Normal¶

class
probnum.random_variables.
Normal
(mean, cov, cov_cholesky=None, random_state=None)¶ Bases:
probnum.random_variables.ContinuousRandomVariable
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 closedform Bayesian inference given linear observations.
 Parameters
mean (
Union
[float
,floating
,ndarray
,LinearOperator
]) – Mean of the random variable.cov (
Union
[float
,floating
,ndarray
,LinearOperator
]) – (Co)variance of the random variable.random_state (
Union
[None
,int
,RandomState
,Generator
]) – Random state of the random variable. If None (or np.random), the globalnumpy.random
state is used. If integer, it is used to seed the localRandomState
instance.
See also
RandomVariable
Class representing random variables.
Examples
>>> from probnum import random_variables as rvs >>> x = rvs.Normal(mean=0.5, cov=1.0, random_state=42) >>> x.sample(size=(2, 2)) array([[0.99671415, 0.3617357 ], [1.14768854, 2.02302986]])
Attributes Summary
Transpose the random variable.
Covariance \(\operatorname{Cov}(X) = \mathbb{E}((X\mathbb{E}(X))(X\mathbb{E}(X))^\top)\) of the random variable.
Cholesky factor \(L\) of the covariance \(\operatorname{Cov}(X) =LL^\top\).
Dense representation of the covariance.
Dense representation of the mean.
Data type of (elements of) a realization of this random variable.
Informationtheoretic entropy \(H(X)\) of the random variable.
Mean \(\mathbb{E}(X)\) of the random variable.
Median of the random variable.
The dtype of the
median
.Mode of the random variable.
The dtype of any (function of a) moment of the random variable, e.g.
Number of dimensions of realizations of the random variable.
Parameters of the associated probability distribution.
Random state of the random variable.
Shape of realizations of the random variable.
Size of realizations of the random variable, defined as the product over all components of
shape()
.Standard deviation of the random variable.
Variance \(\operatorname{Var}(X) = \mathbb{E}((X\mathbb{E}(X))^2)\) of the random variable.
Methods Summary
cdf
(x)Cumulative distribution function.
Compute the Cholesky factorization of the covariance from its dense representation.
in_support
(x)Check whether the random variable takes value
x
with nonzero probability, i.e. ifx
is in the support of its distribution.infer_median_dtype
(value_dtype)Infer the dtype of the median.
infer_moment_dtype
(value_dtype)Infer the dtype of any moment.
logcdf
(x)Logcumulative distribution function.
logpdf
(x)Natural logarithm of the probability density function.
pdf
(x)Probability density function.
quantile
(p)Quantile function.
reshape
(newshape)Give a new shape to a random variable.
sample
([size])Draw realizations from a random variable.
transpose
(*axes)Transpose the random variable.
Attributes Documentation

T
¶ Transpose the random variable.
 Parameters
axes (
int
) – See documentation of numpy.ndarray.transpose. Return type

cov
¶ Covariance \(\operatorname{Cov}(X) = \mathbb{E}((X\mathbb{E}(X))(X\mathbb{E}(X))^\top)\) of the random variable.
To learn about the dtype of the covariance, see
moment_dtype
.

cov_cholesky
¶ Cholesky factor \(L\) of the covariance \(\operatorname{Cov}(X) =LL^\top\).

dense_cov
¶ Dense representation of the covariance.

dense_mean
¶ Dense representation of the mean.

dtype
¶ Data type of (elements of) a realization of this random variable.
 Return type
dtype

entropy
¶ Informationtheoretic entropy \(H(X)\) of the random variable.

mean
¶ Mean \(\mathbb{E}(X)\) of the random variable.
To learn about the dtype of the mean, see
moment_dtype
.

median
¶ Median of the random variable.
To learn about the dtype of the median, see
median_dtype
.

median_dtype
¶ The dtype of the
median
.It will be set to the dtype arising from the multiplication of values with dtypes
dtype
andnp.float_
. This is motivated by the fact that, even for discrete random variables, e.g. integervalued random variables, themedian
might lie in between two values in which case these values are averaged. For example, a uniform random variable on \(\{ 1, 2, 3, 4 \}\) will have a median of \(2.5\). Return type
dtype

mode
¶ Mode of the random variable.

moment_dtype
¶ The dtype of any (function of a) moment of the random variable, e.g. its
mean
,cov
,var
, orstd
. It will be set to the dtype arising from the multiplication of values with dtypesdtype
andnp.float_
. This is motivated by the mathematical definition of a moment as a sum or an integral over products of probabilities and values of the random variable, which are represented as using the dtypesnp.float_
anddtype
, respectively. Return type
dtype

ndim
¶ Number of dimensions of realizations of the random variable.

