5. Fitting Models

PyMC provides three objects that fit models:

  • MCMC, which coordinates Markov chain Monte Carlo algorithms. The actual work of updating stochastic variables conditional on the rest of the model is done by StepMethod objects, which are described in this chapter.
  • MAP, which computes maximum a posteriori estimates.
  • NormApprox, which computes the ‘normal approximation’ [Gelman2004]: the joint distribution of all stochastic variables in a model is approximated as normal using local information at the maximum a posteriori estimate.

All three objects are subclasses of Model, which is PyMC’s base class for fitting methods. MCMC and NormApprox, both of which can produce samples from the posterior, are subclasses of Sampler, which is PyMC’s base class for Monte Carlo fitting methods. Sampler provides a generic sampling loop method and database support for storing large sets of joint samples. These base classes implement some basic methods that are inherited by the three implemented fitting methods, so they are documented at the end of this section.

5.1. Creating models

The first argument to any fitting method’s __init__ method, including that of MCMC, is called input. The input argument can be just about anything; once you have defined the nodes that make up your model, you shouldn’t even have to think about how to wrap them in a Model instance. Some examples of model instantiation using nodes a, b and c follow:

  • M = MCMC(set([a,b,c])) This will create a MCMC model with \(a\), \(b\), and \(c\) as components, each of which will be exposed as attributes of M (e.g. M.a).

  • M = MCMC({`a': a, `d': [b,c]}) In this case, \(M\) will expose \(a\) and \(d\) as attributes: M.a will be \(a\), and M.d will be [b,c].

  • M = MAP([[a,b],c]) This will create a MAP model with \(a\) and \(b\) as a Container object and \(c\) exposed on its own.

  • If file MyModule contains the definitions of a, b and c:

    import MyModule
    M = Model(MyModule)
    

In this case, \(M\) will expose \(a\), \(b\) and \(c\) as attributes.

  • Using a ‘model factory’ function:

    def make_model(x):
        a = pymc.Exponential('a', beta=x, value=0.5)
    
        @pymc.deterministic
        def b(a=a):
            return 100-a
    
        @pymc.stochastic
        def c(value=.5, a=a, b=b):
            return (value-a)**2/b
    
        return locals()
    
    M = pymc.MCMC(make_model(3))
    

In this case, \(M\) will also expose \(a\), \(b\) and \(c\) as attributes.

5.2. The Model class

This class serves as a container for probability models and as a base class for the classes responsible for model fitting, such as MCMC.

Model‘s init method takes the following arguments:

input:
Some collection of PyMC nodes defining a probability model. These may be stored in a list, set, tuple, dictionary, array, module, or any object with a __dict__ attribute.
verbose (optional):
An integer controlling the verbosity of the model’s output.

Models’ useful methods are:

draw_from_prior():
Sets all stochastic variables’ values to new random values, which would be a sample from the joint distribution if all data and Potential instances’ log-probability functions returned zero. If any stochastic variables lack a random() method, PyMC will raise an exception.
seed():
Same as draw_from_prior, but only stochastics whose rseed attribute is not None are changed.

As introduced in the previous chapter, the helper function graph.dag produces graphical representations of models (see [Jordan2004]).

Models have the following important attributes:

  • variables
  • nodes
  • stochastics
  • potentials
  • deterministics
  • observed_stochastics
  • containers
  • value
  • logp

In addition, models expose each node they contain as an attribute. For instance, if model M were produced from model ((?)) M.s would return the switchpoint variable.

Though one may instantiate Model objects directly, most users should pefer to instantiate the Model subclass that they will be using to fit their model. These are each described below.

5.3. Maximum a posteriori estimates

The MAP class sets all stochastic variables to their maximum a posteriori values using functions in SciPy’s optimize package; hence, SciPy must be installed to use it. MAP can only handle variables whose dtype is float, so it will not work, for example, on model ((?)). To fit the model in examples/gelman_bioassay.py using MAP, do the following:

>>> from pymc.examples import gelman_bioassay
>>> M = pymc.MAP(gelman_bioassay)
>>> M.fit()

This call will cause \(M\) to fit the model using modified Powell optimization, which does not require derivatives. The variables in DisasterModel have now been set to their maximum a posteriori values:

