Custom Priors In Pymc

Say I want to put a custom prior on two variables a and b in PyMC, e.g.: p(a,b)∝(a+b)^(−5/2) (for the motivation behind this choice of prior, see this answer) Can this be don

Solution 1:

Yup! It's quite possible, and in fact quite straightforward.

If you're in PyMC 2, check out the documentation on the creation of stochastic variables.

@pymc.stochastic(dtype=int)defswitchpoint(value=1900, t_l=1851, t_h=1962):
    """The switchpoint for the rate of disaster occurrence."""if value > t_h or value < t_l:
        # Invalid valuesreturn -np.inf
    else:
        # Uniform log-likelihoodreturn -np.log(t_h - t_l + 1)

If you're in PyMC 3, have a look at multivariate.py. Keep in mind the values passed in to init and logp are all theano variables, not numpy arrays. Is that enough to get you started?

For example, this is the Multivariate Normal distribution

classMvNormal(Continuous):
    """
    Multivariate normal

    :Parameters:
        mu : vector of means
        tau : precision matrix

    .. math::
        f(x \mid \pi, T) = \frac{|T|^{1/2}}{(2\pi)^{1/2}} \exp\left\{ -\frac{1}{2} (x-\mu)^{\prime}T(x-\mu) \right\}

    :Support:
        2 array of floats
    """def__init__(self, mu, tau, *args, **kwargs):
        super(MvNormal, self).__init__(*args, **kwargs)
        self.mean = self.median = self.mode = self.mu = mu
        self.tau = tau

    deflogp(self, value):
        mu = self.mu
        tau = self.tau

        delta = value - mu
        k = tau.shape[0]

        return1/2. * (-k * log(2*pi) + log(det(tau)) - dot(delta.T, dot(tau, delta)))

Solution 2:

In PyMC2, the trick is to put the a and b parameters together:

# Code that defines the prior: p(a,b)∝(a+b)^(−5/2)@pm.stochasticdefab(power=-2.5, value=[1,1]):
    if np.any(value <= 0):
        return -np.Inf
    return power * np.log(value[0]+value[1])

a = ab[0]
b = ab[1]

This notebook has a full example.