# Source code for probnum.randvars._normal

```"""Normally distributed / Gaussian random variables."""

from typing import Callable, Optional, Union

import numpy as np
import scipy.linalg
import scipy.stats

from probnum import config, linops
from probnum import utils as _utils
from probnum.typing import (
ArrayLikeGetitemArgType,
FloatArgType,
ShapeArgType,
ShapeType,
)

from . import _random_variable

try:
# functools.cached_property is only available in Python >=3.8
from functools import cached_property
except ImportError:
from cached_property import cached_property

_ValueType = Union[np.floating, np.ndarray, linops.LinearOperator]

# pylint: disable="too-complex"

class Normal(_random_variable.ContinuousRandomVariable[_ValueType]):
"""Random variable with a normal distribution.

Gaussian random variables are ubiquitous in probability theory, since the
Gaussian is the equilibrium distribution to which other distributions gravitate
under a wide variety of smooth operations, e.g., convolutions and stochastic
transformations. One example of this is the central limit theorem. Gaussian
random variables are also attractive from a numerical point of view as they
maintain their distribution family through many transformations (e.g. they are
stable). In particular, they allow for efficient closed-form Bayesian inference
given linear observations.

Parameters
----------
mean :
Mean of the random variable.
cov :
(Co-)variance of the random variable.
cov_cholesky :
(Lower triangular) Cholesky factor of the covariance matrix. If None, then the Cholesky factor of the covariance matrix
is computed when :attr:`Normal.cov_cholesky` is called and then cached. If specified, the value is returned by :attr:`Normal.cov_cholesky`.
In this case, its type and data type are compared to the type and data type of the covariance.
If the types do not match, an exception is thrown. If the data types do not match,
the data type of the Cholesky factor is promoted to the data type of the covariance matrix.

--------
RandomVariable : Class representing random variables.

Examples
--------
>>> import numpy as np
>>> from probnum import randvars
>>> x = randvars.Normal(mean=0.5, cov=1.0)
>>> rng = np.random.default_rng(42)
>>> x.sample(rng=rng, size=(2, 2))
array([[ 0.80471708, -0.53998411],
[ 1.2504512 ,  1.44056472]])
"""

# pylint: disable=too-many-locals,too-many-branches,too-many-statements
def __init__(
self,
mean: Union[float, np.floating, np.ndarray, linops.LinearOperator],
cov: Union[float, np.floating, np.ndarray, linops.LinearOperator],
cov_cholesky: Optional[
Union[float, np.floating, np.ndarray, linops.LinearOperator]
] = None,
):
# Type normalization
if np.isscalar(mean):
mean = _utils.as_numpy_scalar(mean)

if np.isscalar(cov):
cov = _utils.as_numpy_scalar(cov)

if np.isscalar(cov_cholesky):
cov_cholesky = _utils.as_numpy_scalar(cov_cholesky)

# Data type normalization
dtype = np.promote_types(mean.dtype, cov.dtype)

if not np.issubdtype(dtype, np.floating):
dtype = np.dtype(np.double)

mean = mean.astype(dtype, order="C", casting="safe", copy=False)
cov = cov.astype(dtype, order="C", casting="safe", copy=False)

# Shape checking
if not 0 <= mean.ndim <= 2:
raise ValueError(
f"Gaussian random variables must either be scalars, vectors, or "
f"matrices (or linear operators), but the given mean is a {mean.ndim}-"
f"dimensional tensor."
)

expected_cov_shape = (np.prod(mean.shape),) * 2 if len(mean.shape) > 0 else ()

if cov.shape != expected_cov_shape:
raise ValueError(
f"The covariance matrix must be of shape {expected_cov_shape}, but "
f"shape {cov.shape} was given."
)

# Method selection
univariate = mean.ndim == 0
dense = isinstance(mean, np.ndarray) and isinstance(cov, np.ndarray)
cov_operator = isinstance(cov, linops.LinearOperator)
compute_cov_cholesky: Callable[[], _ValueType] = None

if univariate:
# Univariate Gaussian
sample = self._univariate_sample
in_support = Normal._univariate_in_support
pdf = self._univariate_pdf
logpdf = self._univariate_logpdf
cdf = self._univariate_cdf
logcdf = self._univariate_logcdf
quantile = self._univariate_quantile

median = lambda: mean
var = lambda: cov
entropy = self._univariate_entropy

compute_cov_cholesky = self._univariate_cov_cholesky

elif dense or cov_operator:
# Multi- and matrixvariate Gaussians
sample = self._dense_sample
in_support = Normal._dense_in_support
pdf = self._dense_pdf
logpdf = self._dense_logpdf
cdf = self._dense_cdf
logcdf = self._dense_logcdf
quantile = None

median = None
var = self._dense_var
entropy = self._dense_entropy

compute_cov_cholesky = self.dense_cov_cholesky

# Ensure that the Cholesky factor has the same type as the covariance,
# and, if necessary, promote data types. Check for (in this order): type, shape, dtype.
if cov_cholesky is not None:

if not isinstance(cov_cholesky, type(cov)):
raise TypeError(
f"The covariance matrix is of type `{type(cov)}`, so its "
f"Cholesky decomposition must be of the same type, but an "
f"object of type `{type(cov_cholesky)}` was given."
