A simple API for defining sample spaces (to run simple statistical simulations)

## Project description

…is a very lightweight Python API for simulating sample spaces, events, random variables, and (conditional) distributions.

## Example

Check out the iPython notebook or read the following:

```
from sample_space import *
class NCoinTosses(Experiment):
def __init__(self, n, p):
self.n = n
self.p = p
def rerun(self):
self.tosses = [Bern(self.p) for _ in range(self.n)]
def heads(self):
return sum(self.tosses)
def there_are_at_least_two_heads(self):
return self.heads() >= 2
def first_toss_heads(self):
return self.tosses[0]
space = SampleSpace(NCoinTosses(10, 0.5), iters=20000)
# ask for probability of any truthy method
print(' P(#H>=2):', space.probability_that('there_are_at_least_two_heads'))
# alias for the above, if it's more grammatical
print(' P(H1):', space.probability_of('first_toss_heads'))
# change the number of iterations
print(' P(H1), 1K iters:', space.probability_of('first_toss_heads', iters=1000))
# ask for probabilities of functions of random variables
print(' P(#H>5):', space.probability_that(['heads', is_greater_than(5)]))
# ask for conditional probabilities
print(' P(#H>5|H1):', space.probability_that(['heads', is_greater_than(5)], given=['first_toss_heads']))
print(' P(H1|#H>5):', space.probability_of('first_toss_heads', given=[['heads', is_greater_than(5)]]))
print(' P(#H>5|H1,H>=2):', space.probability_that(['heads', is_greater_than(5)],
given=['first_toss_heads', 'there_are_at_least_two_heads']))
# ask for expectations, variances, and moments, conditionally or absolutely
print(' E(#H):', space.expected_value_of('heads'))
print(' E(#H|H1):', space.expected_value_of('heads', given=['first_toss_heads']))
print(' Var(#H):', space.variance_of('heads'))
print(' Var(#H|H1):', space.variance_of('heads', given=['first_toss_heads']))
print('1st moment of #H:', space.nth_moment_of('heads', 1))
print('2nd moment of #H:', space.nth_moment_of('heads', 2))
print('3rd moment of #H:', space.nth_moment_of('heads', 3))
print('4th moment of #H:', space.nth_moment_of('heads', 4))
print(' Skewness of #H:', space.nth_moment_of('heads', 3, central=True, normalized=True), '(using nth_moment_of w/ central=True, normalized=True)')
print(' Skewness of #H:', space.skewness_of('heads'), '(using skewness_of)')
print(' Kurtosis of #H:', space.kurtosis_of('heads'))
# some plots
fig = plt.figure(figsize=(14,3))
# plot distribution histograms
fig.add_subplot(121)
space.plot_distribution_of('heads') # pass kwargs
plt.legend()
# plot conditional distribution histograms
fig.add_subplot(122)
space.plot_distribution_of('heads', given=['first_toss_heads'], bins=10) # can pass kwargs
plt.legend()
plt.show()
```

Which should output (plus some plots):

P(#H>=2): 0.98975 P(H1): 0.502 P(H1), 1K iters: 0.48 P(#H>5): 0.37665 P(#H>5|H1): 0.5076294006183305 P(H1|#H>5): 0.6580109757729888 P(#H>5|H1,H>=2): 0.49361831442463533 E(#H): 4.9983 E(#H|H1): 5.48924623116 Var(#H): 2.4486457975 Var(#H|H1): 2.31806506582 1st moment of #H: 4.99245 2nd moment of #H: 27.5097 3rd moment of #H: 163.13055 4th moment of #H: 1015.54155 Skewness of #H: -0.00454435802967 (using nth_moment_of w/ central=True, normalized=True) Skewness of #H: 0.00414054522343 (using skewness_of) Kurtosis of #H: 2.78225928171

## Why?

Mostly to avoid bugs / reduce boilerplate in statistical simulations for sanity-checking homework solutions. But also to get a better understanding of probability theory.

Sample spaces are a core concept in probability theory. They encapsulate the idea of repeatedly running an experiment with random results. Almost every important statistical quantity – the probability of an event, or any moment of a random variable – is always defined relative to a sample space. So if you’re trying to program meaningful simulations (and if you’re more concerned with expressiveness than performance), you might as well organize your code by explicitly defining one.

## Installation / Usage

First run

pip install sample_space

and import the library. Then define a subclass of Experiment that responds to rerun(self). rerun should perform a random experiment and store one or more basic results as instance variables. If you want to define more complex events or random variables, you can express them as instance methods.

Then, initialize a SampleSpace with an instance of your Experiment. You can query your sample space for the probability_that/probability_of an event, or you can query it for the distribution_of, expected_value_of, variance_of, skewness_of, kurtosis_of, or nth_moment_of of random variable (which can also just be an event, in which case it will be interpreted as an indicator). Finally, for any of these methods, you can pass a given keyword argument with a list of events, which will make any results you obtain conditional on all of those events occurring. Behind the scenes, SampleSpace will just rerun your experiment 10000 times and average your random variable or count how often an event occurs (conditionally). You can pass an iters keyword argument to any method or to SampleSpace.__init__ to increase the number of iterations.

To reference events or random variable, pass the string name of an instance variable or instance method of your experiment, or pass an array with a variable/method name and a lambda function. For example:

```
space = SampleSpace(CoinTossExperiment(10))
space.probability_that('first_toss_is_heads')
space.probability_that(['n_heads', lambda h: h > 5])
space.expected_value_of('n_heads')
space.expected_value_of('n_heads', given=['first_toss_is_heads'])
space.probability_that('first_toss_is_heads, given=[['n_heads', lambda h: h > 3], 'last_toss_is_heads'])
```

Additionally, sample_space defines a few helpful lambda-returning methods (is_greater_than(x), is_less_than(x), is_at_least(x), is_at_most(x), equals(x)) for convenience. Of course, you could also define instance methods on your Experiment to accomplish the same goal.

The library also exposes a few basic sampling functions (Bern(p), Bin(n,p), RandomSign(p), and Categ(categories, weights)) to assist with defining experiments.

## Lite Version

If you’d prefer not to define a full Experiment class, you can also just define a random event / random variable function that returns either a boolean or a number, and call probability_that/expected_value_of:

```
import sample_space as ss
def weighted_coin_flip_is_heads(p=0.4):
return ss.Bern(p)
def n_weighted_heads(n=100, p=0.4):
return sum(weighted_coin_flip_is_heads(p) for _ in range(n))
print(ss.probability_that(weighted_coin_flip_is_heads))
print(ss.probability_that(lambda: weighted_coin_flip_is_heads(0.5))
print(ss.expected_value_of(n_weighted_heads))
print(ss.expected_value_of(lambda: n_weighted_heads(200, 0.3)))
```

## License

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