>>> M.alpha.value
array(0.8465892309923545)
>>> M.beta.value
array(7.7488499785334168)

In addition, the AIC and BIC of the model are now available:

>>> M.AIC
7.9648372671389458
>>> M.BIC
6.7374259893787265

MAP has two useful methods:

fit(method='fmin_powell', iterlim=1000, tol=.0001):
The optimization method may be fmin, fmin_l_bfgs_b, fmin_ncg, fmin_cg, or fmin_powell. See the documentation of SciPy’s optimize package for the details of these methods. The tol and iterlim parameters are passed to the optimization function under the appropriate names.
revert_to_max():
If the values of the constituent stochastic variables change after fitting, this function will reset them to their maximum a posteriori values.

If you’re going to use an optimization method that requires derivatives, MAP‘s __init__ method can take additional parameters eps and diff_order. diff_order, which must be an integer, specifies the order of the numerical approximation (see the SciPy function derivative). The step size for numerical derivatives is controlled by eps, which may be either a single value or a dictionary of values whose keys are variables (actual objects, not names).

The useful attributes of MAP are:

logp:
The joint log-probability of the model.
logp_at_max:
The maximum joint log-probability of the model.
AIC:
Akaike’s information criterion for this model ([Akaike1973],[Burnham2002]_).
BIC:
The Bayesian information criterion for this model [Schwarz1978].

One use of the MAP class is finding reasonable initial states for MCMC chains. Note that multiple Model subclasses can handle the same collection of nodes.

5.4. Normal approximations

The NormApprox class extends the MAP class by approximating the posterior covariance of the model using the Fisher information matrix, or the Hessian of the joint log probability at the maximum. To fit the model in examples/gelman_bioassay.py using NormApprox, do:

>>> N = pymc.NormApprox(gelman_bioassay)
>>> N.fit()

The approximate joint posterior mean and covariance of the variables are available via the attributes mu and C:

>>> N.mu[N.alpha]
array([ 0.84658923])
>>> N.mu[N.alpha, N.beta]
array([ 0.84658923,  7.74884998])
>>> N.C[N.alpha]
matrix([[ 1.03854093]])
>>> N.C[N.alpha, N.beta]
matrix([[  1.03854093,   3.54601911],
        [  3.54601911,  23.74406919]])

As with MAP, the variables have been set to their maximum a posteriori values (which are also in the mu attribute) and the AIC and BIC of the model are available.

In addition, it’s now possible to generate samples from the posterior as with MCMC:

>>> N.sample(100)
>>> N.trace('alpha')[::10]
array([-0.85001278,  1.58982854,  1.0388088 ,  0.07626688,  1.15359581,
       -0.25211939,  1.39264616,  0.22551586,  2.69729987,  1.21722872])
>>> N.trace('beta')[::10]
array([  2.50203663,  14.73815047,  11.32166303,   0.43115426,
        10.1182532 ,   7.4063525 ,  11.58584317,   8.99331152,
        11.04720439,   9.5084239 ])

Any of the database backends can be used (chapter Saving and managing sampling results).

In addition to the methods and attributes of MAP, NormApprox provides the following methods:

sample(iter):
Samples from the approximate posterior distribution are drawn and stored.
isample(iter):
An ‘interactive’ version of sample(): sampling can be paused, returning control to the user.
draw:
Sets all variables to random values drawn from the approximate posterior.

It provides the following additional attributes:

mu:
A special dictionary-like object that can be keyed with multiple variables. N.mu[p1, p2, p3] would return the approximate posterior mean values of stochastic variables p1, p2 and p3, raveled and concatenated to form a vector.
C:
Another special dictionary-like object. N.C[p1, p2, p3] would return the approximate posterior covariance matrix of stochastic variables p1, p2 and p3. As with mu, these variables’ values are raveled and concatenated before their covariance matrix is constructed.

5.5. Markov chain Monte Carlo: the MCMC class

The MCMC class implements PyMC’s core business: producing ‘traces’ for a model’s variables which, with careful thinning, can be considered independent joint samples from the posterior. See Tutorial for an example of basic usage.

MCMC‘s primary job is to create and coordinate a collection of ‘step methods’, each of which is responsible for updating one or more variables. The available step methods are described below. Instructions on how to create your own step method are available in Extending PyMC.