)

if cov_cholesky.shape != cov.shape:
raise ValueError(
f"The cholesky decomposition of the covariance matrix must "
f"have the same shape as the covariance matrix, i.e. "
f"{cov.shape}, but shape {cov_cholesky.shape} was given"
)

if cov_cholesky.dtype != cov.dtype:
cov_cholesky = cov_cholesky.astype(
cov.dtype, casting="safe", copy=False
)

if cov_operator:
if isinstance(cov, linops.SymmetricKronecker):
m, n = mean.shape

if m != n or n != cov.A.shape[0] or n != cov.B.shape[1]:
raise ValueError(
"Normal distributions with symmetric Kronecker structured "
"kernels must have square mean and square kernels factors with "
"matching dimensions."
)

if cov.identical_factors:
sample = self._symmetric_kronecker_identical_factors_sample

compute_cov_cholesky = (
self._symmetric_kronecker_identical_factors_cov_cholesky
)
elif isinstance(cov, linops.Kronecker):
compute_cov_cholesky = self._kronecker_cov_cholesky
if mean.ndim == 2:
m, n = mean.shape

if (
m != cov.A.shape[0]
or m != cov.A.shape[1]
or n != cov.B.shape[0]
or n != cov.B.shape[1]
):
raise ValueError(
"Kronecker structured kernels must have factors with the same "
"shape as the mean."
)

else:
# This case handles all linear operators, for which no Cholesky
# factorization is implemented, yet.
# Computes the dense Cholesky and converts it to a LinearOperator.
compute_cov_cholesky = self._dense_cov_cholesky_as_linop

else:
raise ValueError(
f"Cannot instantiate normal distribution with mean of type "
f"{mean.__class__.__name__} and kernels of type "
f"{cov.__class__.__name__}."
)

super().__init__(
shape=mean.shape,
dtype=mean.dtype,
parameters={"mean": mean, "cov": cov},
sample=sample,
in_support=in_support,
pdf=pdf,
logpdf=logpdf,
cdf=cdf,
logcdf=logcdf,
quantile=quantile,
mode=lambda: mean,
median=median,
mean=lambda: mean,
cov=lambda: cov,
var=var,
entropy=entropy,
)

self._compute_cov_cholesky = compute_cov_cholesky
self._cov_cholesky = cov_cholesky

@property
def cov_cholesky(self) -> _ValueType:
"""Cholesky factor :math:`L` of the covariance
:math:`\\operatorname{Cov}(X) =LL^\\top`."""

if not self.cov_cholesky_is_precomputed:
self.precompute_cov_cholesky()
return self._cov_cholesky

[docs]    def precompute_cov_cholesky(
self,
damping_factor: Optional[FloatArgType] = None,
):
"""(P)recompute Cholesky factors (careful: in-place operation!)."""
if damping_factor is None:
damping_factor = config.covariance_inversion_damping
if self.cov_cholesky_is_precomputed:
raise Exception("A Cholesky factor is already available.")
self._cov_cholesky = self._compute_cov_cholesky(damping_factor=damping_factor)

@property
def cov_cholesky_is_precomputed(self) -> bool:
"""Return truth-value of whether the Cholesky factor of the covariance is

This happens if (i) the Cholesky factor is specified during
initialization or if (ii) the property `self.cov_cholesky` has
been called before.
"""
if self._cov_cholesky is None:
return False
return True

@cached_property
def dense_mean(self) -> Union[np.floating, np.ndarray]:
"""Dense representation of the mean."""
if isinstance(self.mean, linops.LinearOperator):
return self.mean.todense()
else:
return self.mean

@cached_property
def dense_cov(self) -> Union[np.floating, np.ndarray]:
"""Dense representation of the covariance."""
if isinstance(self.cov, linops.LinearOperator):
return self.cov.todense()
else:
return self.cov

def __getitem__(self, key: ArrayLikeGetitemArgType) -> "Normal":
"""Marginalization in multi- and matrixvariate normal random variables,

We support all modes of array indexing presented in

https://numpy.org/doc/1.19/reference/arrays.indexing.html.

Note that, currently, this method only works for multi- and matrixvariate
normal distributions.

Parameters
----------
key : int or slice or ndarray or tuple of None, int, slice, or ndarray
Indices, slice objects and/or boolean masks specifying which entries to keep
while marginalizing over all other entries.
