- pymc.Potential(name, var, model=None, dims=None)[source]#
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.
Name of the potential variable to be registered in the model.
Expression to be added to the model joint logp.
The model object to which the potential function is added. If
Noneis provided, the current model in the context stack is used.
Dimension names for the variable.
The registered, named model variable.
Potential functions only influence logp-based sampling. Therefore, they are applicable for sampling with
Have a look at the following example:
In this example, we define a constraint on
xto be greater or equal to 0 via the
pm.Potentialfunction. 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
xshould be greater than or equal to 0. All the cases who do not satisfy the constraint are strictly not considered.
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)))
However, if we use
pm.math.log(pm.math.switch(constraint, 1.0, 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.
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, 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_itemsare likely to be small rather than being large. The prior probability of
max_itemsis defined using a Potential object with the log of the inverse of
max_itemsas its value. This means that larger values of
max_itemshave a lower prior probability density, while smaller values of
max_itemshave a higher prior probability density. When the model is sampled, the posterior distribution of
max_itemsgiven the observed value of
n_itemswill be influenced by the power-law prior defined in the Potential object
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
yto 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.
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.