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# tfp.substrates.jax.distributions.MultivariateNormalDiag

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The multivariate normal distribution on `R^k`.

The Multivariate Normal distribution is defined over `R^k` and parameterized by a (batch of) length-`k` `loc` vector (aka 'mu') and a (batch of) `k x k` `scale` matrix; `covariance = scale @ scale.T` where `@` denotes matrix-multiplication.

#### Mathematical Details

The probability density function (pdf) is,

``````pdf(x; loc, scale) = exp(-0.5 ||y||**2) / Z,
y = inv(scale) @ (x - loc),
Z = (2 pi)**(0.5 k) |det(scale)|,
``````

where:

• `loc` is a vector in `R^k`,
• `scale` is a linear operator in `R^{k x k}`, `cov = scale @ scale.T`,
• `Z` denotes the normalization constant, and,
• `||y||**2` denotes the squared Euclidean norm of `y`.

A (non-batch) `scale` matrix is:

``````scale = diag(scale_diag + scale_identity_multiplier * ones(k))
``````

where:

• `scale_diag.shape = [k]`, and,
• `scale_identity_multiplier.shape = []`.

If both `scale_diag` and `scale_identity_multiplier` are `None`, then `scale` is the Identity matrix.

The MultivariateNormal distribution is a member of the location-scale family, i.e., it can be constructed as,

``````X ~ MultivariateNormal(loc=0, scale=1)   # Identity scale, zero shift.
Y = scale @ X + loc
``````

#### Examples

``````tfd = tfp.distributions

# Initialize a single 2-variate Gaussian.
mvn = tfd.MultivariateNormalDiag(
loc=[1., -1],
scale_diag=[1, 2.])

mvn.mean()
# ==> [1., -1]

mvn.stddev()
# ==> [1., 2]

# Evaluate this on an observation in `R^2`, returning a scalar.
mvn.prob([-1., 0])  # shape: []

# Initialize a 3-batch, 2-variate scaled-identity Gaussian.
mvn = tfd.MultivariateNormalDiag(
loc=[1., -1],
scale_identity_multiplier=[1, 2., 3])

mvn.mean()  # shape: [3, 2]
# ==> [[1., -1]
#      [1, -1],
#      [1, -1]]

mvn.stddev()  # shape: [3, 2]
# ==> [[1., 1],
#      [2, 2],
#      [3, 3]]

# Evaluate this on an observation in `R^2`, returning a length-3 vector.
mvn.prob([-1., 0])  # shape: [3]

# Initialize a 2-batch of 3-variate Gaussians.
mvn = tfd.MultivariateNormalDiag(
loc=[[1., 2, 3],
[11, 22, 33]],           # shape: [2, 3]
scale_diag=[[1., 2, 3],
[0.5, 1, 1.5]])  # shape: [2, 3]

# Evaluate this on a two observations, each in `R^3`, returning a length-2
# vector.
x = [[-1., 0, 1],
[-11, 0, 11.]]   # shape: [2, 3].
mvn.prob(x)    # shape: [2]
``````

`loc` Floating-point `Tensor`. If this is set to `None`, `loc` is implicitly `0`. When specified, may have shape `[B1, ..., Bb, k]` where `b >= 0` and `k` is the event size.
`scale_diag` Non-zero, floating-point `Tensor` representing a diagonal matrix added to `scale`. May have shape `[B1, ..., Bb, k]`, `b >= 0`, and characterizes `b`-batches of `k x k` diagonal matrices added to `scale`. When both `scale_identity_multiplier` and `scale_diag` are `None` then `scale` is the `Identity`.
`scale_identity_multiplier` Non-zero, floating-point `Tensor` representing a scaled-identity-matrix added to `scale`. May have shape `[B1, ..., Bb]`, `b >= 0`, and characterizes `b`-batches of scaled `k x k` identity matrices added to `scale`. When both `scale_identity_multiplier` and `scale_diag` are `None` then `scale` is the `Identity`.
`validate_args` Python `bool`, default `False`. When `True` distribution parameters are checked for validity despite possibly degrading runtime performance. When `False` invalid inputs may silently render incorrect outputs.
`allow_nan_stats` Python `bool`, default `True`. When `True`, statistics (e.g., mean, mode, variance) use the value '`NaN`' to indicate the result is undefined. When `False`, an exception is raised if one or more of the statistic's batch members are undefined.
`experimental_use_kahan_sum` Python `bool`. When `True`, we use Kahan summation to aggregate independent underlying log_prob values as well as when computing the log-determinant of the scale matrix. Doing so improves against the precision of a naive float32 sum. This can be noticeable in particular for large dimensions in float32. See CPU caveat on `tfp.math.reduce_kahan_sum`.
`name` Python `str` name prefixed to Ops created by this class.

