(Prior) Distributions

There are several probability distributions defined in AlTar, serving as prior distributions.

For a prior distribution altar.distributions.distribution, the following methods are required

# model support
@altar.provides
def initializeSample(self, theta):
    """
    Fill my portion of {theta} with initial random values from my distribution.
    """

@altar.provides
def priorLikelihood(self, theta, prior):
    """
    Fill my portion of {prior} with the likelihoods of the samples in {theta}
    """

@altar.provides
def verify(self, theta, mask):
    """
    Check whether my portion of the samples in {theta} are consistent with my constraints, and
    update {mask}, a vector with zeroes for valid samples and non-zero for invalid ones
    """

In AlTar, the logarithmic values of the Bayesian probability densities are processed. Therefore, in addition to priorLikelihood, a verify method is needed for certain ranged distributions to check whether the proposed samples fall outside the range.

For a parameter to be sampled, the distributions for generating the initial samples and computing the prior probability densities during the simulation may be different. For example, in static inversion of earthquakes, you may want to use a Moment Scale distribution to generate (strike or dip) slips whose sum is consistent with a given moment magnitude scale, while during the simulation, while a simple uniform distribution for priorLikelihood and verify.

Please note that AlTar processes samples in batch, where \(\theta\) is a 2d array shape=(samples, parameters). Also, in a specific Model, there may be different parameter sets which observe different prior distributions. Therefore, the methods in a prior distribution are responsible for its own portion of parameters (selected columns of \(\theta\)) and for a batched samples (rows of \(\theta\)).

Uniform

The probability density function (PDF) for a uniform distribution is

\[\begin{split}f(x; a, b) &= \frac {1}{b-a}, \text{for } x \in [a,b] \\ &= 0, \text{otherwise}\end{split}\]

where \([a, b]\) is the support or range.

Example:

prior = uniform
prior:
    support = (0, 1)

Gaussian

The PDF for the Gaussian (normal) distribution is defined as

\[f(x; \mu, \sigma) = \frac {1}{\sqrt{2\pi} \sigma} e^{-\frac{(x-\mu)^2}{2\sigma^2}},\]

where \(\mu\) and \(\sigma^2\) are mean (center) and variance, respectively.

Example:

prior = gaussian
prior:
    mean = 0
    sigma = 2

Truncated Gaussian

The truncated Gaussian distribution is derived from the Gaussian distribution but is only finite within the support range.

Example:

(Currently only implemented in CUDA).

prior = altar.cuda.distributions.tgaussian
prior:
    support = (-1, 1)
    mean = 0
    sigma = 2

Preset

The Preset distribution is used to load initial samples from pre-calculated ones. Therefore, it only serves as a preparation (prep) distribution. The currently support input format is HDF5, as the default output for AlTar simulation results.

Example:

(Currently only implemented in CUDA).

For example, in the earthquake (seismic) inversion, we have samples of strikeslip generated from the static inversion and would like to load them for the kinematic inversion,

prep = altar.cuda.distributions.preset ; load preset samples
prep.input_file = theta_cascaded.h5 ; file name
prep.dataset = ParameterSets/strikeslip ; dataset name in h5

Other Distributions

More prior distributions can be easily added. You may follow the existing distributions as examples. Or please write to us so that we add them to the package.