MCMC provides the following useful methods:

sample(iter, burn, thin, tune_interval, tune_throughout, save_interval, ...):
Runs the MCMC algorithm and produces the traces. The iter argument controls the total number of MCMC iterations. No tallying will be done during the first burn iterations; these samples will be forgotten. After this burn-in period, tallying will be done each thin iterations. Tuning will be done each tune_interval iterations. If tune_throughout=False, no more tuning will be done after the burnin period. The model state will be saved every save_interval iterations, if given.
isample(iter, burn, thin, tune_interval, tune_throughout, save_interval, ...):
An interactive version of sample. The sampling loop may be paused at any time, returning control to the user.
use_step_method(method, *args, **kwargs):
Creates an instance of step method class method to handle some stochastic variables. The extra arguments are passed to the init method of method. Assigning a step method to a variable manually will prevent the MCMC instance from automatically assigning one. However, you may handle a variable with multiple step methods.
goodness():
Calculates goodness-of-fit (GOF) statistics according to [Brooks2000].
save_state():
Saves the current state of the sampler, including all stochastics, to the database. This allows the sampler to be reconstituted at a later time to resume sampling. This is not supported yet for the sqlite backend.
restore_state():
Restores the sampler to the state stored in the database.
stats():
Generate summary statistics for all nodes in the model.
remember(trace_index):
Set all variables’ values from frame trace_index in the database.

MCMC samplers’ step methods can be accessed via the step_method_dict attribute. M.step_method_dict[x] returns a list of the step methods M will use to handle the stochastic variable x.

After sampling, the information tallied by M can be queried via M.db.trace_names. In addition to the values of variables, tuning information for adaptive step methods is generally tallied. These ‘traces’ can be plotted to verify that tuning has in fact terminated.

You can produce ‘traces’ for arbitrary functions with zero arguments as well. If you issue the command M._funs_to_tally['trace_name'] = f before sampling begins, then each time the model variables’ values are tallied, f will be called with no arguments, and the return value will be tallied. After sampling ends you can retrieve the trace as M.trace[’trace_name’].

5.6. The Sampler class

MCMC is a subclass of a more general class called Sampler. Samplers fit models with Monte Carlo fitting methods, which characterize the posterior distribution by approximate samples from it. They are initialized as follows: Sampler(input=None, db='ram', name='Sampler', reinit_model=True, calc_deviance=False, verbose=0). The input argument is a module, list, tuple, dictionary, set, or object that contains all elements of the model, the db argument indicates which database backend should be used to store the samples (see chapter Saving and managing sampling results), reinit_model is a boolean flag that indicates whether the model should be re-initialised before running, and calc_deviance is a boolean flag indicating whether deviance should be calculated for the model at each iteration. Samplers have the following important methods:

sample(iter, length, verbose, ...):
Samples from the joint distribution. The iter argument controls how many times the sampling loop will be run, and the length argument controls the initial size of the database that will be used to store the samples.
isample(iter, length, verbose, ...):
The same as sample, but the sampling is done interactively: you can pause sampling at any point and be returned to the Python prompt to inspect progress and adjust fitting parameters. While sampling is paused, the following methods are useful:
icontinue():
Continue interactive sampling.
halt():
Truncate the database and clean up.
tally():
Write all variables’ current values to the database. The actual write operation depends on the specified database backend.
save_state():
Saves the current state of the sampler, including all stochastics, to the database. This allows the sampler to be reconstituted at a later time to resume sampling. This is not supported yet for the sqlite backend.
restore_state():
Restores the sampler to the state stored in the database.
stats():
Generate summary statistics for all nodes in the model.
remember(trace_index):
Set all variables’ values from frame trace_index in the database. Note that the trace_index is different from the current iteration, since not all samples are necessarily saved due to burning and thinning.

In addition, the sampler attribute deviance is a deterministic variable valued as the model’s deviance at its current state.

5.7. Step methods

Step method objects handle individual stochastic variables, or sometimes groups of them. They are responsible for making the variables they handle take single MCMC steps conditional on the rest of the model. Each subclass of StepMethod implements a method called step(), which is called by MCMC. Step methods with adaptive tuning parameters can optionally implement a method called tune(), which causes them to assess performance (based on the acceptance rates of proposed values for the variable) so far and adjust.