"""

if not isinstance(key, tuple):
key = (key,)

# Select entries from mean
mean = self.dense_mean[key]

# Select submatrix from covariance matrix
cov = self.dense_cov.reshape(self.shape + self.shape)
cov = cov[key][(...,) + key]

if mean.ndim > 0:
cov = cov.reshape(mean.size, mean.size)

return Normal(
mean=mean,
cov=cov,
)

[docs]    def reshape(self, newshape: ShapeArgType) -> "Normal":
try:
reshaped_mean = self.dense_mean.reshape(newshape)
except ValueError as exc:
raise ValueError(
f"Cannot reshape this normal random variable to the given shape: "
f"{newshape}"
) from exc

reshaped_cov = self.dense_cov

if reshaped_mean.ndim > 0 and reshaped_cov.ndim == 0:
reshaped_cov = reshaped_cov.reshape(1, 1)

return Normal(
mean=reshaped_mean,
cov=reshaped_cov,
)

[docs]    def transpose(self, *axes: int) -> "Normal":
if len(axes) == 1 and isinstance(axes[0], tuple):
axes = axes[0]
elif (len(axes) == 1 and axes[0] is None) or len(axes) == 0:
axes = tuple(reversed(range(self.ndim)))

mean_t = self.dense_mean.transpose(*axes).copy()

# Transpose covariance
cov_axes = axes + tuple(mean_t.ndim + axis for axis in axes)
cov_t = self.dense_cov.reshape(self.shape + self.shape)
cov_t = cov_t.transpose(*cov_axes).copy()

if mean_t.ndim > 0:
cov_t = cov_t.reshape(mean_t.size, mean_t.size)

return Normal(
mean=mean_t,
cov=cov_t,
)

# Unary arithmetic operations

def __neg__(self) -> "Normal":
return Normal(
mean=-self.mean,
cov=self.cov,
)

def __pos__(self) -> "Normal":
return Normal(
mean=+self.mean,
cov=self.cov,
)

# TODO: Overwrite __abs__ and add absolute moments of normal
# TODO: (https://arxiv.org/pdf/1209.4340.pdf)

# Binary arithmetic operations

def _add_normal(self, other: "Normal") -> "Normal":
if other.shape != self.shape:
raise ValueError(
"Addition of two normally distributed random variables is only "
"possible if both operands have the same shape."
)

return Normal(
mean=self.mean + other.mean,
cov=self.cov + other.cov,
)

def _sub_normal(self, other: "Normal") -> "Normal":
if other.shape != self.shape:
raise ValueError(
"Subtraction of two normally distributed random variables is only "
"possible if both operands have the same shape."
)

return Normal(
mean=self.mean - other.mean,
cov=self.cov + other.cov,
)

# Univariate Gaussians
def _univariate_cov_cholesky(
self,
damping_factor: FloatArgType,
) -> np.floating:
return np.sqrt(self.cov + damping_factor)

def _univariate_sample(
self,
rng: np.random.Generator,
size: ShapeType = (),
) -> Union[np.floating, np.ndarray]:
sample = scipy.stats.norm.rvs(
loc=self.mean, scale=self.std, size=size, random_state=rng
)

if np.isscalar(sample):
sample = _utils.as_numpy_scalar(sample, dtype=self.dtype)
else:
sample = sample.astype(self.dtype)

assert sample.shape == size

return sample

@staticmethod
def _univariate_in_support(x: _ValueType) -> bool:
return np.isfinite(x)

def _univariate_pdf(self, x: _ValueType) -> np.float_:
return scipy.stats.norm.pdf(x, loc=self.mean, scale=self.std)

def _univariate_logpdf(self, x: _ValueType) -> np.float_:
return scipy.stats.norm.logpdf(x, loc=self.mean, scale=self.std)

def _univariate_cdf(self, x: _ValueType) -> np.float_:
return scipy.stats.norm.cdf(x, loc=self.mean, scale=self.std)

def _univariate_logcdf(self, x: _ValueType) -> np.float_:
return scipy.stats.norm.logcdf(x, loc=self.mean, scale=self.std)

def _univariate_quantile(self, p: FloatArgType) -> np.floating:
return scipy.stats.norm.ppf(p, loc=self.mean, scale=self.std)

def _univariate_entropy(self: _ValueType) -> np.float_:
return _utils.as_numpy_scalar(
scipy.stats.norm.entropy(loc=self.mean, scale=self.std),
dtype=np.float_,
)

# Multi- and matrixvariate Gaussians
[docs]    def dense_cov_cholesky(
self,
damping_factor: Optional[FloatArgType] = None,
) -> np.ndarray:
"""Compute the Cholesky factorization of the covariance from its dense
representation."""
