Distributions

These are the supported distributions based on the conjugate models.

Many have the dist attribute which is a scipy.stats distribution object. From there, you can use the methods from scipy.stats to get the pdf, cdf, etc.

Distributions can be plotted using the plot_pmf or plot_pdf methods of the distribution.

from conjugate.distribution import Beta

beta = Beta(1, 1)
scipy_dist = beta.dist

print(scipy_dist.mean())
# 0.5
print(scipy_dist.ppf([0.025, 0.975]))
# [0.025 0.975]

samples = scipy_dist.rvs(100)

beta.plot_pmf(label="beta distribution")

Distributions like Poisson can be added with other Poissons or multiplied by numerical values in order to scale rate. For instance,

daily_rate = 0.25
daily_pois = Poisson(lam=daily_rate)

two_day_pois = daily_pois + daily_pois
weekly_pois = 7 * daily_pois

Below are the currently supported distributions

Beta dataclass

Bases: ContinuousPlotDistMixin, SliceMixin

Beta distribution.

Parameters:

Name	Type	Description	Default
alpha	NUMERIC	shape parameter	required
beta	NUMERIC	shape parameter	required

Source code in conjugate/distributions.py

@dataclass
class Beta(ContinuousPlotDistMixin, SliceMixin):
    """Beta distribution.

    Args:
        alpha: shape parameter
        beta: shape parameter

    """

    alpha: NUMERIC
    beta: NUMERIC

    def __post_init__(self) -> None:
        self.max_value = 1.0

    @classmethod
    def from_mean(cls, mean: NUMERIC, alpha: NUMERIC) -> "Beta":
        """Alternative constructor from mean and alpha."""
        beta = get_beta_param_from_mean_and_alpha(mean=mean, alpha=alpha)
        return cls(alpha=alpha, beta=beta)

    @classmethod
    def uninformative(cls) -> "Beta":
        return cls(alpha=1, beta=1)

    @classmethod
    def from_successes_and_failures(
        cls, successes: NUMERIC, failures: NUMERIC
    ) -> "Beta":
        """Alternative constructor based on hyperparameter interpretation."""
        alpha = successes + 1
        beta = failures + 1
        return cls(alpha=alpha, beta=beta)

    @property
    def dist(self):
        return stats.beta(self.alpha, self.beta)

from_mean(mean, alpha) classmethod

Alternative constructor from mean and alpha.

Source code in conjugate/distributions.py

@classmethod
def from_mean(cls, mean: NUMERIC, alpha: NUMERIC) -> "Beta":
    """Alternative constructor from mean and alpha."""
    beta = get_beta_param_from_mean_and_alpha(mean=mean, alpha=alpha)
    return cls(alpha=alpha, beta=beta)

from_successes_and_failures(successes, failures) classmethod

Alternative constructor based on hyperparameter interpretation.

Source code in conjugate/distributions.py

@classmethod
def from_successes_and_failures(
    cls, successes: NUMERIC, failures: NUMERIC
) -> "Beta":
    """Alternative constructor based on hyperparameter interpretation."""
    alpha = successes + 1
    beta = failures + 1
    return cls(alpha=alpha, beta=beta)

BetaBinomial dataclass

Bases: DiscretePlotMixin, SliceMixin

Beta binomial distribution.

Parameters:

Name	Type	Description	Default
n	NUMERIC	number of trials	required
alpha	NUMERIC	shape parameter	required
beta	NUMERIC	shape parameter	required

Source code in conjugate/distributions.py

@dataclass
class BetaBinomial(DiscretePlotMixin, SliceMixin):
    """Beta binomial distribution.

    Args:
        n: number of trials
        alpha: shape parameter
        beta: shape parameter

    """

    n: NUMERIC
    alpha: NUMERIC
    beta: NUMERIC

    def __post_init__(self):
        if isinstance(self.n, np.ndarray):
            self.max_value = self.n.max()
        else:
            self.max_value = self.n

    @property
    def dist(self):
        return stats.betabinom(self.n, self.alpha, self.beta)

BetaGeometric dataclass

Bases: DiscretePlotMixin, SliceMixin

Beta geometric distribution.

Parameters:

Name	Type	Description	Default
alpha	NUMERIC	shape parameter	required
beta	NUMERIC	shape parameter	required
one_start	bool	whether to start at 1 or 0. Default is 1.	True

Source code in conjugate/distributions.py

@dataclass
class BetaGeometric(DiscretePlotMixin, SliceMixin):
    """Beta geometric distribution.

    Args:
        alpha: shape parameter
        beta: shape parameter
        one_start: whether to start at 1 or 0. Default is 1.

    """

    alpha: NUMERIC
    beta: NUMERIC
    one_start: bool = True

    @property
    def dist(self):
        return beta_geometric(self.alpha, self.beta, one_start=self.one_start)

BetaNegativeBinomial dataclass

Bases: DiscretePlotMixin, SliceMixin

Beta negative binomial distribution.

Parameters:

Name	Type	Description	Default
n	NUMERIC	number of successes	required
alpha	NUMERIC	shape parameter	required
beta	NUMERIC	shape parameter	required

Source code in conjugate/distributions.py

@dataclass
class BetaNegativeBinomial(DiscretePlotMixin, SliceMixin):
    """Beta negative binomial distribution.

    Args:
        n: number of successes
        alpha: shape parameter
        beta: shape parameter

    """

    n: NUMERIC
    alpha: NUMERIC
    beta: NUMERIC

    def __post_init__(self):
        if isinstance(self.n, np.ndarray):
            self.max_value = self.n.max()
        else:
            self.max_value = self.n

    @property
    def dist(self):
        if version.parse(scipy_version).release < version.parse("1.12.0").release:
            msg = "BetaNegativeBinomial.dist requires scipy >= 1.12.0"
            raise NotImplementedError(msg)

        return stats.betanbinom(self.n, self.alpha, self.beta)

BetaProportional dataclass

Beta proportional distribution.

Parameters:

Name	Type	Description	Default
p	NUMERIC	product of observations	required
q	NUMERIC	product of complements	required
k	NUMERIC	number of observations	required

Source code in conjugate/distributions.py

@dataclass
class BetaProportional:
    """Beta proportional distribution.

    Args:
        p: product of observations
        q: product of complements
        k: number of observations

    """

    p: NUMERIC
    q: NUMERIC
    k: NUMERIC

    def approx_log_likelihood(
        self, alpha: NUMERIC, beta: NUMERIC, ln=np.log, gammaln=gammaln
    ) -> NUMERIC:
        """Approximate log likelihood.

        Args:
            alpha: shape parameter
            beta: shape parameter
            ln: log function
            gammaln: log gamma function

        Returns:
            log likelihood up to a constant

        """
        return (
            self.k * gammaln(alpha + beta)
            + alpha * ln(self.p)
            + beta * ln(self.q)
            - self.k * gammaln(alpha)
            - self.k * gammaln(beta)
        )

approx_log_likelihood(alpha, beta, ln=np.log, gammaln=gammaln)

Approximate log likelihood.