parameters
¶ Parameters of the associated probability distribution.
The parameters of the probability distribution of the random variable, e.g. mean, variance, scale, rate, etc. stored in a
dict
.

random_state
¶ Random state of the random variable.
This attribute defines the RandomState object to use for drawing realizations from this random variable. If None (or np.random), the global np.random state is used. If integer, it is used to seed the local
RandomState
instance. Return type
Union
[RandomState
,Generator
]

size
¶ Size of realizations of the random variable, defined as the product over all components of
shape()
.

std
¶ Standard deviation of the random variable.
To learn about the dtype of the standard deviation, see
moment_dtype
.

var
¶ Variance \(\operatorname{Var}(X) = \mathbb{E}((X\mathbb{E}(X))^2)\) of the random variable.
To learn about the dtype of the variance, see
moment_dtype
.
Methods Documentation

cdf
(x)¶ Cumulative distribution function.
 Parameters
x (~ValueType) – Evaluation points of the cumulative distribution function. The shape of this argument should be
(..., S1, ..., SN)
, where(S1, ..., SN)
is theshape
of the random variable. The cdf evaluation will be broadcast over all additional dimensions. Return type
float64

dense_cov_cholesky
()[source]¶ Compute the Cholesky factorization of the covariance from its dense representation.
 Return type
ndarray

in_support
(x)¶ Check whether the random variable takes value
x
with nonzero probability, i.e. ifx
is in the support of its distribution. Parameters
x (~ValueType) – Input value.
 Return type

static
infer_median_dtype
(value_dtype)¶ Infer the dtype of the median.
Set the dtype to the dtype arising from the multiplication of values with dtypes
dtype
andnp.float_
. This is motivated by the fact that, even for discrete random variables, e.g. integervalued random variables, themedian
might lie in between two values in which case these values are averaged. For example, a uniform random variable on \(\{ 1, 2, 3, 4 \}\) will have a median of \(2.5\).

static
infer_moment_dtype
(value_dtype)¶ Infer the dtype of any moment.
Infers the dtype of any (function of a) moment of the random variable, e.g. its
mean
,cov
,var
, orstd
. Returns the dtype arising from the multiplication of values with dtypesdtype
andnp.float_
. This is motivated by the mathematical definition of a moment as a sum or an integral over products of probabilities and values of the random variable, which are represented as using the dtypesnp.float_
anddtype
, respectively.

logcdf
(x)¶ Logcumulative distribution function.
 Parameters
x (~ValueType) – Evaluation points of the cumulative distribution function. The shape of this argument should be
(..., S1, ..., SN)
, where(S1, ..., SN)
is theshape
of the random variable. The logcdf evaluation will be broadcast over all additional dimensions. Return type
float64

logpdf
(x)¶ Natural logarithm of the probability density function.
 Parameters
x (~ValueType) – Evaluation points of the logprobability density function. The shape of this argument should be
(..., S1, ..., SN)
, where(S1, ..., SN)
is theshape
of the random variable. The logpdf evaluation will be broadcast over all additional dimensions. Return type
float64

pdf
(x)¶ Probability density function.
The area under the curve defined by the probability density function specifies the probability of the random variable \(X\) taking values within that area.
Probability density functions are defined as the RadonNikodym derivative of the pushforward measure \(P \circ X^{1}\) with respect to the Lebesgue measure for a given probability measure \(P\). Following convention we always assume the Lebesgue measure as a base measure unless stated otherwise.
 Parameters
x (~ValueType) – Evaluation points of the probability density function. The shape of this argument should be
(..., S1, ..., SN)
, where(S1, ..., SN)
is theshape
of the random variable. The pdf evaluation will be broadcast over all additional dimensions. Return type
float64

quantile
(p)¶ Quantile function.
The quantile function \(Q \colon [0, 1] \to \mathbb{R}\) of a random variable \(X\) is defined as \(Q(p) = \inf\{ x \in \mathbb{R} \colon p \le F_X(x) \}\), where \(F_X \colon \mathbb{R} \to [0, 1]\) is the
cdf()
of the random variable. From the definition it follows that the quantile function always returns values of the same dtype as the random variable. For instance, for a discrete distribution over the integers, the returned quantiles will also be integers. This means that, in general, \(Q(0.5)\) is not equal to themedian
as it is defined in this class. See https://en.wikipedia.org/wiki/Quantile_function for more details and examples. Return type
~ValueType

sample
(size=())¶ Draw realizations from a random variable.