The major subclasses of StepMethod are Metropolis, AdaptiveMetropolis and Slicer. PyMC provides several flavors of the basic Metropolis steps. There are Gibbs sampling (Gibbs) steps, but they are not ready for use as of the current release, but since it is feasible to write Gibbs step methods for particular applications, the Gibbs base class will be documented here.

5.7.1. Metropolis step methods

Metropolis and subclasses implement Metropolis-Hastings steps. To tell an MCMC object \(M\) to handle a variable x with a Metropolis step method, you might do the following:

M.use_step_method(pymc.Metropolis, x, proposal_sd=1., proposal_distribution='Normal')

Metropolis itself handles float-valued variables, and subclasses DiscreteMetropolis and BinaryMetropolis handle integer- and boolean-valued variables, respectively. Subclasses of Metropolis must implement the following methods:

propose():
Sets the values of the variables handled by the Metropolis step method to proposed values.
reject():
If the Metropolis-Hastings acceptance test fails, this method is called to reset the values of the variables to their values before propose() was called.

Note that there is no accept() method; if a proposal is accepted, the variables’ values are simply left alone. Subclasses that use proposal distributions other than symmetric random-walk may specify the ‘Hastings factor’ by changing the hastings_factor method. See Extending PyMC for an example.

Metropolis__init__ method takes the following arguments:

stochastic:
The variable to handle.
proposal_sd:
A float or array of floats. This sets the proposal standard deviation if the proposal distribution is normal.
scale:

A float, defaulting to 1. If the scale argument is provided but not proposal_sd, proposal_sd is computed as follows:

if all(self.stochastic.value != 0.):
    self.proposal_sd = ones(shape(self.stochastic.value)) * \
                        abs(self.stochastic.value) * scale
else:
    self.proposal_sd = ones(shape(self.stochastic.value)) * scale
proposal_distribution:
A string indicating which distribution should be used for proposals. Current options are 'Normal' and 'Prior'. If proposal_distribution=None, the proposal distribution is chosen automatically. It is set to 'Prior' if the variable has no children and has a random method, and to 'Normal' otherwise.
verbose:
An integer. By convention 0 indicates no output, 1 shows a progress bar only, 2 provides basic feedback about the current MCMC run, while 3 and 4 provide low and high debugging verbosity, respectively.

Alhough the proposal_sd attribute is fixed at creation, Metropolis step methods adjust their initial proposal standard deviations using an attribute called adaptive_scale_factor. When tune() is called, the acceptance ratio of the step method is examined, and this scale factor is updated accordingly. If the proposal distribution is normal, proposals will have standard deviation self.proposal_sd * self.adaptive_scale_factor.

By default, tuning will continue throughout the sampling loop, even after the burnin period is over. This can be changed via the tune_throughout argument to MCMC.sample. If an adaptive step method’s tally flag is set (the default for Metropolis), a trace of its tuning parameters will be kept. If you allow tuning to continue throughout the sampling loop, it is important to verify that the ‘Diminishing Tuning’ condition of [Roberts2007] is satisfied: the amount of tuning should decrease to zero, or tuning should become very infrequent.

If a Metropolis step method handles an array-valued variable, it proposes all elements independently but simultaneously. That is, it decides whether to accept or reject all elements together but it does not attempt to take the posterior correlation between elements into account. The AdaptiveMetropolis class (see below), on the other hand, does make correlated proposals.

5.7.2. The AdaptiveMetropolis class

The AdaptativeMetropolis (AM) step method works like a regular Metropolis step method, with the exception that its variables are block-updated using a multivariate jump distribution whose covariance is tuned during sampling. Although the chain is non-Markovian, it has correct ergodic properties (see [Haario2001]).

