mixle.stats.univariate.continuous.laplace module

Laplace distributions over real values.

Reference: Johnson, Kotz & Balakrishnan, Continuous Univariate Distributions (2nd ed., Wiley, 1994/95).

class LaplaceDistribution(mu, b, name=None, keys=None)[source]

Bases: SequenceEncodableProbabilityDistribution

Laplace distribution with location mu and scale b > 0.

Parameters:
classmethod compute_capabilities()[source]

Describe backend support for generated Laplace kernels.

classmethod compute_declaration()[source]

Return the structured compute declaration for Laplace distributions.

density(x)[source]

Return the probability density or mass at a single observation.

Parameters:

x (float)

Return type:

float

log_density(x)[source]

Return the log-density or log-mass at a single observation.

Parameters:

x (float)

Return type:

float

seq_log_density(x)[source]

Return vectorized log-density values for sequence-encoded observations.

Parameters:

x (ndarray)

Return type:

ndarray

static backend_log_density_from_params(x, mu, b, engine)[source]

Engine-neutral Laplace log-density from explicit parameters.

Parameters:
Return type:

Any

backend_seq_log_density(x, engine)[source]

Engine-neutral vectorized log-density for encoded data.

Parameters:
Return type:

Any

classmethod backend_stacked_params(dists, engine)[source]

Return stacked Laplace parameters for a homogeneous mixture kernel.

Parameters:
Return type:

dict[str, Any]

classmethod backend_stacked_log_density(x, params, engine)[source]

Return an (n, k) matrix of Laplace log densities.

Parameters:
Return type:

Any

classmethod backend_stacked_sufficient_statistics(x, weights, params, engine)[source]

Return per-component raw weighted observations using engine-resident arrays.

Parameters:
Return type:

tuple[tuple[Any, Any], …]

cdf(x)[source]

Cumulative distribution function P(X <= x) (exact). The continuous ‘index of’ a value.

Parameters:

x (float)

Return type:

float

quantile(q)[source]

Inverse CDF F^{-1}(q): the value at cumulative-probability index q (continuous unranking).

Parameters:

q (float)

Return type:

float

mean()[source]

Mean E[X] of the distribution.

Return type:

float

variance()[source]

Variance Var[X] of the distribution.

Return type:

float

entropy()[source]

Differential entropy 1 + log(2b).

Return type:

float

skewness()[source]

Skewness (0).

Return type:

float

kurtosis()[source]

Excess kurtosis (3).

Return type:

float

mode()[source]

Mode (= the location mu).

Return type:

float

sampler(seed=None)[source]

Return a sampler for drawing observations from this distribution.

Parameters:

seed (int | None)

Return type:

LaplaceSampler

estimator(pseudo_count=None)[source]

Return an estimator for fitting this distribution from data.

Parameters:

pseudo_count (float | None)

Return type:

LaplaceEstimator

dist_to_encoder()[source]

Return the data encoder used by this distribution for vectorized methods.

Return type:

LaplaceDataEncoder

class LaplaceSampler(dist, seed=None)[source]

Bases: DistributionSampler

Draw iid Laplace observations.

Parameters:

  • dist (LaplaceDistribution)

  • seed (int | None)

sample(size=None)[source]

Draw one sample or an array of iid samples.

Parameters:

size (int | None)

Return type:

float | ndarray

class LaplaceAccumulator(name=None, keys=None)[source]

Bases: SequenceEncodableStatisticAccumulator

Accumulate weighted observations for exact weighted-median M-step.

Parameters:

  • name (str | None)

  • keys (str | None)

update(x, weight, estimate)[source]

Store one positively weighted observation for the weighted-median M-step.

Parameters:

  • x (float)

  • weight (float)

  • estimate (LaplaceDistribution | None)

Return type:

None

initialize(x, weight, rng)[source]

Initialize statistics from one observation.

Parameters:
Return type:

None

seq_update(x, weights, estimate)[source]

Store positively weighted encoded observations for estimation.

Parameters:
Return type:

None

seq_update_engine(x, weights, estimate, engine)[source]

Engine-aware accumulation. Laplace’s MLE is a weighted median, so the sufficient statistic is the (positively weighted) data itself; this path accepts engine (e.g. torch) weights and stores host arrays. Matches seq_update.

Parameters:

  • x (ndarray)

  • weights (Any)

  • estimate (LaplaceDistribution | None)

  • engine (Any)

Return type:

None

seq_initialize(x, weights, rng)[source]

Initialize statistics from encoded observations.

Parameters:
Return type:

None

combine(suff_stat)[source]

Merge raw weighted observations from another accumulator.

Parameters:

suff_stat (tuple[ndarray, ndarray])

Return type:

LaplaceAccumulator

value()[source]

Return flattened observations and weights.

Return type:

tuple[ndarray, ndarray]

from_value(x)[source]

Replace accumulator contents from raw observations and weights.

Parameters:

x (tuple[ndarray, ndarray])

Return type:

LaplaceAccumulator

scale(c)[source]

Scale weights while preserving the raw observation payload.

Parameters:

c (float)

Return type:

LaplaceAccumulator

key_merge(stats_dict)[source]

Merge keyed statistics into stats_dict when keys are configured.

Parameters:

stats_dict (dict[str, Any])

Return type:

None

key_replace(stats_dict)[source]

Replace this accumulator from keyed statistics when available.

Parameters:

stats_dict (dict[str, Any])

Return type:

None

acc_to_encoder()[source]

Return the encoder used by this accumulator.

Return type:

LaplaceDataEncoder

class LaplaceAccumulatorFactory(name=None, keys=None)[source]

Bases: StatisticAccumulatorFactory

Factory for LaplaceAccumulator.

Parameters:

  • name (str | None)

  • keys (str | None)

make()[source]

Create a fresh Laplace accumulator.

Return type:

LaplaceAccumulator

class LaplaceEstimator(pseudo_count=None, suff_stat=None, min_scale=1.0e-8, name=None, keys=None)[source]

Bases: ParameterEstimator

Exact weighted-MLE estimator for Laplace location and scale.

Parameters:
accumulator_factory()[source]

Return an accumulator factory for Laplace raw-observation statistics.

Return type:

LaplaceAccumulatorFactory

estimate(nobs, suff_stat)[source]

Estimate location and scale by the exact weighted MLE.

Parameters:
Return type:

LaplaceDistribution

class LaplaceDataEncoder[source]

Bases: DataSequenceEncoder

Encode Laplace observations as a float array.

seq_encode(x)[source]

Encode observations as a floating-point array.

Parameters:

x (Sequence[float])

Return type:

ndarray