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Squiggle.c

A self-contained C99 library that provides a subset of Squiggle's functionality in C.

Why C?

  • Because it is fast
  • Because I enjoy it
  • Because C is honest
  • Because it will last long
  • Because it can fit in my head
  • Because if you can implement something in C, you can implement it anywhere else
  • Because it can be made faster if need be
    • e.g., with a multi-threading library like OpenMP,
    • or by implementing faster but more complex algorithms
    • or more simply, by inlining the sampling functions (adding an inline directive before their function declaration)
  • Because there are few abstractions between it and machine code (C => assembly => machine code with gcc, or C => machine code, with tcc), leading to fewer errors beyond the programmer's control.

Getting started

You can follow some example usage in the examples/ folder

  1. In the first example, we define a small model, and draw one sample from it
  2. In the second example, we define a small model, and return many samples
  3. In the third example, we use a gcc extension—nested functions—to rewrite the code from point 2. in a more linear way.
  4. In the fourth example, we define some simple cdfs, and we draw samples from those cdfs. We see that this approach is slower than using the built-in samplers, e.g., the normal sampler.
  5. In the fifth example, we define the cdf for the beta distribution, and we draw samples from it.
  6. In the sixth example, we take samples from simple gamma and beta distributions, using the samplers provided by this library.

Commentary

squiggle.c is short

squiggle.c is less than 500 lines of C. The reader could just read it and grasp its contents.

Core strategy

This library provides some basic building blocks. The recommended strategy is to:

  1. Define sampler functions, which take a seed, and return 1 sample
  2. Compose those sampler functions to define your estimation model
  3. At the end, call the last sampler function many times to generate many samples from your model

Cdf auxiliary functions

To help with the above core strategy, this library provides convenience functions, which take a cdf, and return a sample from the distribution produced by that cdf. This might make it easier to program models, at the cost of a 20x to 60x slowdown in the parts of the code that use it.

Nested functions and compilation with tcc.

GCC has an extension which allows a program to define a function inside another function. This makes squiggle.c code more linear and nicer to read, at the cost of becoming dependent on GCC and hence sacrificing portability and compilation times. Conversely, compiling with tcc (tiny c compiler) is almost instantaneous, but leads to longer execution times and doesn't allow for nested functions.

GCC tcc
slower compilation faster compilation
allows nested functions doesn't allow nested functions
faster execution slower execution

My recommendation would be to use tcc while drawing a small number of samples for fast iteration, and then using gcc for the final version with lots of samples, and possibly with nested functions for ease of reading by others.

Error propagation vs exiting on error

The process of taking a cdf and returning a sample might fail, e.g., it's a Newton method which might fail to converge because of cdf artifacts. The cdf itself might also fail, e.g., if a distribution only accepts a range of parameters, but is fed parameters outside that range.

This library provides two approaches:

  1. Print the line and function in which the error occured, then exit on error
  2. In situations where there might be an error, return a struct containing either the correct value or an error message:
struct box {
    int empty;
    double content;
    char* error_msg;
};

The first approach produces terser programs but might not scale. The second approach seems like it could lead to more robust programmes, but is more verbose.

Behaviour on error can be toggled by the EXIT_ON_ERROR variable. This library also provides a convenient macro, PROCESS_ERROR, to make error handling in either case much terser—see the usage in example 4 in the examples/ folder.

Design choices

This code should be correct, then simple, then fast.

  • It should be correct. The user should be able to rely on it and not think about whether errors come from the library.
    • Nonetheless, the user should understand the limitations of sampling-based methods. See the section on Tests and the long tail of the lognormal for a discussion of how sampling is bad at capturing some aspects of distributions with long tails.
  • It should be clear, conceptually simple. Simple for me to implement, simple for others to understand
  • It should be fast. But when speed conflicts with simplicity, choose simplicity. For example, there might be several possible algorithms to sample a distribution, each of which is faster over part of the domain. In that case, it's conceptually simpler to just pick one algorithm, and pay the—normally small—performance penalty. In any case, though, the code should still be way faster than Python.

Note that being terse, or avoiding verbosity, is a non-goal. This is in part because of the constraints that C imposes. But it also aids with clarity and conceptual simplicity, as the issue of correlated samples illustrates in the next section.

Correlated samples

In the original squiggle language, there is some ambiguity about what this code means:

a = 1 to 10
b = 2 * a
c = b/a
c

Likewise in squigglepy:

import squigglepy as sq
import numpy as np

a = sq.to(1, 3)
b = 2 * a  
c = b / a 

c_samples = sq.sample(c, 10)

print(c_samples)

Should c be equal to 2? or should it be equal to 2 times the expected distribution of the ratio of two independent draws from a (2 * a/a, as it were)?