To tell an MCMC object \(M\) to handle variables x, y and \(z\) with an AdaptiveMetropolis instance, you might do the following:

M.use_step_method(pymc.AdaptiveMetropolis, [x,y,z], \
                   scales={'x':1, 'y':2, 'z':.5}, delay=10000)

AdaptativeMetropolis‘s init method takes the following arguments:

stochastics:
The stochastic variables to handle. These will be updated jointly.
cov (optional):
An initial covariance matrix. Defaults to the identity matrix, adjusted according to the scales argument.
delay (optional):
The number of iterations to delay before computing the empirical covariance matrix.
scales (optional):
The initial covariance matrix will be diagonal, and its diagonal elements will be set to scales times the stochastics’ values, squared.
interval (optional):
The number of iterations between updates of the covariance matrix. Defaults to 1000.
greedy (optional):
If True, only accepted jumps will be counted toward the delay before the covariance is first computed. Defaults to True.
verbose:
An integer from 0 to 4 controlling the verbosity of the step method’s printed output.
shrink_if_necessary (optional):
Whether the proposal covariance should be shrunk if the acceptance rate becomes extremely small.

In this algorithm, jumps are proposed from a multivariate normal distribution with covariance matrix \(\Sigma\). The algorithm first iterates until delay samples have been drawn (if greedy is true, until delay jumps have been accepted). At this point, \(\Sigma\) is given the value of the empirical covariance of the trace so far and sampling resumes. The covariance is then updated each interval iterations throughout the entire sampling run [1]. It is this constant adaptation of the proposal distribution that makes the chain non-Markovian.

5.7.3. The DiscreteMetropolis class

This class is just like Metropolis, but specialized to handle Stochastic instances with dtype int. The jump proposal distribution can either be 'Normal', 'Prior' or 'Poisson' (the default). In the normal case, the proposed value is drawn from a normal distribution centered at the current value and then rounded to the nearest integer.

5.7.4. The BinaryMetropolis class

This class is specialized to handle Stochastic instances with dtype bool.

For array-valued variables, BinaryMetropolis can be set to propose from the prior by passing in dist="Prior". Otherwise, the argument p_jump of the init method specifies how probable a change is. Like Metropolis‘ attribute proposal_sd, p_jump is tuned throughout the sampling loop via adaptive_scale_factor.

For scalar-valued variables, BinaryMetropolis behaves like a Gibbs sampler, since this requires no additional expense. The p_jump and adaptive_scale_factor parameters are not used in this case.

5.7.5. The Slicer class

The Slicer class implements Slice sampling ([Neal2003]). To tell an MCMC object \(M\) to handle a variable x with a Slicer step method, you might do the following:

M.use_step_method(pymc.Slicer, x, w=10, m=10000, doubling=True)

Slicer‘s init method takes the following arguments:

stochastics:
The stochastic variables to handle. These will be updated jointly.
w (optional):
The initial width of the horizontal slice (Defaults to 1). This will be updated via either stepping-out or doubling procedures.
m (optional):
The multiplier defining the maximum slice size as \(mw\) (Defaults to 1000).
tune (optional):
A flag indicating whether to tune the initial slice width (Defaults to True).
doubling (optional):
A flag for using doubling procedure instead of stepping out (Defaults to False)
tally (optional):
Flag for recording values for trace (Defaults to True).
verbose:
An integer from -1 to 4 controlling the verbosity of the step method’s printed output (Defaults to -1).

The *slice sampler* generates posterior samples by alternately drawing “slices” from the vertical (y) and horizontal (x) planes of a distribution. It first samples from the conditional distribution for y given some current value of x, which is uniform over the \((0, f (x))\). Conditional on this value for y, it then samples x, which is uniform on \(S = {x : y < f (x)}\); that is the “slice” defined by the y value. Hence, this algorithm automatically adapts its to the local characteristics of the posterior.

The steps required to perform a single iteration of the slice sampler to update the current value of \(x_i\) is as follows:

  1. Sample y uniformly on \((0,f(x_i))\).
  2. Use this value y to define a horizontal slice \(S = \{x : y < f (x)\}\).
  3. Establish an interval, \(I = (x_{a}, x_{b})\), around \(x_i\) that contains most of the slice.
  4. Sample \(x_{i+1}\) from the region of the slice overlaping I.

Hence, slice sampling employs an auxilliary variable (y) that is not retained at the end of the iteration. Note that in practice one may operate on the log scale such that \(g(x) = \log(f (x))\) to avoid floating-point underflow. In this case, the auxiliary variable becomes \(z = log(y) = g(x_i) − e\), where \(e \sim \text{Exp}(1)\), resulting in the slice \(S = \{x : z < g(x)\}\).