Parameters:

Name	Type	Description	Default
alpha	NUMERIC	shape parameter	required
beta	NUMERIC	shape parameter	required
ln		log function	log
gammaln		log gamma function	gammaln

Returns:

Type	Description
NUMERIC	log likelihood up to a constant

Source code in conjugate/distributions.py

def approx_log_likelihood(
    self, alpha: NUMERIC, beta: NUMERIC, ln=np.log, gammaln=gammaln
) -> NUMERIC:
    """Approximate log likelihood.

    Args:
        alpha: shape parameter
        beta: shape parameter
        ln: log function
        gammaln: log gamma function

    Returns:
        log likelihood up to a constant

    """
    return (
        self.k * gammaln(alpha + beta)
        + alpha * ln(self.p)
        + beta * ln(self.q)
        - self.k * gammaln(alpha)
        - self.k * gammaln(beta)
    )

Binomial dataclass

Bases: DiscretePlotMixin, SliceMixin

Binomial distribution.

Parameters:

Name	Type	Description	Default
n	NUMERIC	number of trials	required
p	NUMERIC	probability of success	required

Source code in conjugate/distributions.py

@dataclass
class Binomial(DiscretePlotMixin, SliceMixin):
    """Binomial distribution.

    Args:
        n: number of trials
        p: probability of success

    """

    n: NUMERIC
    p: NUMERIC

    def __post_init__(self):
        if isinstance(self.n, np.ndarray):
            self.max_value = self.n.max()
        else:
            self.max_value = self.n

    @property
    def dist(self):
        return stats.binom(n=self.n, p=self.p)

CompoundGamma dataclass

Bases: ContinuousPlotDistMixin, SliceMixin

Compound gamma distribution.

Parameters:

Name	Type	Description	Default
alpha	NUMERIC	shape	required
beta	NUMERIC	scale	required
lam	NUMERIC	rate	required

Source code in conjugate/distributions.py

@dataclass
class CompoundGamma(ContinuousPlotDistMixin, SliceMixin):
    """Compound gamma distribution.

    Args:
        alpha: shape
        beta: scale
        lam: rate

    """

    alpha: NUMERIC
    beta: NUMERIC
    lam: NUMERIC

    @property
    def dist(self):
        return compound_gamma(a=self.alpha, b=self.beta, q=self.lam)

Dirichlet dataclass

Bases: DirichletPlotDistMixin

Dirichlet distribution.

Parameters:

Name	Type	Description	Default
alpha	NUMERIC	shape parameter	required

Source code in conjugate/distributions.py

@dataclass
class Dirichlet(DirichletPlotDistMixin):
    """Dirichlet distribution.

    Args:
        alpha: shape parameter

    """

    alpha: NUMERIC

    def __post_init__(self) -> None:
        self.max_value = 1.0

    @classmethod
    def uninformative(cls, n: int) -> "Dirichlet":
        return cls(alpha=np.ones(n))

    @property
    def dist(self):
        if self.alpha.ndim == 1:
            return stats.dirichlet(self.alpha)

        return VectorizedDist(self.alpha, dist=stats.dirichlet)

DirichletMultinomial dataclass

Bases: SliceMixin

Dirichlet multinomial distribution.

Parameters:

Name	Type	Description	Default
alpha	NUMERIC	shape parameter	required
n	NUMERIC	number of trials	required

Source code in conjugate/distributions.py

@dataclass
class DirichletMultinomial(SliceMixin):
    """Dirichlet multinomial distribution.

    Args:
        alpha: shape parameter
        n: number of trials

    """

    alpha: NUMERIC
    n: NUMERIC

    @property
    def dist(self):
        return stats.dirichlet_multinomial(self.alpha, self.n)

Exponential dataclass

Bases: ContinuousPlotDistMixin, SliceMixin

Exponential distribution.

Parameters:

Name	Type	Description	Default
lam	NUMERIC	rate parameter	required

Source code in conjugate/distributions.py

@dataclass
class Exponential(ContinuousPlotDistMixin, SliceMixin):
    """Exponential distribution.

    Args:
        lam: rate parameter

    """

    lam: NUMERIC

    @property
    def dist(self):
        return stats.expon(scale=self.lam)

    def __mul__(self, other):
        return Gamma(alpha=other, beta=1 / self.lam)

    __rmul__ = __mul__

Gamma dataclass

Bases: ContinuousPlotDistMixin, SliceMixin

Gamma distribution.

Gamma Distribution Scipy Docmentation

Parameters:

Name	Type	Description	Default
alpha	NUMERIC	shape parameter	required
beta	NUMERIC	rate parameter	required

Source code in conjugate/distributions.py

@dataclass
class Gamma(ContinuousPlotDistMixin, SliceMixin):
    """Gamma distribution.

    <a href=https://en.wikipedia.org/wiki/Gamma_distribution>Gamma Distribution</a>
    <a href=https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.gamma.html>Scipy Docmentation</a>

    Args:
        alpha: shape parameter
        beta: rate parameter
    """

    alpha: NUMERIC
    beta: NUMERIC

    @classmethod
    def from_occurrences_in_intervals(cls, occurrences: NUMERIC, intervals: NUMERIC):
        return cls(alpha=occurrences, beta=intervals)

    @property
    def dist(self):
        return stats.gamma(a=self.alpha, scale=1 / self.beta)

    def __mul__(self, other):
        return Gamma(alpha=self.alpha * other, beta=self.beta)

    __rmul__ = __mul__

GammaKnownRateProportional dataclass

Gamma known rate proportional distribution.

Parameters:

Name	Type	Description	Default
a	NUMERIC	prod of observations	required
b	NUMERIC	number of observations	required
c	NUMERIC	number of observations	required

Source code in conjugate/distributions.py

@dataclass
class GammaKnownRateProportional:
    """Gamma known rate proportional distribution.

    Args:
        a: prod of observations
        b: number of observations
        c: number of observations

    """

    a: NUMERIC
    b: NUMERIC
    c: NUMERIC

    def approx_log_likelihood(
        self, alpha: NUMERIC, beta: NUMERIC, ln=np.log, gammaln=gammaln
    ) -> NUMERIC:
        """Approximate log likelihood.

        Args:
            alpha: shape parameter
            beta: known rate parameter
            ln: log function
            gammaln: log gamma function

        Returns:
            log likelihood up to a constant

        """
        return (
            (alpha - 1) * ln(self.a)
            + alpha * self.c * ln(beta)
            - self.b * gammaln(alpha)
        )

approx_log_likelihood(alpha, beta, ln=np.log, gammaln=gammaln)

Approximate log likelihood.

Parameters:

Name	Type	Description	Default
alpha	NUMERIC	shape parameter	required
beta	NUMERIC	known rate parameter	required
ln		log function	log
gammaln		log gamma function	gammaln

Returns:

Type	Description
NUMERIC	log likelihood up to a constant

Source code in conjugate/distributions.py

def approx_log_likelihood(
    self, alpha: NUMERIC, beta: NUMERIC, ln=np.log, gammaln=gammaln
) -> NUMERIC:
    """Approximate log likelihood.

    Args:
        alpha: shape parameter
        beta: known rate parameter
        ln: log function
        gammaln: log gamma function

    Returns:
        log likelihood up to a constant

    """
    return (
        (alpha - 1) * ln(self.a)
        + alpha * self.c * ln(beta)
        - self.b * gammaln(alpha)
    )

GammaProportional dataclass

Gamma proportional distribution.

