- 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.
2. In the 2nd example, we define a small model, and return many samples
3. In the 3rd example, we use a gcc extension—nested functions—to rewrite the code from point 2. in a more linear way.
4. In the 4th 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 5th example, we define the cdf for the beta distribution, and we draw samples from it.
6. In the 6th example, we take samples from simple gamma and beta distributions, using the samplers provided by this library.
7. In the 7th example, we get the 90% confidence interval of a beta distribution
8. The 8th example translates the models from Eli and Nuño from [Samotsvety Nuclear Risk Forecasts — March 2022](https://forum.nunosempere.com/posts/KRFXjCqqfGQAYirm5/samotsvety-nuclear-risk-forecasts-march-2022#Nu_o_Sempere) into squiggle.c, then creates a mixture from both, and returns the mean probability of death per month and the 90% confidence interval.
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.
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.
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.
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:
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.
- Nonetheless, the user should understand the limitations of sampling-based methods. See the section on [Tests and the long tail of the lognormal](https://git.nunosempere.com/personal/squiggle.c#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.
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)?
`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
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
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.
