mixle.stats.univariate.continuous.student_t module

Create, estimate, and sample from a location-scale Student’s t distribution.

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:
classmethod compute_capabilities()[source]
classmethod compute_declaration()[source]
static backend_legacy_sufficient_statistics(x, params, engine)[source]

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

Parameters:
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:
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 Student-t 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 Student-t log densities.

Parameters:
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 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 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]
Parameters:

  • x (float)

  • weight (float)

  • estimate (StudentTDistribution | None)

Return type:

None

initialize(x, weight, rng)[source]
Parameters:
Return type:

None

seq_update(x, weights, estimate)[source]
Parameters:

  • x (ndarray)

  • weights (ndarray)

  • estimate (StudentTDistribution | None)

Return type:

None

seq_initialize(x, weights, rng)[source]
Parameters:
Return type:

None

combine(suff_stat)[source]
Parameters:

suff_stat (tuple[float, float, float])

Return type:

StudentTAccumulator

value()[source]
Return type:

tuple[float, float, float]

from_value(x)[source]
Parameters:

x (tuple[float, float, float])

Return type:

StudentTAccumulator

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:

StudentTDataEncoder

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

Bases: StatisticAccumulatorFactory

Factory for StudentTAccumulator.

Parameters:

  • name (str | None)

  • keys (str | None)

make()[source]
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:
accumulator_factory()[source]
Return type:

StudentTAccumulatorFactory

estimate(nobs, suff_stat)[source]
Parameters:
Return type:

StudentTDistribution

class StudentTDataEncoder[source]

Bases: DataSequenceEncoder

Encode Student’s t 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