Policies.Posterior.Gamma module

Manipulate a Gamma posterior. No need for tricks to handle non-binary rewards.

• See https://en.wikipedia.org/wiki/Gamma_distribution#Conjugate_prior

• And https://en.wikipedia.org/wiki/Conjugate_prior#Continuous_distributions

Policies.Posterior.Gamma.gammavariate()

gamma(shape, scale=1.0, size=None)

Draw samples from a Gamma distribution.

Samples are drawn from a Gamma distribution with specified parameters, shape (sometimes designated “k”) and scale (sometimes designated “theta”), where both parameters are > 0.

shapefloat or array_like of floats

The shape of the gamma distribution. Must be non-negative.

scalefloat or array_like of floats, optional

The scale of the gamma distribution. Must be non-negative. Default is equal to 1.

sizeint or tuple of ints, optional

Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. If size is None (default), a single value is returned if shape and scale are both scalars. Otherwise, np.broadcast(shape, scale).size samples are drawn.

outndarray or scalar

Drawn samples from the parameterized gamma distribution.

scipy.stats.gammaprobability density function, distribution or

cumulative density function, etc.

The probability density for the Gamma distribution is

\[p(x) = x^{k-1}\frac{e^{-x/\theta}}{\theta^k\Gamma(k)},\]

where \(k\) is the shape and \(\theta\) the scale, and \(\Gamma\) is the Gamma function.

The Gamma distribution is often used to model the times to failure of electronic components, and arises naturally in processes for which the waiting times between Poisson distributed events are relevant.

1

Weisstein, Eric W. “Gamma Distribution.” From MathWorld–A Wolfram Web Resource. http://mathworld.wolfram.com/GammaDistribution.html

2

Wikipedia, “Gamma distribution”, https://en.wikipedia.org/wiki/Gamma_distribution

Draw samples from the distribution:

>>> shape, scale = 2., 2.  # mean=4, std=2*sqrt(2)
>>> s = np.random.gamma(shape, scale, 1000)

Display the histogram of the samples, along with the probability density function:

>>> import matplotlib.pyplot as plt
>>> import scipy.special as sps  
>>> count, bins, ignored = plt.hist(s, 50, density=True)
>>> y = bins**(shape-1)*(np.exp(-bins/scale) /  
...                      (sps.gamma(shape)*scale**shape))
>>> plt.plot(bins, y, linewidth=2, color='r')  
>>> plt.show()

class Policies.Posterior.Gamma.Gamma(k=1, lmbda=1)

Bases: Policies.Posterior.Posterior.Posterior

Manipulate a Gamma posterior.

__init__(k=1, lmbda=1)

Create a Gamma posterior, \(\Gamma(k, \lambda)\), with \(k=1\) and \(\lambda=1\) by default.

k = None

Parameter \(k\)

lmbda = None

Parameter \(\lambda\)

__str__()

Return str(self).

reset(k=None, lmbda=None)

Reset k and lmbda, both to 1 as when creating a new default Gamma.

sample()

Get a random sample from the Beta posterior (using numpy.random.gammavariate()).

• Used only by Thompson Sampling and AdBandits so far.

quantile(p)

Return the p quantile of the Gamma posterior (using scipy.stats.gdtrix()).

• Used only by BayesUCB and AdBandits so far.

mean()

Compute the mean of the Gamma posterior (should be useless).

forget(obs)

Forget the last observation.

update(obs)

Add an observation: increase k by k0, and lmbda by obs (do not have to be normalized).

__module__ = 'Policies.Posterior.Gamma'