[mlpack-git] [mlpack/mlpack] GammaDistribution: Adds functionality to solve #749 (#751)

[Ryan Curtin]

> +  // Combine into one 2-dimensional distribution.
> +  const arma::vec a3("2.0 3.1"), b3("0.9 1.4");
> +  arma::mat x3(2, 2);
> +  x3
> +    << 2.0 << 2.94 << arma::endr
> +    << 2.0 << 2.94;
> +  arma::vec prob3;
> +
> +  // Expect that the 2-dimensional distribution returns the product of the
> +  // 1-dimensional distributions (evaluated at wolfram|alpha).
> +  GammaDistribution d3(a3, b3);
> +  d3.LogProbability(x3, prob3);
> +  BOOST_REQUIRE_CLOSE(prob3(0), std::log(0.04408), 1e-3);
> +  BOOST_REQUIRE_CLOSE(prob3(1), std::log(0.026165), 1e-3);
> +}
> +

Can you add a test for the `Random()` function?  Here's some idea:

```
BOOST_AUTO_TEST_CASE(RandomTest)
{
  arma::vec alphas("1.0 2.0 3.0");
  arma::vec beta("1.5 2.5 3.5");

  GammaDistribution d(alphas, betas);

  arma::mat observations(1000, 3);
  for (size_t i = 0; i < 1000; ++i)
    observations.col(i) = d.Random();

  // Train a second distribution, then ensure it is similar to the first.
  GammaDistribution d2(observations);

  // Check the values, with a fairly large tolerance (3%).
  BOOST_REQUIRE_CLOSE(d.Alpha(0), d2.Alpha(0), 3.0);
  BOOST_REQUIRE_CLOSE(d.Alpha(1), d2.Alpha(1), 3.0);
  BOOST_REQUIRE_CLOSE(d.Alpha(2), d2.Alpha(2), 3.0);
  BOOST_REQUIRE_CLOSE(d.Beta(0), d2.Beta(0), 3.0);
  BOOST_REQUIRE_CLOSE(d.Beta(1), d2.Beta(1), 3.0);
  BOOST_REQUIRE_CLOSE(d.Beta(2), d2.Beta(2), 3.0);
  // Should these all be Alphas()[i] and Betas()[i]?
}
```

To test the tolerance, add a `math::RandomSeed(std::time(NULL))` at the top, recompile, and run this: `while(true); do bin/mlpack_test -t GammaDistributionTest/RandomTest; sleep 1; done` and make sure that over a decent period of time, you don't see any failures.  If so, 5% or maybe even 10% are reasonable tolerances.  We just want to check that it's giving us stuff that's "reasonably close" to a Gamma distribution; checking exact precision is quite a large amount of work and here I am not sure that effort is warranted.

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