Parameters:

Name	Type	Description	Default
p	NUMERIC	product of r observations	required
q	NUMERIC	sum of s observations	required
r	NUMERIC	number of observations for p	required
s	NUMERIC	number of observations for q	required

Source code in conjugate/distributions.py

@dataclass
class GammaProportional:
    """Gamma proportional distribution.

    Args:
        p: product of r observations
        q: sum of s observations
        r: number of observations for p
        s: number of observations for q

    """

    p: NUMERIC
    q: NUMERIC
    r: NUMERIC
    s: NUMERIC

    def approx_log_likelihood(
        self, alpha: NUMERIC, beta: NUMERIC, ln=np.log, gammaln=gammaln
    ) -> NUMERIC:
        """Approximate log likelihood.

        Args:
            alpha: shape parameter
            beta: rate parameter
            ln: log function
            gammaln: log gamma function

        Returns:
            log likelihood up to a constant

        """
        return (
            (alpha - 1) * ln(self.p)
            - self.q * beta
            - self.r * gammaln(alpha)
            + self.s * alpha * ln(beta)
        )

approx_log_likelihood(alpha, beta, ln=np.log, gammaln=gammaln)

Approximate log likelihood.

Parameters:

Name	Type	Description	Default
alpha	NUMERIC	shape parameter	required
beta	NUMERIC	rate parameter	required
ln		log function	log
gammaln		log gamma function	gammaln

Returns:

Type	Description
NUMERIC	log likelihood up to a constant

Source code in conjugate/distributions.py

def approx_log_likelihood(
    self, alpha: NUMERIC, beta: NUMERIC, ln=np.log, gammaln=gammaln
) -> NUMERIC:
    """Approximate log likelihood.

    Args:
        alpha: shape parameter
        beta: rate parameter
        ln: log function
        gammaln: log gamma function

    Returns:
        log likelihood up to a constant

    """
    return (
        (alpha - 1) * ln(self.p)
        - self.q * beta
        - self.r * gammaln(alpha)
        + self.s * alpha * ln(beta)
    )

Geometric dataclass

Bases: DiscretePlotMixin, SliceMixin

Geometric distribution.

Parameters:

Name	Type	Description	Default
p	NUMERIC	probability of success	required
one_start	bool	whether to start at 1 or 0. Default is 1.	True

Source code in conjugate/distributions.py

@dataclass
class Geometric(DiscretePlotMixin, SliceMixin):
    """Geometric distribution.

    Args:
        p: probability of success
        one_start: whether to start at 1 or 0. Default is 1.

    """

    p: NUMERIC
    one_start: bool = True

    @property
    def dist(self):
        loc = 0 if self.one_start else -1
        return stats.geom(self.p, loc=loc)

Hypergeometric dataclass

Bases: DiscretePlotMixin, SliceMixin

Hypergeometric distribution.

Parameters:

Name	Type	Description	Default
N	NUMERIC	population size	required
k	NUMERIC	number of successes in the population	required
n	NUMERIC	number of draws	required

Source code in conjugate/distributions.py

@dataclass
class Hypergeometric(DiscretePlotMixin, SliceMixin):
    """Hypergeometric distribution.

    Args:
        N: population size
        k: number of successes in the population
        n: number of draws

    """

    N: NUMERIC
    k: NUMERIC
    n: NUMERIC

    def __post_init__(self) -> None:
        if isinstance(self.N, np.ndarray):
            self.max_value = self.N.max()
        else:
            self.max_value = self.N

    @property
    def dist(self):
        return stats.hypergeom(self.N, self.k, self.n)

InverseGamma dataclass

Bases: ContinuousPlotDistMixin, SliceMixin

InverseGamma distribution.

Parameters:

Name	Type	Description	Default
alpha	NUMERIC	shape	required
beta	NUMERIC	scale	required

Source code in conjugate/distributions.py

@dataclass
class InverseGamma(ContinuousPlotDistMixin, SliceMixin):
    """InverseGamma distribution.

    Args:
        alpha: shape
        beta: scale

    """

    alpha: NUMERIC
    beta: NUMERIC

    @property
    def dist(self):
        return stats.invgamma(a=self.alpha, scale=self.beta)

InverseWishart dataclass

Inverse Wishart distribution.

Parameters:

Name	Type	Description	Default
nu	NUMERIC	degrees of freedom	required
psi	NUMERIC	scale matrix	required

Source code in conjugate/distributions.py

@dataclass
class InverseWishart:
    """Inverse Wishart distribution.

    Args:
        nu: degrees of freedom
        psi: scale matrix

    """

    nu: NUMERIC
    psi: NUMERIC

    @property
    def dist(self):
        return stats.invwishart(df=self.nu, scale=self.psi)

LogNormal dataclass

Bases: ContinuousPlotDistMixin, SliceMixin

Log normal distribution.

Parameters:

Name	Type	Description	Default
mu	NUMERIC	mean	required
sigma	NUMERIC	standard deviation	required

Source code in conjugate/distributions.py

@dataclass
class LogNormal(ContinuousPlotDistMixin, SliceMixin):
    """Log normal distribution.

    Args:
        mu: mean
        sigma: standard deviation

    """

    mu: NUMERIC
    sigma: NUMERIC

    @property
    def dist(self):
        return stats.lognorm(s=self.sigma, loc=self.mu)

Lomax dataclass

Bases: ContinuousPlotDistMixin, SliceMixin

Lomax distribution.

Parameters:

Name	Type	Description	Default
alpha	NUMERIC	shape	required
lam	NUMERIC	scale	required

Source code in conjugate/distributions.py

@dataclass
class Lomax(ContinuousPlotDistMixin, SliceMixin):
    """Lomax distribution.

    Args:
        alpha: shape
        lam: scale

    """

    alpha: NUMERIC
    lam: NUMERIC

    @property
    def dist(self):
        return stats.lomax(c=self.alpha, scale=self.lam)

Multinomial dataclass

Bases: SliceMixin

Multinomial distribution.

Parameters:

Name	Type	Description	Default
n	NUMERIC	number of trials	required
p	NUMERIC	probability of success	required

Source code in conjugate/distributions.py

@dataclass
class Multinomial(SliceMixin):
    """Multinomial distribution.

    Args:
        n: number of trials
        p: probability of success

    """

    n: NUMERIC
    p: NUMERIC

    @property
    def dist(self):
        return stats.multinomial(n=self.n, p=self.p)

MultivariateNormal dataclass

Multivariate normal distribution.

Parameters:

Name	Type	Description	Default
mu	NUMERIC	mean	required
cov	NUMERIC	covariance matrix	required

Source code in conjugate/distributions.py

@dataclass
class MultivariateNormal:
    """Multivariate normal distribution.

    Args:
        mu: mean
        cov: covariance matrix

    """

    mu: NUMERIC
    cov: NUMERIC

    @property
    def dist(self):
        return stats.multivariate_normal(mean=self.mu, cov=self.cov)

    def __getitem__(self, key):
        if isinstance(key, int):
            return Normal(mu=self.mu[key], sigma=self.cov[key, key] ** 0.5)

        return MultivariateNormal(mu=self.mu[key], cov=self.cov[key][:, key])

MultivariateStudentT dataclass

MultivariateStudentT distribution.

