doc(gh-2187): add math explanations for Normal, Cauchy, and Exponential distributions (#2188)

* doc: add math explanations for Normal, Cauchy, and Exponential distributions (#2187)

Add LaTeX math explanations, parameter documentation, and method
docstrings following the format established in #2185 (Beta/BetaProportion).

Covers PDF, log_prob, CDF, ICDF, mean, variance, entropy, and sample
for each distribution. Cauchy docstrings note the undefined mean/variance.

* Address review feedback: add log_cdf equation and use \frac

- Add equation for Normal.log_cdf with note about jax.scipy.stats.norm.logcdf implementation
- Replace \tfrac with \frac in Cauchy.icdf equation

Addresses review comments from @Qazalbash in PR #2188

* add icdf equation for Normal distribution

* fix: documenting equations as per the implementation

---------

Co-authored-by: Meesum Qazalbash <meesumqazalbash@gmail.com>

Author: kimjune01

numpyro/distributions/continuous.py

@@ -334,6 +334,27 @@ def entropy(self) -> ArrayLike:
 
 
 class Cauchy(Distribution):
+    r"""Cauchy distribution parameterized by location (:attr:`loc`) and
+    scale (:attr:`scale`).
+
+    The probability density function (PDF) is defined as:
+
+    .. math::
+       f(x; x_0, \gamma) = \frac{1}{\pi \gamma \left[1 +
+       \left(\frac{x - x_0}{\gamma}\right)^2\right]}
+
+    where :math:`x \in \mathbb{R}`, :math:`x_0 \in \mathbb{R}` is the location,
+    and :math:`\gamma > 0` is the scale. The Cauchy distribution has no finite
+    mean or variance.
+
+    :param loc: Location parameter (:math:`x_0`).
+    :type loc: ArrayLike
+    :param scale: Scale parameter (:math:`\gamma`).
+    :type scale: ArrayLike
+    :param validate_args: Whether to validate input constraints, defaults to None.
+    :type validate_args: bool, optional
+    """
+
     arg_constraints = {"loc": constraints.real, "scale": constraints.positive}
     support = constraints.real
     reparametrized_params = ["loc", "scale"]
@@ -352,12 +373,32 @@ def __init__(
         )
 
     def sample(self, key: jax.Array, sample_shape: tuple[int, ...] = ()) -> ArrayLike:
+        r"""Generates samples using the inverse CDF method via :func:`~jax.random.cauchy`.
+
+        :param key: JAX PRNGKey for reproducibility.
+        :type key: jax.Array
+        :param sample_shape: The shape of the samples to be generated.
+        :type sample_shape: tuple[int, ...]
+        :return: Samples from the Cauchy distribution of shape ``sample_shape + batch_shape``.
+        :rtype: ArrayLike
+        """
         assert is_prng_key(key)
         eps = random.cauchy(key, shape=sample_shape + self.batch_shape)
         return self.loc + eps * self.scale
 
     @validate_sample
     def log_prob(self, value: ArrayLike) -> ArrayLike:
+        r"""Calculates the log of the probability density function.
+
+        .. math::
+           \log f(x; x_0, \gamma) = -\log(\pi) - \log(\gamma)
+           - \log\!\left[1 + \left(\frac{x - x_0}{\gamma}\right)^2\right]
+
+        :param value: Values at which to evaluate the log density.
+        :type value: ArrayLike
+        :return: Log probability density.
+        :rtype: ArrayLike
+        """
         return (
             -jnp.log(jnp.pi)
             - jnp.log(self.scale)
@@ -366,20 +407,51 @@ def log_prob(self, value: ArrayLike) -> ArrayLike:
 
     @property
     def mean(self) -> ArrayLike:
+        r"""The mean of the Cauchy distribution is undefined.
+
+        Returns ``NaN`` for all batch elements.
+        """
         return jnp.full(self.batch_shape, jnp.nan)
 
     @property
     def variance(self) -> ArrayLike:
+        r"""The variance of the Cauchy distribution is undefined.
+
+        Returns ``NaN`` for all batch elements.
+        """
         return jnp.full(self.batch_shape, jnp.nan)
 
     def cdf(self, value: ArrayLike) -> ArrayLike:
+        r"""Cumulative distribution function.
+
+        .. math::
+           F(x; x_0, \gamma) = \frac{1}{\pi}\arctan\!\left(\frac{x - x_0}{\gamma}\right)
+           + \frac{1}{2}
+
+        :param value: Value to evaluate.
+        :type value: ArrayLike
+        """
         scaled = (value - self.loc) / self.scale
         return jnp.arctan(scaled) / jnp.pi + 0.5
 
     def icdf(self, q: ArrayLike) -> ArrayLike:
+        r"""Inverse cumulative distribution function (Quantile function).
+
+        .. math::
+           F^{-1}(q; x_0, \gamma) = x_0 + \gamma \tan\!\left[\pi\!\left(q
+           - \frac{1}{2}\right)\right]
+
+        :param q: Probability value in :math:`[0,1]`.
+        :type q: ArrayLike
+        """
         return self.loc + self.scale * jnp.tan(jnp.pi * (q - 0.5))
 
     def entropy(self) -> ArrayLike:
+        r"""Entropy of the Cauchy distribution.
+
+        .. math::
+           H(X) = \log(4\pi\gamma)
+        """
         return jnp.broadcast_to(jnp.log(4 * np.pi * self.scale), self.batch_shape)
 
