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Mean, covariance, and input shape
A multivariate normal distribution describes a vector of values. Its mean is a location vector, and its cov matrix describes each component’s variance and the covariance between pairs of components. For a two-component distribution, for example, mean has two entries and cov is a 2×2 matrix.
SciPy accepts covariance as a scalar, a vector of diagonal entries, a two-dimensional array, or a Covariance object. If mean is omitted, it defaults to the zero vector. The last axis of the input x always identifies the components: a point has shape (d,), a batch has shape (n, d), and a grid may have shape (..., d), where d is the number of dimensions.
The SciPy v1.18.0 reference describes the object as “A multivariate normal random variable.” Its density uses the mean μ, covariance Σ, and rank k:
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f(x) = 1 / sqrt((2π)^k det(Σ)) × exp(-1/2 (x - μ)ᵀ Σ⁻¹ (x - μ))
For singular covariance, SciPy extends the density definition using the pseudo-determinant and pseudo-inverse; the ordinary full-rank formula above is not a complete account of that degenerate case. See the SciPy v1.18.0 multivariate_normal reference for the API and its cited treatment of the degenerate case.
Direct calls or a frozen distribution?
Use the module’s direct methods when parameters vary from call to call. Construct a frozen distribution when the mean and covariance stay fixed across repeated evaluations or draws. A frozen object stores those parameters; its methods then take the point or sampling options rather than requiring the parameters again.
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import numpy as np
from scipy.stats import multivariate_normal
mean = np.array([0.0, 1.0])
cov = np.array([[1.0, 0.3],
[0.3, 2.0]])
# Direct call: parameters are supplied to the method.
value = multivariate_normal.pdf([0.5, 1.5], mean=mean, cov=cov)
# Frozen distribution: parameters are retained for repeated calls.
rv = multivariate_normal(mean=mean, cov=cov)
values = rv.pdf([[0.5, 1.5], [1.0, 2.0]])The point passed to pdf has shape (2,); the batch has shape (2, 2). In both cases, the final axis contains the two component coordinates.
pdf: evaluate density, not point probability
pdf(x, mean=None, cov=1, allow_singular=False) returns a probability density at each supplied point. A density is not the probability that a continuous random variable equals that exact point; probabilities are associated with regions and can be computed with a cumulative distribution function.
For a two-dimensional grid, use the final axis for coordinates. Here each grid location becomes one two-component point, so the resulting density array has the same two leading grid dimensions.
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x, y = np.mgrid[-3:3:0.1, -3:3:0.1]
points = np.dstack((x, y)) # shape (..., 2); last axis is (x, y)
pdf_values = rv.pdf(points)For calculations that are more convenient on a logarithmic scale, use logpdf with the same point-axis convention.
cdf: cumulative and rectangular probabilities
cdf evaluates cumulative probability up to an upper point. In SciPy v1.18.0, its signature is cdf(x, mean=None, cov=1, allow_singular=False, maxpts=1000000*dim, abseps=1e-5, releps=1e-5, lower_limit=None). The final axis of x still contains the component coordinates.
For the probability of a rectangle, supply the upper corner as x and the lower corner as lower_limit. In two dimensions, the probability is for points within both component bounds:
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lower = np.array([-1.0, -0.5])
upper = np.array([1.0, 1.5])
probability = rv.cdf(upper, lower_limit=lower)The CDF calculation exposes a work budget and error controls: maxpts limits points used in the calculation, while abseps and releps specify absolute and relative error tolerances. Adjust them when the default calculation is insufficient for your application, balancing desired numerical accuracy against computational work. These are numerical controls, not a guarantee that every calculation attains a particular error.
rvs: draw random observations
rvs generates random draws; it does not evaluate density or estimate parameters. The direct method signature is rvs(mean=None, cov=1, size=1, random_state=None). The constructor instead accepts seed, which may be None, an integer, a RandomState, or a Generator. The following uses a seeded generator and a frozen distribution:
rng = np.random.default_rng(42)
rv = multivariate_normal(mean=mean, cov=cov, seed=rng)
samples = rv.rvs(size=1000)
# Each sample is a two-component point: shape (1000, 2).For repeatability, retain the generator state or recreate the generator with the intended seed and call sequence. Reusing a generator advances its state, so later draws need not match the first batch.
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fit: estimate parameters from data
The SciPy v1.18.0 API lists fit(x, fix_mean=None, fix_cov=None), but that signature alone does not establish the orientation of the input data, the estimator, the return structure, or the exact behavior of the fixing options. Do not infer those details from the separate univariate scipy.stats.fit API or assume that rows versus columns represent observations.
Before relying on fit in code, consult the implementation and documentation for the exact SciPy release installed, then verify the expected input orientation and outputs with a small known dataset. This is especially important when using fix_mean or fix_cov, since the listed signature does not explain precisely what each option constrains.
Covariance validity and singular cases
With an ordinary array covariance, allow_singular=False is the default, and the covariance must be strictly positive definite. If the model is intentionally rank deficient, pass allow_singular=True; the covariance must still be positive semidefinite. SciPy then uses the pseudo-determinant and pseudo-inverse for the singular case.
SciPy does not check that an array covariance is symmetric, and only its lower triangular portion is used. Supply a valid symmetric covariance yourself rather than relying on the function to detect an asymmetric matrix. If cov is a Covariance object, allow_singular is ignored.
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| Need | Use | What it returns or controls |
|---|---|---|
| Evaluate density at point(s) | pdf or logpdf |
Density values, not point probabilities |
| Calculate cumulative or rectangular probability | cdf |
Cumulative probability; work and error settings include maxpts, abseps, and releps |
| Generate random observations | rvs |
Draws controlled by size and a seed or random-state object |
| Estimate distribution parameters from data | fit |
Consult the target release’s implementation for data orientation, estimator, returns, and fixed-parameter semantics |
For all methods that accept points, check the final axis first: it must contain the components of each observation, while earlier axes can represent multiple points or a grid.
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