Parameters:

Name	Type	Description	Default
mu	NUMERIC	mean	required
sigma	NUMERIC	covariance matrix	required
nu	NUMERIC	degrees of freedom	required

Source code in conjugate/distributions.py

@dataclass
class MultivariateStudentT:
    """MultivariateStudentT distribution.

    Args:
        mu: mean
        sigma: covariance matrix
        nu: degrees of freedom

    """

    mu: NUMERIC
    sigma: NUMERIC
    nu: NUMERIC

    @property
    def dist(self):
        return stats.multivariate_t(loc=self.mu, shape=self.sigma, df=self.nu)

    def __getitem__(self, key):
        if isinstance(key, int):
            return StudentT(mu=self.mu[key], sigma=self.sigma[key, key], nu=self.nu)

        return MultivariateStudentT(
            mu=self.mu[key], sigma=self.sigma[key][:, key], nu=self.nu
        )

NegativeBinomial dataclass

Bases: DiscretePlotMixin, SliceMixin

Negative binomial distribution.

Parameters:

Name	Type	Description	Default
n	NUMERIC	number of successes	required
p	NUMERIC	probability of success	required

Source code in conjugate/distributions.py

@dataclass
class NegativeBinomial(DiscretePlotMixin, SliceMixin):
    """Negative binomial distribution.

    Args:
        n: number of successes
        p: probability of success

    """

    n: NUMERIC
    p: NUMERIC

    @property
    def dist(self):
        return stats.nbinom(n=self.n, p=self.p)

    def __mul__(self, other):
        return NegativeBinomial(n=self.n * other, p=self.p)

    __rmul__ = __mul__

Normal dataclass

Bases: ContinuousPlotDistMixin, SliceMixin

Normal distribution.

Parameters:

Name	Type	Description	Default
mu	NUMERIC	mean	required
sigma	NUMERIC	standard deviation	required

Source code in conjugate/distributions.py

@dataclass
class Normal(ContinuousPlotDistMixin, SliceMixin):
    """Normal distribution.

    Args:
        mu: mean
        sigma: standard deviation

    """

    mu: NUMERIC
    sigma: NUMERIC

    @property
    def dist(self):
        return stats.norm(self.mu, self.sigma)

    @classmethod
    def uninformative(cls, sigma: NUMERIC = 1) -> "Normal":
        """Uninformative normal distribution."""
        return cls(mu=0, sigma=sigma)

    @classmethod
    def from_mean_and_variance(cls, mean: NUMERIC, variance: NUMERIC) -> "Normal":
        """Alternative constructor from mean and variance."""
        return cls(mu=mean, sigma=variance**0.5)

    @classmethod
    def from_mean_and_precision(cls, mean: NUMERIC, precision: NUMERIC) -> "Normal":
        """Alternative constructor from mean and precision."""
        return cls(mu=mean, sigma=precision**-0.5)

    def __mul__(self, other):
        sigma = ((self.sigma**2) * other) ** 0.5
        return Normal(mu=self.mu * other, sigma=sigma)

    __rmul__ = __mul__

from_mean_and_precision(mean, precision) classmethod

Alternative constructor from mean and precision.

Source code in conjugate/distributions.py

@classmethod
def from_mean_and_precision(cls, mean: NUMERIC, precision: NUMERIC) -> "Normal":
    """Alternative constructor from mean and precision."""
    return cls(mu=mean, sigma=precision**-0.5)

from_mean_and_variance(mean, variance) classmethod

Alternative constructor from mean and variance.

Source code in conjugate/distributions.py

@classmethod
def from_mean_and_variance(cls, mean: NUMERIC, variance: NUMERIC) -> "Normal":
    """Alternative constructor from mean and variance."""
    return cls(mu=mean, sigma=variance**0.5)

uninformative(sigma=1) classmethod

Uninformative normal distribution.

Source code in conjugate/distributions.py

@classmethod
def uninformative(cls, sigma: NUMERIC = 1) -> "Normal":
    """Uninformative normal distribution."""
    return cls(mu=0, sigma=sigma)

NormalGamma dataclass

Normal gamma distribution.

Parameters:

Name	Type	Description	Default
mu	NUMERIC	mean	required
lam	NUMERIC	precision	required
alpha	NUMERIC	shape	required
beta	NUMERIC	scale	required

Source code in conjugate/distributions.py

@dataclass
class NormalGamma:
    """Normal gamma distribution.

    Args:
        mu: mean
        lam: precision
        alpha: shape
        beta: scale

    """

    mu: NUMERIC
    lam: NUMERIC
    alpha: NUMERIC
    beta: NUMERIC

    @property
    def gamma(self) -> Gamma:
        return Gamma(alpha=self.alpha, beta=self.beta)

    def sample_variance(self, size: int, random_state=None) -> NUMERIC:
        """Sample precision from gamma distribution and invert.

        Args:
            size: number of samples
            random_state: random state

        Returns:
            samples from the inverse gamma distribution

        """
        precision = self.lam * self.gamma.dist.rvs(size=size, random_state=random_state)

        return 1 / precision

    def sample_mean(
        self,
        size: int,
        return_variance: bool = False,
        random_state=None,
    ) -> NUMERIC | tuple[NUMERIC, NUMERIC]:
        """Sample mean from the normal distribution.

        Args:
            size: number of samples
            return_variance: whether to return variance as well
            random_state: random state

        Returns:
            samples from the normal distribution

        """
        return self.sample_beta(
            size=size, return_variance=return_variance, random_state=random_state
        )

    def sample_beta(
        self, size: int, return_variance: bool = False, random_state=None
    ) -> NUMERIC | tuple[NUMERIC, NUMERIC]:
        """Sample beta from the normal distribution.

        Args:
            size: number of samples
            return_variance: whether to return variance as well
            random_state: random state

        Returns:
            samples from the normal distribution

        """
        variance = self.sample_variance(size=size, random_state=random_state)
        sigma = variance**0.5
        beta = stats.norm(loc=self.mu, scale=sigma).rvs(
            size=size, random_state=random_state
        )

        if return_variance:
            return beta, variance

        return beta

sample_beta(size, return_variance=False, random_state=None)

Sample beta from the normal distribution.

Parameters:

Name	Type	Description	Default
size	int	number of samples	required
return_variance	bool	whether to return variance as well	False
random_state		random state	None

Returns:

Type	Description
NUMERIC | tuple[NUMERIC, NUMERIC]	samples from the normal distribution

Source code in conjugate/distributions.py

def sample_beta(
    self, size: int, return_variance: bool = False, random_state=None
) -> NUMERIC | tuple[NUMERIC, NUMERIC]:
    """Sample beta from the normal distribution.

    Args:
        size: number of samples
        return_variance: whether to return variance as well
        random_state: random state

    Returns:
        samples from the normal distribution

    """
    variance = self.sample_variance(size=size, random_state=random_state)
    sigma = variance**0.5
    beta = stats.norm(loc=self.mu, scale=sigma).rvs(
        size=size, random_state=random_state
    )

    if return_variance:
        return beta, variance

    return beta

sample_mean(size, return_variance=False, random_state=None)

Sample mean from the normal distribution.

