Updated docstring for Potential function (#6559)

* Updated docstring for Potential function

* Updating docstring according to suggestions

* Updated Potential docstring as per suggestion

Author: Dhruvanshu-Joshi

pymc/model.py

@@ -2042,16 +2042,95 @@ def Deterministic(name, var, model=None, dims=None):
 
 
 def Potential(name, var, model=None):
-    """Add an arbitrary factor potential to the model likelihood
+    """
+    Add an arbitrary factor potential to the model likelihood
+
+    The Potential function is used to add arbitrary factors (such as constraints or other likelihood components) to adjust the probability density of the model.
+
+    Warnings
+    --------
+    Potential functions only influence logp based sampling, like the one used by ``pm.sample``.
+    Potentials, modify the log-probability of the model by adding a contribution to the logp which is used by sampling algorithms which rely on the information about the observed data to generate posterior samples.
+    Potentials are not applicable in the context of forward sampling because they don't affect the prior distribution itself, only the computation of the logp.
+    Forward sampling algorithms generate sample points from the prior distribution of the model, without taking into account the likelihood function.
+    In other words, it does not use the information about the observed data.
+    Hence, Potentials do not affect forward sampling, which is used by ``sample_prior_predictive`` and ``sample_posterior_predictive``.
+    A warning saying "The effect of Potentials on other parameters is ignored during prior predictive sampling" is always emitted to alert user of this.
 
     Parameters
     ----------
-    name: str
-    var: PyTensor variables
+    name : str
+        Name of the potential variable to be registered in the model.
+    var : tensor_like
+        Expression to be added to the model joint logp.
+    model : Model, optional
+        The model object to which the potential function is added.
+        If ``None`` is provided, the current model is used.
 
     Returns
     -------
-    var: var, with name attribute
+    var : tensor_like
+        The registered, named model variable.
+
+    Examples
+    --------
+    Have a look at the following example:
+
+    In this example, we define a constraint on ``x`` to be greater or equal to 0 via the ``pm.Potential`` function.
+    We pass ``-pm.math.log(pm.math.switch(constraint, 1, 0))`` as second argument which will return an expression depending on if the constraint is met or not and which will be added to the likelihood of the model.
+    The probablity density that this model produces agrees strongly with the constraint that ``x`` should be greater than or equal to 0. All the cases who do not satisfy the constraint are strictly not considered.
+
+    .. code:: python
+
+        with pm.Model() as model:
+            x = pm.Normal("x", mu=0, sigma=1)
+            y = pm.Normal("y", mu=x, sigma=1, observed=data)
+            constraint = x >= 0
+            potential = pm.Potential("x_constraint", pm.math.log(pm.math.switch(constraint, 1, 0.0)))
+
+    However, if we use ``-pm.math.log(pm.math.switch(constraint, 1, 0.5))`` the potential again penalizes the likelihood when constraint is not met but with some deviations allowed.
+    Here, Potential function is used to pass a soft constraint.
+    A soft constraint is a constraint that is only partially satisfied.
+    The effect of this is that the posterior probability for the parameters decreases as they move away from the constraint, but does not become exactly zero.
+    This allows the sampler to generate values that violate the constraint, but with lower probability.
+
+    .. code:: python
+
+        with pm.Model() as model:
+            x = pm.Normal("x", mu=0.1, sigma=1)
+            y = pm.Normal("y", mu=x, sigma=1, observed=data)
+            constraint = x >= 0
+            potential = pm.Potential("x_constraint", pm.math.log(pm.math.switch(constraint, 1, 0.5)))
+
+    In this example, Potential is used to obtain an arbitrary prior.
+    This prior distribution refers to the prior knowledge that the values of ``max_items`` are likely to be small rather than being large.
+    The prior probability of ``max_items`` is defined using a Potential object with the log of the inverse of ``max_items`` as its value.
+    This means that larger values of ``max_items`` have a lower prior probability density, while smaller values of ``max_items`` have a higher prior probability density.
+    When the model is sampled, the posterior distribution of ``max_items`` given the observed value of ``n_items`` will be influenced by the power-law prior defined in the Potential object
+
+    .. code:: python
+
+        with pm.Model():
+            # p(max_items) = 1 / max_items
+            max_items = pm.Uniform("max_items", lower=1, upper=100)
+            pm.Potential("power_prior", pm.math.log(1/max_items))
+
+            n_items = pm.Uniform("n_items", lower=1, upper=max_items, observed=60)
+
+    In the next example, the ``soft_sum_constraint`` potential encourages ``x`` and ``y`` to have a small sum, effectively adding a soft constraint on the relationship between the two variables.
+    This can be useful in cases where you want to ensure that the sum of multiple variables stays within a certain range, without enforcing an exact value.
+    In this case, the larger the deviation, larger will be the negative value (-((x + y)**2)) which the MCMC sampler will attempt to minimize.
+    However, the sampler might generate values for some small deviations but with lower probability hence this is a soft constraint.
+
+    .. code:: python
+
+        with pm.Model() as model:
+            x = pm.Normal("x", mu=0.1, sigma=1)
+            y = pm.Normal("y", mu=x, sigma=1, observed=data)
+            soft_sum_constraint = pm.Potential("soft_sum_constraint", -((x + y)**2))
+
+    The potential value is incorporated into the model log-probability, so it should be -inf (or very negative) when a constraint is violated, so that those draws are rejected. 0 won't have any effect and positive values will make the proposals more likely to be accepted.
+
     """
     model = modelcontext(model)
     var.name = model.name_for(name)