`ValueError` if at most `scale_identity_multiplier` is specified.

`allow_nan_stats` Python `bool` describing behavior when a stat is undefined.

Stats return +/- infinity when it makes sense. E.g., the variance of a Cauchy distribution is infinity. However, sometimes the statistic is undefined, e.g., if a distribution's pdf does not achieve a maximum within the support of the distribution, the mode is undefined. If the mean is undefined, then by definition the variance is undefined. E.g. the mean for Student's T for df = 1 is undefined (no clear way to say it is either + or - infinity), so the variance = E[(X - mean)**2] is also undefined.

`batch_shape` Shape of a single sample from a single event index as a `TensorShape`.

May be partially defined or unknown.

The batch dimensions are indexes into independent, non-identical parameterizations of this distribution.

`bijector` Function transforming x => y.
`distribution` Base distribution, p(x).
`dtype` The `DType` of `Tensor`s handled by this `Distribution`.
`event_shape` Shape of a single sample from a single batch as a `TensorShape`.

May be partially defined or unknown.

`experimental_shard_axis_names` The list or structure of lists of active shard axis names.
`loc` The `loc` `Tensor` in `Y = scale @ X + loc`.
`name` Name prepended to all ops created by this `Distribution`.
`parameters` Dictionary of parameters used to instantiate this `Distribution`.
`reparameterization_type` Describes how samples from the distribution are reparameterized.

Currently this is one of the static instances `tfd.FULLY_REPARAMETERIZED` or `tfd.NOT_REPARAMETERIZED`.

`scale` The `scale` `LinearOperator` in `Y = scale @ X + loc`.
`trainable_variables`

`validate_args` Python `bool` indicating possibly expensive checks are enabled.
`variables`

## Methods

### `batch_shape_tensor`

View source

Shape of a single sample from a single event index as a 1-D `Tensor`.

The batch dimensions are indexes into independent, non-identical parameterizations of this distribution.

Args
`name` name to give to the op

Returns
`batch_shape` `Tensor`.

### `cdf`

View source

Cumulative distribution function.

Given random variable `X`, the cumulative distribution function `cdf` is:

``````cdf(x) := P[X <= x]
``````

Args
`value` `float` or `double` `Tensor`.
`name` Python `str` prepended to names of ops created by this function.
`**kwargs` Named arguments forwarded to subclass implementation.

Returns
`cdf` a `Tensor` of shape `sample_shape(x) + self.batch_shape` with values of type `self.dtype`.

### `copy`

View source

Creates a deep copy of the distribution.

Args
`**override_parameters_kwargs` String/value dictionary of initialization arguments to override with new values.

Returns
`distribution` A new instance of `type(self)` initialized from the union of self.parameters and override_parameters_kwargs, i.e., `dict(self.parameters, **override_parameters_kwargs)`.

### `covariance`

View source

Covariance.

Covariance is (possibly) defined only for non-scalar-event distributions.

For example, for a length-`k`, vector-valued distribution, it is calculated as,

``````Cov[i, j] = Covariance(X_i, X_j) = E[(X_i - E[X_i]) (X_j - E[X_j])]
``````

where `Cov` is a (batch of) `k x k` matrix, `0 <= (i, j) < k`, and `E` denotes expectation.

Alternatively, for non-vector, multivariate distributions (e.g., matrix-valued, Wishart), `Covariance` shall return a (batch of) matrices under some vectorization of the events, i.e.,

``````Cov[i, j] = Covariance(Vec(X)_i, Vec(X)_j) = [as above]
``````

where `Cov` is a (batch of) `k' x k'` matrices, `0 <= (i, j) < k' = reduce_prod(event_shape)`, and