There are many ways of establishing and sampling from the interval I, with the only restriction being that the resulting Markov chain leaves \(f(x)\) invariant. The objective is to include as much of the slice as possible, so that the potential step size can be large, but not (much) larger than the slice, so that the sampling of invalid points is minimized. Ideally, we would like it to be the slice itself, but it may not always be feasible to determine (and certainly not automatically).

One method for determining a sampling interval for \(x_{i+1}\) involves specifying an initial “guess” at the slice width w, and iteratively moving the endpoints out (growing the interval) until either (1) the interval reaches a maximum pre-specified width or (2) y is less than the \(f(x)\) evaluated both at the left and the right interval endpoints. This is the stepping out method. The efficiency of stepping out depends largely on the ability to pick a reasonable interval w from which to sample. Otherwise, the doubling procedure may be preferable, as it can be expanded faster. It simply doubles the size of the interval until both endpoints are outside the slice.

5.8. Gibbs step methods

Gibbs step methods handle conjugate submodels. These models usually have two components: the `parent’ and the `children’. For example, a gamma-distributed variable serving as the precision of several normally-distributed variables is a conjugate submodel; the gamma variable is the parent and the normal variables are the children.

This section describes PyMC’s current scheme for Gibbs step methods, several of which are in a semi-working state in the sandbox. It is meant to be as generic as possible to minimize code duplication, but it is admittedly complicated. Feel free to subclass StepMethod directly when writing Gibbs step methods if you prefer.

Gibbs step methods that subclass PyMC’s Gibbs should define the following class attributes:

child_class:
The class of the children in the submodels the step method can handle.
parent_class:
The class of the parent.
parent_label:
The label the children would apply to the parent in a conjugate submodel. In the gamma-normal example, this would be tau.
linear_OK:
A flag indicating whether the children can use linear combinations involving the parent as their actual parent without destroying the conjugacy.

A subclass of Gibbs that defines these attributes only needs to implement a propose() method, which will be called by Gibbs.step(). The resulting step method will be able to handle both conjugate and ‘non-conjugate’ cases. The conjugate case corresponds to an actual conjugate submodel. In the nonconjugate case all the children are of the required class, but the parent is not. In this case the parent’s value is proposed from the likelihood and accepted based on its prior. The acceptance rate in the nonconjugate case will be less than one.

The inherited class method Gibbs.competence will determine the new step method’s ability to handle a variable x by checking whether:

  • all x‘s children are of class child_class, and either apply parent_label to x directly or (if linear_OK=True) to a LinearCombination object (Building models), one of whose parents contains x.
  • x is of class parent_class

If both conditions are met, pymc.conjugate_Gibbs_competence will be returned. If only the first is met, pymc.nonconjugate_Gibbs_competence will be returned.

5.8.1. Granularity of step methods: one-at-a-time vs. block updating

There is currently no way for a stochastic variable to compute individual terms of its log-probability; it is computed all together. This means that updating the elements of a array-valued variable individually would be inefficient, so all existing step methods update array-valued variables together, in a block update.

To update an array-valued variable’s elements individually, simply break it up into an array of scalar-valued variables. Instead of this:

A = pymc.Normal('A', value=zeros(100), mu=0., tau=1.)

do this:

A = [pymc.Normal('A_%i'%i, value=0., mu=0., tau=1.) for i in range(100)]

An individual step method will be assigned to each element of A in the latter case, and the elements will be updated individually. Note that A can be broken up into larger blocks if desired.

5.8.2. Automatic assignment of step methods

Every step method subclass (including user-defined ones) that does not require any __init__ arguments other than the stochastic variable to be handled adds itself to a list called StepMethodRegistry in the PyMC namespace. If a stochastic variable in an MCMC object has not been explicitly assigned a step method, each class in StepMethodRegistry is allowed to examine the variable.

To do so, each step method implements a class method called competence(stochastic), whose only argument is a single stochastic variable. These methods return values from 0 to 3; 0 meaning the step method cannot safely handle the variable and 3 meaning it will most likely perform well for variables like this. The MCMC object assigns the step method that returns the highest competence value to each of its stochastic variables.

Footnotes

[1]The covariance is estimated recursively from the previous value and the last interval samples, instead of computing it each time from the entire trace.