mixle.stats.univariate.continuous.student_t module¶

Location-scale Student’s t distributions over real values.

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

class StudentTDistribution(df, loc=0.0, scale=1.0, name=None, keys=None)[source]

Bases: SequenceEncodableProbabilityDistribution

Student’s t distribution with degrees of freedom df, location loc, and scale > 0.

Parameters:

df (float)
loc (float)
scale (float)
name (str | None)
keys (str | None)

classmethod compute_capabilities()[source]: Describe backend support for generated Student-t kernels.

classmethod compute_declaration()[source]: Return the structured compute declaration for Student-t distributions.

static backend_legacy_sufficient_statistics(x, params, engine)[source]

Return per-row Student-t sufficient statistics in accumulator order.

Parameters:

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

Return type:

tuple[Any, …]

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, df, loc, scale, engine)[source]

Engine-neutral Student-t log-density from explicit parameters.

Parameters:

x (Any)
df (Any)
loc (Any)
scale (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 Student-t parameters for a homogeneous mixture kernel.

Parameters:

dists (Sequence[StudentTDistribution])
engine (Any)

Return type:

dict[str, Any]

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

Return an (n, k) matrix of Student-t log densities.

Parameters:

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

Return type:

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 (loc) for df > 1, else inf (undefined).

Return type:: float

variance()[source]

Variance scale^2 * df/(df-2) for df > 2, else inf.

Return type:: float

sampler(seed=None)[source]

Return a sampler for drawing observations from this distribution.

Parameters:: seed (int | None)
Return type:: StudentTSampler

estimator(pseudo_count=None)[source]

Return an estimator for fitting this distribution from data.

Parameters:: pseudo_count (float | None)
Return type:: StudentTEstimator

dist_to_encoder()[source]

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

Return type:: StudentTDataEncoder

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

Bases: DistributionSampler

Draw iid Student’s t observations.

Parameters:

dist (StudentTDistribution)
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 StudentTAccumulator(name=None, keys=None)[source]

Bases: SequenceEncodableStatisticAccumulator

Accumulate weighted first and second moments for fixed-df t estimation.

Parameters:

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

update(x, weight, estimate)[source]

Accumulate weighted first and second moments for one observation.

Parameters:

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

Return type:

None

initialize(x, weight, rng)[source]

Initialize statistics from one observation.

Parameters:

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

Return type:

None

seq_update(x, weights, estimate)[source]

Accumulate weighted first and second moments from encoded data.

Parameters:

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

Return type:

None

seq_initialize(x, weights, rng)[source]

Initialize statistics from encoded observations.

Parameters:

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

Return type:

None

combine(suff_stat)[source]

Merge another Student-t sufficient-statistic tuple.

Parameters:: suff_stat (tuple[float, float, float])
Return type:: StudentTAccumulator

value()[source]

Return accumulated sum, second moment sum, and count.

Return type:: tuple[float, float, float]

from_value(x)[source]

Replace accumulator contents from a sufficient-statistic tuple.

Parameters:: x (tuple[float, float, float])
Return type:: StudentTAccumulator

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:: StudentTDataEncoder

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

Bases: StatisticAccumulatorFactory

Factory for StudentTAccumulator.

Parameters:

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

make()[source]

Create a fresh Student-t accumulator.

Return type:: StudentTAccumulator

class StudentTEstimator(df=5.0, pseudo_count=None, suff_stat=None, min_scale=1.0e-8, name=None, keys=None)[source]

Bases: ParameterEstimator

Moment-style fixed-df estimator for Student’s t location and scale.

The exact MLE has no simple closed-form update. This estimator keeps df fixed and uses weighted moments, while generic gradient optimizers such as mixle.inference.gradient_fit.fit_mle / fit_map can fit all three parameters through distribution-owned backend math.

Parameters:

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

accumulator_factory()[source]

Return an accumulator factory for fixed-df Student-t moments.

Return type:: StudentTAccumulatorFactory

estimate(nobs, suff_stat)[source]

Estimate location and scale while keeping degrees of freedom fixed.

Parameters:

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

Return type:

StudentTDistribution

class StudentTDataEncoder[source]

Bases: DataSequenceEncoder

Encode Student’s t observations as a float array.

seq_encode(x)[source]

Encode observations as a floating-point array.

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