Distributions Reference¶

Pre-built likelihood models for common statistical distributions.

All functions return a LikelihoodModel instance ready for fitting.

exponential¶

from symlik.distributions import exponential

model = exponential(data_var='x', param='lambda')

Distribution: \(f(x; \lambda) = \lambda e^{-\lambda x}\)

Parameters:

data_var (str): Name of data variable. Default: 'x'
param (str): Name of rate parameter. Default: 'lambda'

MLE: \(\hat\lambda = 1/\bar{x}\)

normal¶

from symlik.distributions import normal

model = normal(data_var='x', mean='mu', var='sigma2')

Distribution: \(f(x; \mu, \sigma^2) = \frac{1}{\sqrt{2\pi\sigma^2}} e^{-(x-\mu)^2/(2\sigma^2)}\)

Parameters:

data_var (str): Name of data variable. Default: 'x'
mean (str): Name of mean parameter. Default: 'mu'
var (str): Name of variance parameter. Default: 'sigma2'

MLE: \(\hat\mu = \bar{x}\), \(\hat\sigma^2 = \frac{1}{n}\sum(x_i - \bar{x})^2\)

Note: Use bounds to keep variance positive: bounds={'sigma2': (0.01, None)}

normal_mean¶

from symlik.distributions import normal_mean

model = normal_mean(data_var='x', mean='mu', known_var=1.0)

Normal distribution with known variance. Estimates mean only.

Parameters:

data_var (str): Name of data variable. Default: 'x'
mean (str): Name of mean parameter. Default: 'mu'
known_var (float): Known variance value. Default: 1.0

MLE: \(\hat\mu = \bar{x}\)

poisson¶

from symlik.distributions import poisson

model = poisson(data_var='x', param='lambda')

Distribution: \(P(X = k) = \frac{\lambda^k e^{-\lambda}}{k!}\)

Parameters:

data_var (str): Name of data variable. Default: 'x'
param (str): Name of rate parameter. Default: 'lambda'

MLE: \(\hat\lambda = \bar{x}\)

Note: The log-likelihood ignores the \(\log(k!)\) term (constant w.r.t. \(\lambda\)).

bernoulli¶

from symlik.distributions import bernoulli

model = bernoulli(data_var='x', param='p')

Distribution: \(P(X = 1) = p\), \(P(X = 0) = 1 - p\)

Parameters:

data_var (str): Name of data variable (0/1 values). Default: 'x'
param (str): Name of probability parameter. Default: 'p'

MLE: \(\hat{p} = \bar{x}\) (proportion of 1s)

Note: Use bounds to avoid log(0): bounds={'p': (0.01, 0.99)}

binomial¶

from symlik.distributions import binomial

model = binomial(successes='k', trials='n', param='p')

Binomial distribution for a single observation of k successes in n trials.

Distribution: \(P(K = k) = \binom{n}{k} p^k (1-p)^{n-k}\)

Parameters:

successes (str): Name of success count variable. Default: 'k'
trials (str): Name of trials count variable. Default: 'n'
param (str): Name of probability parameter. Default: 'p'

MLE: \(\hat{p} = k/n\)

gamma¶

from symlik.distributions import gamma

model = gamma(data_var='x', shape='alpha', rate='beta')

Distribution: \(f(x; \alpha, \beta) = \frac{\beta^\alpha}{\Gamma(\alpha)} x^{\alpha-1} e^{-\beta x}\)

Uses the rate parameterization (not scale).

Parameters:

data_var (str): Name of data variable. Default: 'x'
shape (str): Name of shape parameter \(\alpha\). Default: 'alpha'
rate (str): Name of rate parameter \(\beta\). Default: 'beta'

Note: No closed-form MLE. Use numerical optimization with good starting values.

weibull¶

from symlik.distributions import weibull

model = weibull(data_var='x', shape='k', scale='lambda')

Distribution: \(f(x; k, \lambda) = \frac{k}{\lambda}\left(\frac{x}{\lambda}\right)^{k-1} e^{-(x/\lambda)^k}\)

Parameters:

data_var (str): Name of data variable. Default: 'x'
shape (str): Name of shape parameter \(k\). Default: 'k'
scale (str): Name of scale parameter \(\lambda\). Default: 'lambda'

beta¶

from symlik.distributions import beta

model = beta(data_var='x', alpha='alpha', beta_param='beta')

Distribution: \(f(x; \alpha, \beta) = \frac{\Gamma(\alpha+\beta)}{\Gamma(\alpha)\Gamma(\beta)} x^{\alpha-1} (1-x)^{\beta-1}\)

For data in the interval \((0, 1)\).

Parameters:

data_var (str): Name of data variable. Default: 'x'
alpha (str): First shape parameter. Default: 'alpha'
beta_param (str): Second shape parameter. Default: 'beta'

Note: Ensure data is strictly in \((0, 1)\), not including endpoints.