Normal¶

class probnum.randvars.Normal(mean, cov, cov_cholesky=None)¶

Bases: ContinuousRandomVariable[Union[floating, ndarray, LinearOperator]]

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 Normal.cov_cholesky is called and then cached. If specified, the value is returned by 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.

See also

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]])

Attributes Summary

`T`	Transpose the random variable.
`cov`	Covariance \(\operatorname{Cov}(X) = \mathbb{E}((X-\mathbb{E}(X))(X-\mathbb{E}(X))^\top)\) of the random variable.
`cov_cholesky`	Cholesky factor \(L\) of the covariance \(\operatorname{Cov}(X) =LL^\top\).
`cov_cholesky_is_precomputed`	Return truth-value of whether the Cholesky factor of the covariance is readily available.
`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.
`entropy`	Information-theoretic entropy \(H(X)\) of the random variable.
`mean`	Mean \(\mathbb{E}(X)\) of the random variable.
`median`	Median of the random variable.
`median_dtype`	The dtype of the `median`.
`mode`	Mode of the random variable.
`moment_dtype`	The dtype of any (function of a) moment of the random variable, e.g.
`ndim`	Number of dimensions of realizations of the random variable.
`parameters`	Parameters of the associated probability distribution.
`shape`	Shape of realizations of the random variable.
`size`	Size of realizations of the random variable, defined as the product over all components of `shape()`.
`std`	Standard deviation of the random variable.
`var`	Variance \(\operatorname{Var}(X) = \mathbb{E}((X-\mathbb{E}(X))^2)\) of the random variable.

Methods Summary

`cdf`(x)	Cumulative distribution function.
`dense_cov_cholesky`([damping_factor])	Compute the Cholesky factorization of the covariance from its dense representation.
`in_support`(x)	Check whether the random variable takes value `x` with non-zero probability, i.e. if `x` 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)	Log-cumulative distribution function.
`logpdf`(x)	Natural logarithm of the probability density function.
`pdf`(x)	Probability density function.
`precompute_cov_cholesky`([damping_factor])	(P)recompute Cholesky factors (careful: in-place operation!).
`quantile`(p)	Quantile function.
`reshape`(newshape)	Give a new shape to a random variable.
`sample`(rng[, size])	Draw realizations from a random variable.
`transpose`(*axes)	Transpose the random variable.

Attributes Documentation

T¶

Transpose the random variable.

Parameters: axes – See documentation of numpy.ndarray.transpose().

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 \(\operatorname{Cov}(X) =LL^\top\).

cov_cholesky_is_precomputed¶

Return truth-value of whether the Cholesky factor of the covariance is readily available.

This happens if (i) the Cholesky factor is specified during initialization or if (ii) the property self.cov_cholesky has been called before.

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.

entropy¶: Information-theoretic 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 and numpy.float_. This is motivated by the fact that, even for discrete random variables, e.g. integer-valued random variables, the median 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 \(2.5\).

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, or std. It will be set to the dtype arising from the multiplication of values with dtypes dtype and numpy.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 dtypes numpy.float_ and dtype, respectively.

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.

shape¶: Shape of realizations of the random variable.

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 the shape of the random variable. The cdf evaluation will be broadcast over all additional dimensions.
Return type: float64

dense_cov_cholesky(damping_factor=None)[source]¶

Compute the Cholesky factorization of the covariance from its dense representation.

Parameters: damping_factor (Optional[FloatLike]) –
Return type: np.ndarray

in_support(x)¶

Check whether the random variable takes value x with non-zero probability, i.e. if x is in the support of its distribution.

Parameters: x (ValueType) – Input value.
Return type: bool

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 and numpy.float_. This is motivated by the fact that, even for discrete random variables, e.g. integer-valued random variables, the median 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 \(2.5\).

Parameters: value_dtype (DTypeLike) – Dtype of a value.
Return type: dtype

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, or std. Returns the dtype arising from the multiplication of values with dtypes dtype and numpy.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 dtypes numpy.float_ and dtype, respectively.

Parameters: value_dtype (DTypeLike) – Dtype of a value.
Return type: dtype

logcdf(x)¶

Log-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 the shape 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 log-probability density function. The shape of this argument should be (..., S1, ..., SN), where (S1, ..., SN) is the shape 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 Radon-Nikodym 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 the shape of the random variable. The pdf evaluation will be broadcast over all additional dimensions.
Return type: float64

precompute_cov_cholesky(damping_factor=None)[source]¶

(P)recompute Cholesky factors (careful: in-place operation!).

Parameters: damping_factor (Optional[FloatLike]) –

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 the median as it is defined in this class. See https://en.wikipedia.org/wiki/Quantile_function for more details and examples.

Parameters: p (FloatLike) –
Return type: ValueType

reshape(newshape)[source]¶

Give a new shape to a random variable.

Parameters: newshape (ShapeLike) – New shape for the random variable. It must be compatible with the original shape.
Return type: Normal

sample(rng, size=())¶

Draw realizations from a random variable.

Parameters

rng (Generator) – Random number generator used for sampling.
size (ShapeLike) – Size of the drawn sample of realizations.

Return type

ValueType

transpose(*axes)[source]¶

Transpose the random variable.

Parameters: axes (int) – See documentation of numpy.ndarray.transpose().
Return type: Normal