Parameters:

Name	Type	Description	Default
size	int	number of samples	required
return_variance	bool	whether to return variance as well	False
random_state		random state	None

Returns:

Type	Description
NUMERIC | tuple[NUMERIC, NUMERIC]	samples from the normal distribution

Source code in conjugate/distributions.py

def sample_mean(
    self,
    size: int,
    return_variance: bool = False,
    random_state=None,
) -> NUMERIC | tuple[NUMERIC, NUMERIC]:
    """Sample mean from the normal distribution.

    Args:
        size: number of samples
        return_variance: whether to return variance as well
        random_state: random state

    Returns:
        samples from the normal distribution

    """
    return self.sample_beta(
        size=size, return_variance=return_variance, random_state=random_state
    )

sample_variance(size, random_state=None)

Sample precision from gamma distribution and invert.

Parameters:

Name	Type	Description	Default
size	int	number of samples	required
random_state		random state	None

Returns:

Type	Description
NUMERIC	samples from the inverse gamma distribution

Source code in conjugate/distributions.py

def sample_variance(self, size: int, random_state=None) -> NUMERIC:
    """Sample precision from gamma distribution and invert.

    Args:
        size: number of samples
        random_state: random state

    Returns:
        samples from the inverse gamma distribution

    """
    precision = self.lam * self.gamma.dist.rvs(size=size, random_state=random_state)

    return 1 / precision

NormalInverseGamma dataclass

Normal inverse gamma distribution.

Supports both 1 dimensional and multivariate cases.

Parameters:

Name	Type	Description	Default
mu	NUMERIC	mean	required
alpha	NUMERIC	shape	required
beta	NUMERIC	scale	required
delta_inverse	NUMERIC | None	covariance matrix, 2d array for multivariate case	None
nu	NUMERIC | None	alternative precision parameter for 1 dimensional case	None

Source code in conjugate/distributions.py

@dataclass
class NormalInverseGamma:
    """Normal inverse gamma distribution.

    Supports both 1 dimensional and multivariate cases.

    Args:
        mu: mean
        alpha: shape
        beta: scale
        delta_inverse: covariance matrix, 2d array for multivariate case
        nu: alternative precision parameter for 1 dimensional case

    """

    mu: NUMERIC
    alpha: NUMERIC
    beta: NUMERIC
    delta_inverse: NUMERIC | None = None
    nu: NUMERIC | None = None

    def __post_init__(self) -> None:
        if self.delta_inverse is None and self.nu is None:
            raise ValueError("Either delta_inverse or nu must be provided.")

        if self.delta_inverse is not None and self.nu is not None:
            raise ValueError("Only one of delta_inverse or nu must be provided.")

    @classmethod
    def from_inverse_gamma(
        cls,
        mu: NUMERIC,
        inverse_gamma: InverseGamma,
        delta_inverse: NUMERIC | None = None,
        nu: NUMERIC | None = None,
    ) -> "NormalInverseGamma":
        return cls(
            mu=mu,
            alpha=inverse_gamma.alpha,
            beta=inverse_gamma.beta,
            delta_inverse=delta_inverse,
            nu=nu,
        )

    @property
    def inverse_gamma(self) -> InverseGamma:
        return InverseGamma(alpha=self.alpha, beta=self.beta)

    def sample_variance(self, size: int, random_state=None) -> NUMERIC:
        """Sample variance from the inverse gamma distribution.

        Args:
            size: number of samples
            random_state: random state

        Returns:
            samples from the inverse gamma distribution

        """
        return self.inverse_gamma.dist.rvs(size=size, random_state=random_state)

    def _sample_beta_1d(self, variance, size: int, random_state=None) -> NUMERIC:
        sigma = (variance / self.nu) ** 0.5
        return stats.norm(self.mu, sigma).rvs(size=size, random_state=random_state)

    def _sample_beta_nd(self, variance, size: int, random_state=None) -> NUMERIC:
        variance = (self.delta_inverse[None, ...].T * variance).T
        return np.stack(
            [
                stats.multivariate_normal(self.mu, v).rvs(
                    size=1, random_state=random_state
                )
                for v in variance
            ]
        )

    def sample_mean(
        self,
        size: int,
        return_variance: bool = False,
        random_state=None,
    ) -> NUMERIC | tuple[NUMERIC, NUMERIC]:
        """Sample the mean from the normal distribution.

        Args:
            size: number of samples
            return_variance: whether to return variance as well
            random_state: random state

        Returns:
            samples from the normal distribution and optionally variance

        """
        return self.sample_beta(
            size=size, return_variance=return_variance, random_state=random_state
        )

    def sample_beta(
        self, size: int, return_variance: bool = False, random_state=None
    ) -> NUMERIC | tuple[NUMERIC, NUMERIC]:
        """Sample beta from the normal distribution.

        Args:
            size: number of samples
            return_variance: whether to return variance as well
            random_state: random state

        Returns:
            samples from the normal distribution and optionally variance

        """
        variance = self.sample_variance(size=size, random_state=random_state)

        sample_beta = (
            self._sample_beta_1d if self.delta_inverse is None else self._sample_beta_nd
        )
        beta = sample_beta(variance=variance, size=size, random_state=random_state)

        if return_variance:
            return beta, variance

        return beta

sample_beta(size, return_variance=False, random_state=None)

Sample beta from the normal distribution.

Parameters:

Name	Type	Description	Default
size	int	number of samples	required
return_variance	bool	whether to return variance as well	False
random_state		random state	None

Returns:

Type	Description
NUMERIC | tuple[NUMERIC, NUMERIC]	samples from the normal distribution and optionally variance

Source code in conjugate/distributions.py

def sample_beta(
    self, size: int, return_variance: bool = False, random_state=None
) -> NUMERIC | tuple[NUMERIC, NUMERIC]:
    """Sample beta from the normal distribution.

    Args:
        size: number of samples
        return_variance: whether to return variance as well
        random_state: random state

    Returns:
        samples from the normal distribution and optionally variance

    """
    variance = self.sample_variance(size=size, random_state=random_state)

    sample_beta = (
        self._sample_beta_1d if self.delta_inverse is None else self._sample_beta_nd
    )
    beta = sample_beta(variance=variance, size=size, random_state=random_state)

    if return_variance:
        return beta, variance

    return beta

sample_mean(size, return_variance=False, random_state=None)

Sample the mean from the normal distribution.

Parameters:

Name	Type	Description	Default
size	int	number of samples	required
return_variance	bool	whether to return variance as well	False
random_state		random state	None

Returns:

Type	Description
NUMERIC | tuple[NUMERIC, NUMERIC]	samples from the normal distribution and optionally variance

Source code in conjugate/distributions.py

def sample_mean(
    self,
    size: int,
    return_variance: bool = False,
    random_state=None,
) -> NUMERIC | tuple[NUMERIC, NUMERIC]:
    """Sample the mean from the normal distribution.

    Args:
        size: number of samples
        return_variance: whether to return variance as well
        random_state: random state

    Returns:
        samples from the normal distribution and optionally variance

    """
    return self.sample_beta(
        size=size, return_variance=return_variance, random_state=random_state
    )

sample_variance(size, random_state=None)

Sample variance from the inverse gamma distribution.