if damping_factor is None:
damping_factor = config.covariance_inversion_damping
dense_cov = self.dense_cov

return scipy.linalg.cholesky(
dense_cov + damping_factor * np.eye(self.size, dtype=self.dtype),
lower=True,
)

def _dense_cov_cholesky_as_linop(
self, damping_factor: FloatArgType
) -> linops.LinearOperator:
return linops.aslinop(self.dense_cov_cholesky(damping_factor=damping_factor))

def _dense_sample(
self, rng: np.random.Generator, size: ShapeType = ()
) -> np.ndarray:
sample = scipy.stats.multivariate_normal.rvs(
mean=self.dense_mean.ravel(),
cov=self.dense_cov,
size=size,
random_state=rng,
)

return sample.reshape(sample.shape[:-1] + self.shape)

@staticmethod
def _arg_todense(x: Union[np.ndarray, linops.LinearOperator]) -> np.ndarray:
if isinstance(x, linops.LinearOperator):
return x.todense()
elif isinstance(x, np.ndarray):
return x
else:
raise ValueError(f"Unsupported argument type {type(x)}")

@staticmethod
def _dense_in_support(x: _ValueType) -> bool:
return np.all(np.isfinite(Normal._arg_todense(x)))

def _dense_pdf(self, x: _ValueType) -> np.float_:
return scipy.stats.multivariate_normal.pdf(
Normal._arg_todense(x).reshape(x.shape[: -self.ndim] + (-1,)),
mean=self.dense_mean.ravel(),
cov=self.dense_cov,
)

def _dense_logpdf(self, x: _ValueType) -> np.float_:
return scipy.stats.multivariate_normal.logpdf(
Normal._arg_todense(x).reshape(x.shape[: -self.ndim] + (-1,)),
mean=self.dense_mean.ravel(),
cov=self.dense_cov,
)

def _dense_cdf(self, x: _ValueType) -> np.float_:
return scipy.stats.multivariate_normal.cdf(
Normal._arg_todense(x).reshape(x.shape[: -self.ndim] + (-1,)),
mean=self.dense_mean.ravel(),
cov=self.dense_cov,
)

def _dense_logcdf(self, x: _ValueType) -> np.float_:
return scipy.stats.multivariate_normal.logcdf(
Normal._arg_todense(x).reshape(x.shape[: -self.ndim] + (-1,)),
mean=self.dense_mean.ravel(),
cov=self.dense_cov,
)

def _dense_var(self) -> np.ndarray:
return np.diag(self.dense_cov).reshape(self.shape)

def _dense_entropy(self) -> np.float_:
return _utils.as_numpy_scalar(
scipy.stats.multivariate_normal.entropy(
mean=self.dense_mean.ravel(),
cov=self.dense_cov,
),
dtype=np.float_,
)

# Matrixvariate Gaussian with Kronecker covariance
def _kronecker_cov_cholesky(
self,
damping_factor: FloatArgType,
) -> linops.Kronecker:
assert isinstance(self.cov, linops.Kronecker)

A = self.cov.A.todense()
B = self.cov.B.todense()

return linops.Kronecker(
A=scipy.linalg.cholesky(
A + damping_factor * np.eye(A.shape[0], dtype=self.dtype),
lower=True,
),
B=scipy.linalg.cholesky(
B + damping_factor * np.eye(B.shape[0], dtype=self.dtype),
lower=True,
),
)

# Matrixvariate Gaussian with symmetric Kronecker covariance from identical
# factors
def _symmetric_kronecker_identical_factors_cov_cholesky(
self,
damping_factor: FloatArgType,
) -> linops.SymmetricKronecker:
assert (
isinstance(self.cov, linops.SymmetricKronecker)
and self.cov.identical_factors
)

A = self.cov.A.todense()

return linops.SymmetricKronecker(
A=scipy.linalg.cholesky(
A + damping_factor * np.eye(A.shape[0], dtype=self.dtype),
lower=True,
),
)

def _symmetric_kronecker_identical_factors_sample(
self, rng: np.random.Generator, size: ShapeType = ()
) -> np.ndarray:
assert (
isinstance(self.cov, linops.SymmetricKronecker)
and self.cov.identical_factors
)

n = self.mean.shape[1]

# Draw standard normal samples
size_sample = (n * n,) + size

stdnormal_samples = scipy.stats.norm.rvs(size=size_sample, random_state=rng)

# Appendix E: Bartels, S., Probabilistic Linear Algebra, PhD Thesis 2019
samples_scaled = linops.Symmetrize(n) @ (self.cov_cholesky @ stdnormal_samples)

# TODO: can we avoid todense here and just return operator samples?
return self.dense_mean[None, :, :] + samples_scaled.T.reshape(-1, n, n)
```