 
@@ -589,6 +661,21 @@ def log_prob(self, value: ArrayLike) -> ArrayLike:
 
 
 class Exponential(Distribution):
+    r"""Exponential distribution parameterized by rate (:attr:`rate`).
+
+    The probability density function (PDF) is defined as:
+
+    .. math::
+       f(x; \lambda) = \lambda e^{-\lambda x}
+
+    where :math:`x \geq 0` and :math:`\lambda > 0` is the rate parameter.
+
+    :param rate: Rate parameter (:math:`\lambda`), the inverse of the mean.
+    :type rate: ArrayLike
+    :param validate_args: Whether to validate input constraints, defaults to None.
+    :type validate_args: bool, optional
+    """
+
     reparametrized_params = ["rate"]
     arg_constraints = {"rate": constraints.positive}
     support = constraints.positive
@@ -605,30 +692,79 @@ def __init__(
         )
 
     def sample(self, key: jax.Array, sample_shape: tuple[int, ...] = ()) -> ArrayLike:
+        r"""Generates samples by scaling standard exponential draws by the
+        inverse rate: :math:`X = E / \lambda`, where :math:`E \sim \mathrm{Exp}(1)`.
+
+        :param key: JAX PRNGKey for reproducibility.
+        :type key: jax.Array
+        :param sample_shape: The shape of the samples to be generated.
+        :type sample_shape: tuple[int, ...]
+        :return: Samples from the Exponential distribution of shape ``sample_shape + batch_shape``.
+        :rtype: ArrayLike
+        """
         assert is_prng_key(key)
         return (
             random.exponential(key, shape=sample_shape + self.batch_shape) / self.rate
         )
 
     @validate_sample
     def log_prob(self, value: ArrayLike) -> ArrayLike:
+        r"""Calculates the log of the probability density function.
+
+        .. math::
+           \log f(x; \lambda) = \log \lambda - \lambda x
+
+        :param value: Values at which to evaluate the log density.
+        :type value: ArrayLike
+        :return: Log probability density.
+        :rtype: ArrayLike
+        """
         return jnp.log(self.rate) - self.rate * value
 
     @property
     def mean(self) -> ArrayLike:
+        r"""Calculates the analytical mean.
+
+        .. math:: E[X] = \frac{1}{\lambda}
+        """
         return jnp.reciprocal(self.rate)
 
     @property
     def variance(self) -> ArrayLike:
+        r"""Calculates the analytical variance.
+
+        .. math:: \mathrm{Var}(X) = \frac{1}{\lambda^2}
+        """
         return jnp.reciprocal(self.rate**2)
 
     def cdf(self, value: ArrayLike) -> ArrayLike:
+        r"""Cumulative distribution function.
+
+        .. math::
+           F(x; \lambda) = 1 - e^{-\lambda x}
+
+        :param value: Value to evaluate.
+        :type value: ArrayLike
+        """
         return -jnp.expm1(-self.rate * value)
 
     def icdf(self, q: ArrayLike) -> ArrayLike:
+        r"""Inverse cumulative distribution function (Quantile function).
+
+        .. math::
+           F^{-1}(q; \lambda) = -\frac{\ln(1 - q)}{\lambda}
+
+        :param q: Probability value in :math:`[0,1]`.
+        :type q: ArrayLike
+        """
         return -jnp.log1p(-q) / self.rate
 
     def entropy(self) -> ArrayLike:
+        r"""Entropy of the Exponential distribution.
+
+        .. math::
+           H(X) = 1 - \ln \lambda
+        """
         return 1 - jnp.log(self.rate)
 
