mixle.stats.univariate.continuous.laplace module¶

Create, estimate, and sample from a Laplace distribution.

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:

mu (float)
b (float)
name (str | None)
keys (str | None)

classmethod compute_capabilities()[source]

classmethod compute_declaration()[source]

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:

x (Any)
mu (Any)
b (Any)
engine (Any)

Return type:

Any

backend_seq_log_density(x, engine)[source]

Engine-neutral vectorized log-density for encoded data.

Parameters:

x (Any)
engine (Any)

Return type:

Any

classmethod backend_stacked_params(dists, engine)[source]

Return stacked Laplace parameters for a homogeneous mixture kernel.

Parameters:

dists (Sequence[LaplaceDistribution])
engine (Any)

Return type:

dict[str, Any]

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

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

Parameters:

x (Any)
params (dict[str, Any])
engine (Any)

Return type:

Any

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

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

Parameters:

x (Any)
weights (Any)
params (dict[str, Any])
engine (Any)

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 observations.

Combinator samplers (mixture/sequence/…) accept batched. With batched=True (the default) each child stream is drawn in one vectorized call instead of a per-draw Python loop – far faster. Because every child sampler owns an independent RandomState, batching consumes each stream in the same order as the loop, so the draws are identical to the legacy path. batched=False forces that legacy per-draw loop as a guaranteed- stable reference. Leaf samplers are already vectorized and ignore the flag.

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]

Parameters:

x (float)
weight (float)
estimate (LaplaceDistribution | None)

Return type:

None

initialize(x, weight, rng)[source]

Parameters:

x (float)
weight (float)
rng (RandomState | None)

Return type:

None

seq_update(x, weights, estimate)[source]

Parameters:

x (ndarray)
weights (ndarray)
estimate (LaplaceDistribution | None)

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]

Parameters:

x (ndarray)
weights (ndarray)
rng (RandomState | None)

Return type:

None

combine(suff_stat)[source]

Parameters:: suff_stat (tuple[ndarray, ndarray])
Return type:: LaplaceAccumulator

value()[source]

Return type:: tuple[ndarray, ndarray]

from_value(x)[source]

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]

Pool this accumulator’s statistics into stats_dict under its merge key.

The structural default implements the common single-key pattern: store the accumulator under self.keys the first time the key is seen, else combine into the one already there. Accumulators with several named keys (e.g. an HMM’s init/trans/state keys) or a non-accumulator stats payload override this. A keys of None (the default) is a no-op.

Parameters:: stats_dict (dict[str, Any])
Return type:: None

key_replace(stats_dict)[source]

Replace this accumulator’s statistics from the pooled stats_dict entry (see key_merge).

Parameters:: stats_dict (dict[str, Any])
Return type:: None

acc_to_encoder()[source]

Return type:: LaplaceDataEncoder

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

Bases: StatisticAccumulatorFactory

Factory for LaplaceAccumulator.

Parameters:

name (str | None)
keys (str | None)

make()[source]

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:

pseudo_count (float | None)
suff_stat (tuple[float, float] | None)
min_scale (float)
name (str | None)
keys (str | None)

accumulator_factory()[source]

Return type:: LaplaceAccumulatorFactory

estimate(nobs, suff_stat)[source]

Parameters:

nobs (float | None)
suff_stat (tuple[ndarray, ndarray])

Return type:

LaplaceDistribution

class LaplaceDataEncoder[source]

Bases: DataSequenceEncoder

Encode Laplace observations as a float array.

seq_encode(x)[source]

Encode the iid observation sequence x for vectorized evaluation.

Parameters:: x (Sequence[float])
Return type:: ndarray