Parameters:

Name	Type	Description	Default
size	int	number of samples	required
random_state		random state	None

Returns:

Type	Description
NUMERIC	samples from the inverse gamma distribution

Source code in conjugate/distributions.py

def sample_variance(self, size: int, random_state=None) -> NUMERIC:
    """Sample variance from the inverse gamma distribution.

    Args:
        size: number of samples
        random_state: random state

    Returns:
        samples from the inverse gamma distribution

    """
    return self.inverse_gamma.dist.rvs(size=size, random_state=random_state)

NormalInverseWishart dataclass

Normal inverse Wishart distribution.

Parameters:

Name	Type	Description	Default
mu	NUMERIC	mean	required
kappa	NUMERIC	precision	required
nu	NUMERIC	degrees of freedom	required
psi	NUMERIC	scale matrix	required

Source code in conjugate/distributions.py

@dataclass
class NormalInverseWishart:
    """Normal inverse Wishart distribution.

    Args:
        mu: mean
        kappa: precision
        nu: degrees of freedom
        psi: scale matrix

    """

    mu: NUMERIC
    kappa: NUMERIC
    nu: NUMERIC
    psi: NUMERIC

    @property
    def inverse_wishart(self):
        """Inverse wishart distribution."""
        return InverseWishart(nu=self.nu, psi=self.psi)

    @classmethod
    def from_inverse_wishart(
        cls,
        mu: NUMERIC,
        kappa: NUMERIC,
        inverse_wishart: InverseWishart,
    ):
        return cls(mu=mu, kappa=kappa, nu=inverse_wishart.nu, psi=inverse_wishart.psi)

    def sample_variance(
        self, size: int, random_state: np.random.Generator | None = None
    ) -> NUMERIC:
        """Sample precision from gamma distribution and invert.

        Args:
            size: number of samples
            random_state: random state

        Returns:
            samples from the inverse wishart distribution

        """
        variance = (
            self.inverse_wishart.dist.rvs(size=size, random_state=random_state)
            / self.kappa
        )
        if size == 1:
            variance = variance[None, ...]

        return variance

    def sample_mean(
        self,
        size: int,
        return_variance: bool = False,
        random_state: np.random.Generator | None = None,
    ) -> NUMERIC:
        """Sample the mean from the normal distribution.

        Args:
            size: number of samples
            return_variance: whether to return variance as well
            random_state: random state

        Returns:
            samples from the normal distribution and optionally variance

        """
        variance = self.sample_variance(size=size, random_state=random_state)

        mean = np.stack(
            [
                stats.multivariate_normal(self.mu, cov=cov).rvs(
                    size=1, random_state=random_state
                )
                for cov in variance
            ]
        )

        if return_variance:
            return mean, variance

        return mean

inverse_wishart property

Inverse wishart distribution.

sample_mean(size, return_variance=False, random_state=None)

Sample the mean from the normal distribution.

Parameters:

Name	Type	Description	Default
size	int	number of samples	required
return_variance	bool	whether to return variance as well	False
random_state	Generator | None	random state	None

Returns:

Type	Description
NUMERIC	samples from the normal distribution and optionally variance

Source code in conjugate/distributions.py

def sample_mean(
    self,
    size: int,
    return_variance: bool = False,
    random_state: np.random.Generator | None = None,
) -> NUMERIC:
    """Sample the mean from the normal distribution.

    Args:
        size: number of samples
        return_variance: whether to return variance as well
        random_state: random state

    Returns:
        samples from the normal distribution and optionally variance

    """
    variance = self.sample_variance(size=size, random_state=random_state)

    mean = np.stack(
        [
            stats.multivariate_normal(self.mu, cov=cov).rvs(
                size=1, random_state=random_state
            )
            for cov in variance
        ]
    )

    if return_variance:
        return mean, variance

    return mean

sample_variance(size, random_state=None)

Sample precision from gamma distribution and invert.

Parameters:

Name	Type	Description	Default
size	int	number of samples	required
random_state	Generator | None	random state	None

Returns:

Type	Description
NUMERIC	samples from the inverse wishart distribution

Source code in conjugate/distributions.py

def sample_variance(
    self, size: int, random_state: np.random.Generator | None = None
) -> NUMERIC:
    """Sample precision from gamma distribution and invert.

    Args:
        size: number of samples
        random_state: random state

    Returns:
        samples from the inverse wishart distribution

    """
    variance = (
        self.inverse_wishart.dist.rvs(size=size, random_state=random_state)
        / self.kappa
    )
    if size == 1:
        variance = variance[None, ...]

    return variance

NormalWishart dataclass

Normal Wishart distribution.

Parameterization from Wikipedia.

Parameters:

Name	Type	Description	Default
mu	NUMERIC	mean	required
lam	NUMERIC	precision	required
W	NUMERIC	scale matrix	required
nu	NUMERIC	degrees of freedom	required

Source code in conjugate/distributions.py

@dataclass
class NormalWishart:
    """Normal Wishart distribution.

    Parameterization from <a href=https://en.wikipedia.org/wiki/Normal-Wishart_distribution>Wikipedia</a>.

    Args:
        mu: mean
        lam: precision
        W: scale matrix
        nu: degrees of freedom

    """

    mu: NUMERIC
    lam: NUMERIC
    W: NUMERIC
    nu: NUMERIC

    @property
    def wishart(self):
        return Wishart(nu=self.nu, V=self.W)

    def sample_variance(
        self,
        size: int = 1,
        random_state: np.random.Generator | None = None,
        inv: Callable = np.linalg.inv,
    ) -> np.ndarray:
        """Sample variance

        Args:
            size: number of samples
            random_state: random state
            inv: matrix inversion function

        Returns:
            samples from the inverse wishart distribution

        """

        variance = inv(
            self.lam * self.wishart.dist.rvs(size=size, random_state=random_state)
        )

        if size == 1:
            variance = variance[None, ...]

        return variance

    def sample_mean(
        self,
        size: int = 1,
        return_variance: bool = False,
        random_state: np.random.Generator | None = None,
    ) -> np.ndarray | tuple[np.ndarray, np.ndarray]:
        """Sample mean

        Args:
            size: number of samples
            return_variance: whether to return variance as well
            random_state: random state

        Returns:
            samples from the normal distribution and optionally variance

        """

        variance = self.sample_variance(size=size, random_state=random_state)

        mean = np.stack(
            [
                stats.multivariate_normal(self.mu, cov=cov).rvs(
                    size=1,
                    random_state=random_state,
                )
                for cov in variance
            ]
        )

        if return_variance:
            return mean, variance

        return mean

sample_mean(size=1, return_variance=False, random_state=None)

Sample mean

Parameters:

Name	Type	Description	Default
size	int	number of samples	1
return_variance	bool	whether to return variance as well	False
random_state	Generator | None	random state	None

Returns:

Type	Description
ndarray | tuple[ndarray, ndarray]	samples from the normal distribution and optionally variance