 
@@ -2513,6 +2649,26 @@ def infer_shapes(loc, cov_factor, cov_diag):
 
 
 class Normal(Distribution):
+    r"""Normal (Gaussian) distribution parameterized by mean (:attr:`loc`) and
+    standard deviation (:attr:`scale`).
+
+    The probability density function (PDF) is defined as:
+
+    .. math::
+       f(x; \mu, \sigma) = \frac{1}{\sigma \sqrt{2\pi}}
+       \exp\!\left( -\frac{(x - \mu)^2}{2\sigma^2} \right)
+
+    where :math:`x \in \mathbb{R}`, :math:`\mu \in \mathbb{R}` is the mean,
+    and :math:`\sigma > 0` is the standard deviation.
+
+    :param loc: Mean of the distribution (:math:`\mu`).
+    :type loc: ArrayLike
+    :param scale: Standard deviation of the distribution (:math:`\sigma`).
+    :type scale: ArrayLike
+    :param validate_args: Whether to validate input constraints, defaults to None.
+    :type validate_args: bool, optional
+    """
+
     arg_constraints = {"loc": constraints.real, "scale": constraints.positive}
     support = constraints.real
     reparametrized_params = ["loc", "scale"]
@@ -2531,6 +2687,16 @@ def __init__(
         )
 
     def sample(self, key: jax.Array, sample_shape: tuple[int, ...] = ()) -> ArrayLike:
+        r"""Generates samples via the reparameterization trick:
+        :math:`X = \mu + \sigma \epsilon`, where :math:`\epsilon \sim \mathcal{N}(0,1)`.
+
+        :param key: JAX PRNGKey for reproducibility.
+        :type key: jax.Array
+        :param sample_shape: The shape of the samples to be generated.
+        :type sample_shape: tuple[int, ...]
+        :return: Samples from the Normal distribution of shape ``sample_shape + batch_shape``.
+        :rtype: ArrayLike
+        """
         assert is_prng_key(key)
         eps = random.normal(
             key, shape=sample_shape + self.batch_shape + self.event_shape
@@ -2539,29 +2705,83 @@ def sample(self, key: jax.Array, sample_shape: tuple[int, ...] = ()) -> ArrayLik
 
     @validate_sample
     def log_prob(self, value: ArrayLike) -> ArrayLike:
+        r"""Calculates the log of the probability density function.
+
+        .. math::
+           \log f(x; \mu, \sigma) = -\frac{(x - \mu)^2}{2\sigma^2}
+           - \log(\sigma \sqrt{2\pi})
+
+        :param value: Values at which to evaluate the log density.
+        :type value: ArrayLike
+        :return: Log probability density.
+        :rtype: ArrayLike
+        """
         normalize_term = jnp.log(jnp.sqrt(2 * jnp.pi) * self.scale)
         value_scaled = (value - self.loc) / self.scale
         return -0.5 * value_scaled**2 - normalize_term
 
     def cdf(self, value: ArrayLike) -> ArrayLike:
+        r"""Cumulative distribution function.
+
+        .. math::
+           F(x; \mu, \sigma) = \Phi\!\left(\frac{x-\mu}{\sigma}\right)
+
+        where, :math:`\Phi` is the
+        `cumulative distribution function of standard normal distribution <https://en.wikipedia.org/wiki/Normal_distribution#Cumulative_distribution_function>`_.
+        Implementation uses :func:`jax.scipy.special.ndtr` for :math:`\Phi`.
+
+        :param value: Value to evaluate.
+        :type value: ArrayLike
+        """
         scaled = (value - self.loc) / self.scale
         return ndtr(scaled)
 
     def log_cdf(self, value: ArrayLike) -> ArrayLike:
+        r"""Log of the cumulative distribution function. Implementation
+        calls :func:`jax.scipy.stats.norm.logcdf`.
+
+        :param value: Value to evaluate.
+        :type value: ArrayLike
+        """
         return jax_norm.logcdf(value, loc=self.loc, scale=self.scale)
 
     def icdf(self, q: ArrayLike) -> ArrayLike:
+        r"""Inverse cumulative distribution function (Quantile function).
+
+        .. math::
+           F^{-1}(q; \mu, \sigma) = \mu + \sigma\,\Phi^{-1}(q)
+
+        where, :math:`\mathrm{\Phi^{-1}}` is inverse
+        `cumulative distribution function of standard normal distribution <https://en.wikipedia.org/wiki/Normal_distribution#Cumulative_distribution_function>`_.
+        Implementation uses :func:`jax.scipy.special.ndtri` for :math:`\Phi^{-1}`.
+
+        :param q: Probability value in :math:`[0,1]`.
+        :type q: ArrayLike
+        """
         return self.loc + self.scale * ndtri(q)
 
     @property
     def mean(self) -> ArrayLike:
+        r"""Calculates the analytical mean.
+
+        .. math:: E[X] = \mu
+        """
         return jnp.broadcast_to(self.loc, self.batch_shape)
 
     @property
     def variance(self) -> ArrayLike:
+        r"""Calculates the analytical variance.
+
+        .. math:: \mathrm{Var}(X) = \sigma^2
+        """
         return jnp.broadcast_to(self.scale**2, self.batch_shape)
 
     def entropy(self) -> ArrayLike:
+        r"""Entropy of the Normal distribution.
+
+        .. math::
+           H(X) = \frac{1}{2} \ln(2\pi e \sigma^2)
+        """
         return jnp.broadcast_to(
             (jnp.log(2 * np.pi * self.scale**2) + 1) / 2, self.batch_shape
         )