How to Combine Distributions

There are two ways to combine distributions in Ciw: arithmetically, and probabilistically.

Arithmetically Combining Distributions

As distribution objects inherit from the generic distirbution function, they can be combined using the operations +, -, *, and /.

For example, let’s combine an Exponential distribution with a Deterministic distribution in all four ways:

>>> import ciw
>>> Exp_add_Det = ciw.dists.Exponential(rate=0.05) + ciw.dists.Deterministic(value=3.0)
>>> Exp_sub_Det = ciw.dists.Exponential(rate=0.05) - ciw.dists.Deterministic(value=3.0)
>>> Exp_mul_Det = ciw.dists.Exponential(rate=0.05) * ciw.dists.Deterministic(value=3.0)
>>> Exp_div_Det = ciw.dists.Exponential(rate=0.05) / ciw.dists.Deterministic(value=3.0)

These combined distributions return the combined sampled values:

>>> ciw.seed(10)
>>> Ex = ciw.dists.Exponential(rate=0.05)
>>> Dt = ciw.dists.Deterministic(value=3.0)
>>> [round(Ex.sample(), 2) for _ in range(5)]
[16.94, 11.2, 17.26, 4.62, 33.57]
>>> [round(Dt.sample(), 2) for _ in range(5)]
[3.0, 3.0, 3.0, 3.0, 3.0]

>>> # Addition
>>> ciw.seed(10)
>>> [round(Exp_add_Det.sample(), 2) for _ in range(5)]
[19.94, 14.2, 20.26, 7.62, 36.57]

>>> # Subtraction
>>> ciw.seed(10)
>>> [round(Exp_sub_Det.sample(), 2) for _ in range(5)]
[13.94, 8.2, 14.26, 1.62, 30.57]

>>> # Multiplication
>>> ciw.seed(10)
>>> [round(Exp_mul_Det.sample(), 2) for _ in range(5)]
[50.83, 33.61, 51.78, 13.85, 100.7]

>>> # Division
>>> ciw.seed(10)
>>> [round(Exp_div_Det.sample(), 2) for _ in range(5)]
[5.65, 3.73, 5.75, 1.54, 11.19]

Probabilistically Combining Distributions

Distributions can also be combined probabilistically. A countable and finite mixture distriubtion probabilistically chooses to sample from one of a number of given distributions. This is given by the MixtureDistribution object. Given a number of distributions each with PDF \(D_i(x)\), each with a probability \(p_i\), such that \(\sum_i p_i = 1\), then the Mixture distribution has PMF \(f(x) = \sum_i p_i D_i(x)\)

For example, say let’s make a mixture distribution that samples from an Exponential distribution rate 1 with probability 0.5, another Exponential distribution rate 2 with probability 0.2, a Uniform distribution between 0.2 and 0.8 with probability 0.2, and returns a Deterministic value of 0.5 with probability 0.1. We can do this with:

>>> Exp1 = ciw.dists.Exponential(rate=1)
>>> Exp2 = ciw.dists.Exponential(rate=2)
>>> Unif = ciw.dists.Uniform(lower=0.2, upper=0.8)
>>> Det5 = ciw.dists.Deterministic(0.5)

>>> M = ciw.dists.MixtureDistribution(dists=[Exp1, Exp2, Unif, Det5], probs=[0.5, 0.2, 0.2, 0.1])