Source code in conjugate/distributions.py

def sample_mean(
    self,
    size: int = 1,
    return_variance: bool = False,
    random_state: np.random.Generator | None = None,
) -> np.ndarray | tuple[np.ndarray, np.ndarray]:
    """Sample mean

    Args:
        size: number of samples
        return_variance: whether to return variance as well
        random_state: random state

    Returns:
        samples from the normal distribution and optionally variance

    """

    variance = self.sample_variance(size=size, random_state=random_state)

    mean = np.stack(
        [
            stats.multivariate_normal(self.mu, cov=cov).rvs(
                size=1,
                random_state=random_state,
            )
            for cov in variance
        ]
    )

    if return_variance:
        return mean, variance

    return mean

sample_variance(size=1, random_state=None, inv=np.linalg.inv)

Sample variance

Parameters:

Name	Type	Description	Default
size	int	number of samples	1
random_state	Generator | None	random state	None
inv	Callable	matrix inversion function	inv

Returns:

Type	Description
ndarray	samples from the inverse wishart distribution

Source code in conjugate/distributions.py

def sample_variance(
    self,
    size: int = 1,
    random_state: np.random.Generator | None = None,
    inv: Callable = np.linalg.inv,
) -> np.ndarray:
    """Sample variance

    Args:
        size: number of samples
        random_state: random state
        inv: matrix inversion function

    Returns:
        samples from the inverse wishart distribution

    """

    variance = inv(
        self.lam * self.wishart.dist.rvs(size=size, random_state=random_state)
    )

    if size == 1:
        variance = variance[None, ...]

    return variance

Pareto dataclass

Bases: ContinuousPlotDistMixin, SliceMixin

Pareto distribution.

Parameters:

Name	Type	Description	Default
x_m	NUMERIC	minimum value	required
alpha	NUMERIC	scale parameter	required

Source code in conjugate/distributions.py

@dataclass
class Pareto(ContinuousPlotDistMixin, SliceMixin):
    """Pareto distribution.

    Args:
        x_m: minimum value
        alpha: scale parameter

    """

    x_m: NUMERIC
    alpha: NUMERIC

    @property
    def dist(self):
        return stats.pareto(self.alpha, scale=self.x_m)

Poisson dataclass

Bases: DiscretePlotMixin, SliceMixin

Poisson distribution.

Parameters:

Name	Type	Description	Default
lam	NUMERIC	rate parameter	required

Source code in conjugate/distributions.py

@dataclass
class Poisson(DiscretePlotMixin, SliceMixin):
    """Poisson distribution.

    Args:
        lam: rate parameter

    """

    lam: NUMERIC

    @property
    def dist(self):
        return stats.poisson(self.lam)

    def __mul__(self, other) -> "Poisson":
        return Poisson(lam=self.lam * other)

    __rmul__ = __mul__

    def __add__(self, other) -> "Poisson":
        return Poisson(self.lam + other.lam)

    __radd__ = __add__

ScaledInverseChiSquared dataclass

Bases: ContinuousPlotDistMixin, SliceMixin

Scaled inverse chi squared distribution.

Parameters:

Name	Type	Description	Default
nu	NUMERIC	degrees of freedom	required
sigma2	NUMERIC	scale parameter	required

Source code in conjugate/distributions.py

@dataclass
class ScaledInverseChiSquared(ContinuousPlotDistMixin, SliceMixin):
    """Scaled inverse chi squared distribution.

    Args:
        nu: degrees of freedom
        sigma2: scale parameter

    """

    nu: NUMERIC
    sigma2: NUMERIC

    @classmethod
    def from_inverse_gamma(
        cls,
        inverse_gamma: InverseGamma,
    ) -> "ScaledInverseChiSquared":
        """Alternative constructor from inverse gamma distribution.

        Args:
            inverse_gamma: inverse gamma distribution

        Returns:
            scaled inverse chi squared distribution

        """
        nu = inverse_gamma.alpha * 2
        sigma2 = inverse_gamma.beta * 2 / nu

        return cls(nu=nu, sigma2=sigma2)

    def to_inverse_gamma(self) -> InverseGamma:
        """Convert to inverse gamma distribution.

        Returns:
            inverse gamma distribution

        """
        return InverseGamma(alpha=self.nu / 2, beta=self.nu * self.sigma2 / 2)

    @property
    def dist(self):
        return stats.invgamma(a=self.nu / 2, scale=self.nu * self.sigma2 / 2)

from_inverse_gamma(inverse_gamma) classmethod

Alternative constructor from inverse gamma distribution.

Parameters:

Name	Type	Description	Default
inverse_gamma	InverseGamma	inverse gamma distribution	required

Returns:

Type	Description
ScaledInverseChiSquared	scaled inverse chi squared distribution

Source code in conjugate/distributions.py

@classmethod
def from_inverse_gamma(
    cls,
    inverse_gamma: InverseGamma,
) -> "ScaledInverseChiSquared":
    """Alternative constructor from inverse gamma distribution.

    Args:
        inverse_gamma: inverse gamma distribution

    Returns:
        scaled inverse chi squared distribution

    """
    nu = inverse_gamma.alpha * 2
    sigma2 = inverse_gamma.beta * 2 / nu

    return cls(nu=nu, sigma2=sigma2)

to_inverse_gamma()

Convert to inverse gamma distribution.

Returns:

Type	Description
InverseGamma	inverse gamma distribution

Source code in conjugate/distributions.py

def to_inverse_gamma(self) -> InverseGamma:
    """Convert to inverse gamma distribution.

    Returns:
        inverse gamma distribution

    """
    return InverseGamma(alpha=self.nu / 2, beta=self.nu * self.sigma2 / 2)

StudentT dataclass

Bases: ContinuousPlotDistMixin, SliceMixin

StudentT distribution.

Parameters:

Name	Type	Description	Default
mu	NUMERIC	mean	required
sigma	NUMERIC	standard deviation	required
nu	NUMERIC	degrees of freedom	required

Source code in conjugate/distributions.py

@dataclass
class StudentT(ContinuousPlotDistMixin, SliceMixin):
    """StudentT distribution.

    Args:
        mu: mean
        sigma: standard deviation
        nu: degrees of freedom

    """

    mu: NUMERIC
    sigma: NUMERIC
    nu: NUMERIC

    @property
    def dist(self):
        return stats.t(self.nu, self.mu, self.sigma)

Uniform dataclass

Bases: ContinuousPlotDistMixin, SliceMixin

Uniform distribution.

Parameters:

Name	Type	Description	Default
low	NUMERIC	lower bound	required
high	NUMERIC	upper bound	required

Source code in conjugate/distributions.py

@dataclass
class Uniform(ContinuousPlotDistMixin, SliceMixin):
    """Uniform distribution.

    Args:
        low: lower bound
        high: upper bound

    """

    low: NUMERIC
    high: NUMERIC

    def __post_init__(self):
        self.min_value = self.low
        self.max_value = self.high

    @property
    def dist(self):
        return stats.uniform(self.low, self.high)

VectorizedDist

Vectorized distribution to handle scipy distributions that don't support vectorization.

