PreliZ: A tool-box for prior elicitation

A Hurdle distribution is not a standalone distribution but a modifier of univariate PreliZ distributions. It creates a mixture distribution that combines a point mass at zero with a continuous (or discrete) distribution.

This is used to model data generated by two processes: one that generates zeros and another that generates non-zero values (e.g., healthcare costs or insurance claims with excess zeros).

Parameters:

• dist (PreliZ distribution): The base univariate PreliZ distribution, which will be truncated at zero.
• psi (float): The expected proportion of the base distribution ($0 < \psi < 1$).

The Logistic distribution in PreliZ is a continuous, symmetric, and unbounded distribution defined by two parameters:

• mu ($\mu$): The mean of the distribution (center).
• s: The scale parameter ($s > 0$), which controls the width.

It is characterized by an S-shaped cumulative distribution function (the logistic function) and has heavier tails than a normal distribution. It is useful for modeling growth processes or as a latent variable model where error terms follow a logistic distribution.

Properties:

• Support: $x \in \mathbb{R}$
• Mean: $\mu$
• Variance: $\frac{\pi^2}{3}s^2$

from preliz import Logistic

# Initialize a Logistic distribution with mean 0 and scale 1
dist = Logistic(mu=0.0, s=1.0)

The Von Mises distribution is a continuous probability distribution on the unit circle, used for modeling circular data like wind directions or compass bearings. It is characterized by two parameters:

• mu ($\mu$): Mean direction, where $-\pi \leq \mu \leq \pi$.
• kappa ($\kappa$): Concentration parameter, where $\kappa \geq 0$.

When $\kappa \to \infty$, it approximates a Normal distribution. When $\kappa = 0$, it converges to a Uniform distribution on the interval $(-\pi, \pi)$.

from preliz import VonMises

# Initialize with mean direction (mu) and concentration (kappa)
dist = VonMises(mu=0.0, kappa=4.0)

The Geometric distribution is a discrete, univariate, non-negative, and asymmetric distribution used to model the number of Bernoulli trials required to achieve the first success. It is characterized by its memoryless property.

Mathematical Properties:

• Support: $x ∈ ℕ_{>0}$
• Mean: $\frac{1}{p}$
• Variance: $\frac{1 - p}{p^2}$
• PMF: $f(x \mid p) = p(1-p)^{x-1}$
• CDF: $F(x \mid p) = 1 - (1-p)^x$

The Beta prime distribution has the following mathematical properties:

• Support: $x \in (0, \infty)$
• Mean: $\frac{\alpha}{\beta - 1}$ (valid for $\beta > 1$)
• Variance: $\frac{\alpha(\alpha + \beta - 1)}{(\beta - 1)^2(\beta - 2)}$ (valid for $\beta > 2$)

It is closely related to the Beta distribution: if $X \sim \text{Beta}(\alpha, \beta)$, then $Y = X / (1 - X) \sim \text{Beta-prime}(\alpha, \beta)$.

The Uniform distribution is a continuous, symmetric, and bounded probability distribution defined by a lower and upper bound. The probability density is constant within the interval $[lower, upper]$ and zero elsewhere. It is characterized by the following properties:

• Support: $x ext{ } ∈ [lower, upper]$
• Mean: $\frac{lower + upper}{2}$
• Variance: $\frac{(upper - lower)^2}{12}$

Parameters:

• lower (float): The lower bound of the distribution.
• upper (float): The upper bound of the distribution ($upper > lower$).

from preliz import Uniform

# Example initialization
dist = Uniform(lower=1.0, upper=6.0)

The ppa tool returns two types of suggested priors:

1. Family-restricted priors: These are restricted to the same distribution families specified in your original PreliZ model (e.g., if your model used pz.Normal, the suggestion will also be a Normal distribution).
2. Flexible family priors: These can suggest entirely new distribution families. Note that these results may require more selections to be considered reliable and are currently an area of active exploration.

The Gumbel distribution (type-I Generalized Extreme Value distribution) is a continuous distribution used to model the maximum or minimum of various random variables.

Parameters:

• mu (float): Location parameter.
• beta (float): Scale parameter.

Mathematical Properties:

• Support: $x \in (-\infty, \infty)$
• Mean: $\mu + \beta \gamma$ (where $\gamma$ is the Euler-Mascheroni constant)
• Variance: $\frac{\pi^2}{6} \beta^2$

Related Distributions:

• Weibull Distribution: If a random variable follows a Weibull distribution, its logarithm follows a Gumbel distribution.
• Exponential Distribution: The maximum of exponentially distributed variables follows a Gumbel distribution.

The Kumaraswamy distribution is a continuous probability distribution bounded between 0 and 1. It is defined by two positive shape parameters: $a$ and $b$ ($a > 0, b > 0$). It can represent various shapes such as uniform, U-shape, or exponential-like distributions within the unit interval [0, 1].

Mathematical Properties:

• Support: $x \in (0, 1)$
• PDF: $f(x|a, b) = a b x^{a-1} (1 - x^{a})^{b-1}$
• CDF: $F(x|a, b) = 1 - (1 - x^{a})^{b}$

Parameters:

• a (float): First shape parameter, must be $> 0$.
• b (float): Second shape parameter, must be $> 0$.

from preliz import Kumaraswamy

# Initialize with shape parameters a and b
dist = Kumaraswamy(a=2.0, b=5.0)

PreliZ uses a preliz.rcParams object to manage default values. This object behaves similarly to a dictionary but uses fixed keys and includes validation functions to ensure settings are valid. You can inspect the current settings by accessing preliz.rcParams.

import preliz as pz

# View all current settings
print(pz.rcParams)