpha.com/input?i=log%281-exp%28-exp%28-16%29%29%29>
*/ */
if (log_rate_of_life_formation_in_habitable_planets < -16) {
log_fraction_of_habitable_planets_in_which_any_life_appears = log_rate_of_life_formation_in_habitable_planets;
} else {
double rate_of_life_formation_in_habitable_planets = exp(log_rate_of_life_formation_in_habitable_planets);
double fraction_of_habitable_planets_in_which_any_life_appears = -expm1(-rate_of_life_formation_in_habitable_planets);
log_fraction_of_habitable_planets_in_which_any_life_appears = log(fraction_of_habitable_planets_in_which_any_life_appears);
} }
double log_fraction_of_planets_with_life_in_which_intelligent_life_appears = sample_uniform(log(0.001), log(1), seed);
double log_fraction_of_intelligent_planets_which_are_detectable_as_such = sample_uniform(log(0.01), log(1), seed);
double log_longevity_of_detectable_civilizations = sample_uniform(log(100), log(10000000000), seed);
double log_n = log_rate_of_star_formation + log_fraction_of_stars_with_planets + log_number_of_habitable_planets_per_star_system + log_fraction_of_habitable_planets_in_which_any_life_appears + log_fraction_of_planets_with_life_in_which_intelligent_life_appears + log_fraction_of_intelligent_planets_which_are_detectable_as_such + log_longevity_of_detectable_civilizations;
return log_n;
}
double sample_are_we_alone_logspace(uint64_t * seed)
{
double log_n = sample_fermi_logspace(seed);
return ((log_n > 0) ? 1 : 0);
// log_n > 0 => n > 1
}
double logspace_fermi_proportion = 0;
int n_samples = 1000 * 1000;
for (int i = 0; i < n_samples; i++) {
double result = sample_are_we_alone_logspace(seed);
logspace_fermi_proportion += result;
}
double p_not_alone = logspace_fermi_proportion / n_samples;
printf("Probability that we are not alone: %lf (%.lf%%)\n", p_not_alone, p_not_alone * 100);
free(seed);
}

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examples/core/06_dissolving_fermi_paradox/naive.c Normal file
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@ -0,0 +1,79 @@
#include "../../../squiggle.h"
#include <math.h>
#include <stdint.h>
#include <stdio.h>
#include <stdlib.h>
#define VERBOSE 0
double sample_loguniform(double a, double b, uint64_t* seed)
{
return exp(sample_uniform(log(a), log(b), seed));
}
int main()
{
// Replicate <https://arxiv.org/pdf/1806.02404.pdf>, and in particular the red line in page 11.
// Could also be interesting to just produce and save many samples.
// set randomness seed
uint64_t* seed = malloc(sizeof(uint64_t));
*seed = UINT64_MAX / 64; // xorshift can't start with a seed of 0
// Do this naïvely, without worrying that much about numerical precision
double sample_fermi_naive(uint64_t * seed)
{
double rate_of_star_formation = sample_loguniform(1, 100, seed);
double fraction_of_stars_with_planets = sample_loguniform(0.1, 1, seed);
double number_of_habitable_planets_per_star_system = sample_loguniform(0.1, 1, seed);
double rate_of_life_formation_in_habitable_planets = sample_lognormal(1, 50, seed);
double fraction_of_habitable_planets_in_which_any_life_appears = -expm1(-rate_of_life_formation_in_habitable_planets);
// double fraction_of_habitable_planets_in_which_any_life_appears = 1-exp(-rate_of_life_formation_in_habitable_planets);
// but with more precision
double fraction_of_planets_with_life_in_which_intelligent_life_appears = sample_loguniform(0.001, 1, seed);
double fraction_of_intelligent_planets_which_are_detectable_as_such = sample_loguniform(0.01, 1, seed);
double longevity_of_detectable_civilizations = sample_loguniform(100, 10000000000, seed);
if(VERBOSE) printf(" rate_of_star_formation = %lf\n", rate_of_star_formation);
if(VERBOSE) printf(" fraction_of_stars_with_planets = %lf\n", fraction_of_stars_with_planets);
if(VERBOSE) printf(" number_of_habitable_planets_per_star_system = %lf\n", number_of_habitable_planets_per_star_system);
if(VERBOSE) printf(" rate_of_life_formation_in_habitable_planets = %.16lf\n", rate_of_life_formation_in_habitable_planets);
if(VERBOSE) printf(" fraction_of_habitable_planets_in_which_any_life_appears = %lf\n", fraction_of_habitable_planets_in_which_any_life_appears);
if(VERBOSE) printf(" fraction_of_planets_with_life_in_which_intelligent_life_appears = %lf\n", fraction_of_planets_with_life_in_which_intelligent_life_appears);
if(VERBOSE) printf(" fraction_of_intelligent_planets_which_are_detectable_as_such = %lf\n", fraction_of_intelligent_planets_which_are_detectable_as_such);
if(VERBOSE) printf(" longevity_of_detectable_civilizations = %lf\n", longevity_of_detectable_civilizations);
// Expected number of civilizations in the Milky way;
// see footnote 3 (p. 5)
double n = rate_of_star_formation * fraction_of_stars_with_planets * number_of_habitable_planets_per_star_system * fraction_of_habitable_planets_in_which_any_life_appears * fraction_of_planets_with_life_in_which_intelligent_life_appears * fraction_of_intelligent_planets_which_are_detectable_as_such * longevity_of_detectable_civilizations;
return n;
}
double sample_are_we_alone_naive(uint64_t * seed)
{
double n = sample_fermi_naive(seed);
return ((n > 1) ? 1 : 0);
}
double n = 1000000;
double naive_fermi_proportion = 0;
for (int i = 0; i < n; i++) {
double result = sample_are_we_alone_naive(seed);
if(VERBOSE) printf("result: %lf\n", result);
naive_fermi_proportion += result;
}
printf("Naïve %% that we are not alone: %lf\n", naive_fermi_proportion / n);
free(seed);
/*
double invert(double x){
return log(1-exp(-exp(-x)));
}
for(int i=0; i<64; i++){
double j = i;
printf("for %lf, log(1-exp(-exp(-x))) is calculated as... %lf\n", j, invert(j));
}
*/
}