Source code in conjugate/distributions.py

class VectorizedDist:
    """Vectorized distribution to handle scipy distributions that don't support vectorization."""

    def __init__(self, params: np.ndarray, dist: Any):
        self.params = params
        self.dist = dist

    def rvs(self, size: int = 1) -> np.ndarray:
        return np.stack([self.dist(param).rvs(size=size) for param in self.params])

    def mean(self) -> np.ndarray:
        return np.stack([self.dist(param).mean() for param in self.params])

VonMises dataclass

Bases: ContinuousPlotDistMixin, SliceMixin

Von Mises distribution.

Parameters:

Name	Type	Description	Default
mu	NUMERIC	mean	required
kappa	NUMERIC	concentration	required

Source code in conjugate/distributions.py

@dataclass
class VonMises(ContinuousPlotDistMixin, SliceMixin):
    """Von Mises distribution.

    Args:
        mu: mean
        kappa: concentration

    """

    mu: NUMERIC
    kappa: NUMERIC

    def __post_init__(self) -> None:
        self.min_value = -np.pi
        self.max_value = np.pi

    @property
    def dist(self):
        return stats.vonmises(loc=self.mu, kappa=self.kappa)

VonMisesKnownConcentration dataclass

Von Mises known concentration distribution.

Taken from Section 2.13.1.

Parameters:

Name	Type	Description	Default
a	NUMERIC	positive value	required
b	NUMERIC	value between 0 and 2 pi	required

Source code in conjugate/distributions.py

@dataclass
class VonMisesKnownConcentration:
    """Von Mises known concentration distribution.

    Taken from <a href=https://web.archive.org/web/20090529203101/http://www.people.cornell.edu/pages/df36/CONJINTRnew%20TEX.pdf>Section 2.13.1</a>.

    Args:
        a: positive value
        b: value between 0 and 2 pi

    """

    a: NUMERIC
    b: NUMERIC

    def log_likelihood(self, mu: NUMERIC, cos=np.cos, ln=np.log, i0=i0) -> NUMERIC:
        """Approximate log likelihood.

        Args:
            mu: mean
            cos: cosine function
            ln: log function
            i0: modified bessel function of order 0

        Returns:
            log likelihood

        """
        return self.a + cos(mu - self.b) - ln(i0(self.a))

log_likelihood(mu, cos=np.cos, ln=np.log, i0=i0)

Approximate log likelihood.

Parameters:

Name	Type	Description	Default
mu	NUMERIC	mean	required
cos		cosine function	cos
ln		log function	log
i0		modified bessel function of order 0	i0

Returns:

Type	Description
NUMERIC	log likelihood

Source code in conjugate/distributions.py

def log_likelihood(self, mu: NUMERIC, cos=np.cos, ln=np.log, i0=i0) -> NUMERIC:
    """Approximate log likelihood.

    Args:
        mu: mean
        cos: cosine function
        ln: log function
        i0: modified bessel function of order 0

    Returns:
        log likelihood

    """
    return self.a + cos(mu - self.b) - ln(i0(self.a))

VonMisesKnownDirectionProportional dataclass

Von Mises known direction proportional distribution.

Taken from Section 2.13.2.

Args: c: NUMERIC r: NUMERIC

Source code in conjugate/distributions.py

@dataclass
class VonMisesKnownDirectionProportional:
    """Von Mises known direction proportional distribution.

    Taken from <a href=https://web.archive.org/web/20090529203101/http://www.people.cornell.edu/pages/df36/CONJINTRnew%20TEX.pdf>Section 2.13.2</a>.

    Args:
    c: NUMERIC
    r: NUMERIC
    """

    c: NUMERIC
    r: NUMERIC

    def approx_log_likelihood(self, kappa: NUMERIC, ln=np.log, i0=i0) -> NUMERIC:
        """Approximate log likelihood.

        Args:
            kappa: concentration
            ln: log function
            i0: modified bessel function of order 0

        Returns:
            log likelihood up to a constant

        """
        return kappa * self.r - self.c * ln(i0(kappa))

approx_log_likelihood(kappa, ln=np.log, i0=i0)

Approximate log likelihood.

Parameters:

Name	Type	Description	Default
kappa	NUMERIC	concentration	required
ln		log function	log
i0		modified bessel function of order 0	i0

Returns:

Type	Description
NUMERIC	log likelihood up to a constant

Source code in conjugate/distributions.py

def approx_log_likelihood(self, kappa: NUMERIC, ln=np.log, i0=i0) -> NUMERIC:
    """Approximate log likelihood.

    Args:
        kappa: concentration
        ln: log function
        i0: modified bessel function of order 0

    Returns:
        log likelihood up to a constant

    """
    return kappa * self.r - self.c * ln(i0(kappa))

Weibull dataclass

Bases: ContinuousPlotDistMixin, SliceMixin

Weibull distribution.

Parameterization from Section 2.11 of paper.

Parameters:

Name	Type	Description	Default
beta	NUMERIC	shape parameter	required
theta	NUMERIC	scale parameter	required

Example

Recreation of the plot on Wikipedia.

import matplotlib.pyplot as plt
import numpy as np

from conjugate.distributions import Weibull

lam = 1
k = np.array([0.5, 1.0, 1.5, 5.0])

beta = k
theta = lam ** beta

distribution = Weibull(beta=beta, theta=theta)
ax = distribution.set_bounds(0, 2.5).plot_pdf(
    label=["k=0.5", "k=1.0", "k=1.5", "k=5.0"],
    color=["blue", "red", "pink", "green"],
)
ax.legend()

Source code in conjugate/distributions.py

@dataclass
class Weibull(ContinuousPlotDistMixin, SliceMixin):
    """Weibull distribution.

    Parameterization from Section 2.11 of <a href="https://web.archive.org/web/20090529203101/http://www.people.cornell.edu/pages/df36/CONJINTRnew%20TEX.pdf">paper</a>.

    Args:
        beta: shape parameter
        theta: scale parameter

    Example:
        Recreation of the plot on <a href=https://en.wikipedia.org/wiki/Weibull_distribution>Wikipedia</a>.

        ```python
        import matplotlib.pyplot as plt
        import numpy as np

        from conjugate.distributions import Weibull

        lam = 1
        k = np.array([0.5, 1.0, 1.5, 5.0])

        beta = k
        theta = lam ** beta

        distribution = Weibull(beta=beta, theta=theta)
        ax = distribution.set_bounds(0, 2.5).plot_pdf(
            label=["k=0.5", "k=1.0", "k=1.5", "k=5.0"],
            color=["blue", "red", "pink", "green"],
        )
        ax.legend()
        ```
        <!--
        plt.savefig("plot-check.png")
        plt.close()
        -->

    """

    beta: NUMERIC
    theta: NUMERIC

    @property
    def dist(self):
        k = self.beta
        lam = self.theta ** (1 / self.beta)
        return stats.weibull_min(c=k, scale=lam)

Wishart dataclass

Wishart distribution.

Parameters:

Name	Type	Description	Default
nu	NUMERIC	degrees of freedom	required
V	NUMERIC	scale matrix	required

Source code in conjugate/distributions.py

@dataclass
class Wishart:
    """Wishart distribution.

    Args:
        nu: degrees of freedom
        V: scale matrix

    """

    nu: NUMERIC
    V: NUMERIC

    @property
    def dist(self):
        return stats.wishart(df=self.nu, scale=self.V)

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