In squiggle.c, this ambiguity doesn't exist, at the cost of much greater overhead & verbosity:

// correlated samples
// gcc -O3  correlated.c squiggle.c -lm -o correlated

#include "squiggle.h"
#include <stdint.h>
#include <stdlib.h>
#include <stdio.h>

int main(){
    // set randomness seed
    uint64_t* seed = malloc(sizeof(uint64_t));
    *seed = 1000; // xorshift can't start with a seed of 0

    double a = sample_to(1, 10, seed);
    double b = 2 * a;
    double c = b / a;

    printf("a: %f, b: %f, c: %f\n", a, b, c);
    // a: 0.607162, b: 1.214325, c: 0.500000

    free(seed);
}

vs

// uncorrelated samples
// gcc -O3    uncorrelated.c ../../squiggle.c -lm -o uncorrelated

#include "squiggle.h"
#include <stdint.h>
#include <stdlib.h>
#include <stdio.h>

double draw_xyz(uint64_t* seed){
    // function could also be placed inside main with gcc nested functions extension.
    return sample_to(1, 20, seed);
}


int main(){
    // set randomness seed
    uint64_t* seed = malloc(sizeof(uint64_t));
    *seed = 1000; // xorshift can't start with a seed of 0

    double a = draw_xyz(seed);
    double b = 2 * draw_xyz(seed);
    double c = b / a;

    printf("a: %f, b: %f, c: %f\n", a, b, c);
    // a: 0.522484, b: 10.283501, c: 19.681936

    free(seed)
}

Tests and the long tail of the lognormal

Distribution functions can be tested with:

cd tests
make && make run

make verify is an alias that runs all the tests and just displays the ones that are failing.

These tests are somewhat rudimentary: they get between 1M and 10M samples from a given sampling function, and check that their mean and standard deviations correspond to what they should theoretically should be.

If you run make run (or make verify), you will see errors such as these:

[-] Mean test for normal(47211.047473, 682197.019012) NOT passed.
Mean of normal(47211.047473, 682197.019012): 46933.673278, vs expected mean: 47211.047473
delta: -277.374195, relative delta: -0.005910

[-] Std test for lognormal(4.584666, 2.180816) NOT passed.
Std of lognormal(4.584666, 2.180816): 11443.588861, vs expected std: 11342.434900
delta: 101.153961, relative delta: 0.008839

[-] Std test for to(13839.861856, 897828.354318) NOT passed.
Std of to(13839.861856, 897828.354318): 495123.630575, vs expected std: 498075.002499
delta: -2951.371925, relative delta: -0.005961

These tests I wouldn't worry about. Due to luck of the draw, their relative error is a bit over 0.005, or 0.5%, and so the test fails. But it would surprise me if that had some meaningful practical implication.

The errors that should raise some worry are:

[-] Mean test for lognormal(1.210013, 4.766882) NOT passed.
Mean of lognormal(1.210013, 4.766882): 342337.257677, vs expected mean: 288253.061628
delta: 54084.196049, relative delta: 0.157985
[-] Std test for lognormal(1.210013, 4.766882) NOT passed.
Std of lognormal(1.210013, 4.766882): 208107782.972184, vs expected std: 24776840217.604111
delta: -24568732434.631927, relative delta: -118.057730

[-] Mean test for lognormal(-0.195240, 4.883106) NOT passed.
Mean of lognormal(-0.195240, 4.883106): 87151.733198, vs expected mean: 123886.818303
delta: -36735.085104, relative delta: -0.421507
[-] Std test for lognormal(-0.195240, 4.883106) NOT passed.
Std of lognormal(-0.195240, 4.883106): 33837426.331671, vs expected std: 18657000192.914921
delta: -18623162766.583248, relative delta: -550.371727

[-] Mean test for lognormal(0.644931, 4.795860) NOT passed.
Mean of lognormal(0.644931, 4.795860): 125053.904456, vs expected mean: 188163.894101
delta: -63109.989645, relative delta: -0.504662
[-] Std test for lognormal(0.644931, 4.795860) NOT passed.
Std of lognormal(0.644931, 4.795860): 39976300.711166, vs expected std: 18577298706.170452
delta: -18537322405.459286, relative delta: -463.707799

What is happening in this case is that you are taking a normal, like normal(-0.195240, 4.883106), and you are exponentiating it to arrive at a lognormal. But normal(-0.195240, 4.883106) is going to have some noninsignificant weight on, say, 18. But exp(18) = 39976300, and points like it are going to end up a nontrivial amount to the analytical mean and standard deviation, even though they have little probability mass.

The reader can also check that for more plausible real-world values, like those fitting a lognormal to a really wide 90% confidence interval from 10 to 10k, errors aren't eggregious:

[x] Mean test for to(10.000000, 10000.000000) PASSED
[-] Std test for to(10.000000, 10000.000000) NOT passed.
Std of to(10.000000, 10000.000000): 23578.091775, vs expected std: 25836.381819
delta: -2258.290043, relative delta: -0.095779

Overall, I would caution that if you really care about the very far tails of distributions, you might want to instead use tools which can do some of the analytical manipulations for you, like the original Squiggle, Simple Squiggle (both linked below), or even doing lognormal multiplication by hand, relying on the fact that two lognormals multiplied together result in another lognormal with known shape.

Related projects

To do list

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