The variational inference (VI) API is focused on approximating posterior distributions for Bayesian models. Common use cases to which this module can be applied include:
eval)Sounds good, doesn't it?
The module provides an interface to a variety of inference methods, so you are free to choose what is most appropriate for the problem.
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%matplotlib inline
import matplotlib.pyplot as plt
import pymc3 as pm
import theano
import numpy as np
np.random.seed(42)
pm.set_tt_rng(42)
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w = pm.floatX([.2, .8])
mu = pm.floatX([-.3, .5])
sd = pm.floatX([.1, .1])
with pm.Model() as model:
x = pm.NormalMixture('x', w=w, mu=mu, sd=sd, dtype=theano.config.floatX)
x2 = x ** 2
sin_x = pm.math.sin(x)
We can't compute analytical expectations for this model. However, we can obtain an approximation using Markov chain Monte Carlo methods; let's use NUTS first.
To allow samples of the expressions to be saved, we need to wrap them in Deterministic objects:
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with model:
pm.Deterministic('x2', x2)
pm.Deterministic('sin_x', sin_x)
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with model:
trace = pm.sample(50000)
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pm.traceplot(trace);
Above are traces for $x^2$ and $sin(x)$. We can see there is clear multi-modality in this model. One drawback, is that you need to know in advance what exactly you want to see in trace and wrap it with Deterministic.
The VI API takes an alternate approach: You obtain inference from model, then calculate expressions based on this model afterwards.
Let's use the same model:
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with pm.Model() as model:
x = pm.NormalMixture('x', w=w, mu=mu, sd=sd, dtype=theano.config.floatX)
x2 = x ** 2
sin_x = pm.math.sin(x)
Here we will use automatic differentiation variational inference (ADVI).
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with model:
mean_field = pm.fit(method='advi')
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pm.plot_posterior(mean_field.sample(1000), color='LightSeaGreen');
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help(pm.callbacks.CheckParametersConvergence)
Let's use the default arguments for CheckParametersConvergence as they seem to be reasonable.
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from pymc3.variational.callbacks import CheckParametersConvergence
with model:
mean_field = pm.fit(method='advi', callbacks=[CheckParametersConvergence()])
We can access inference history via .hist attribute.
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plt.plot(mean_field.hist);
This is not a good convergence plot, despite the fact that we ran many iterations. The reason is that the mean of the ADVI approximation is close to zero, and therefore taking the relative difference (the default method) is unstable for checking convergence.
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with model:
mean_field = pm.fit(method='advi', callbacks=[pm.callbacks.CheckParametersConvergence(diff='absolute')])
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plt.plot(mean_field.hist);
That's much better! We've reached convergence after less than 5000 iterations.
Another usefull callback allows users to track parameters. It allows for the tracking of arbitrary statistics during inference, though it can be memory-hungry. Using the fit function, we do not have direct access to the approximation before inference. However, tracking parameters requires access to the approximation. We can get around this constraint by using the object-oriented (OO) API for inference.
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with model:
advi = pm.ADVI()
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advi.approx
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Different approximations have different hyperparameters. In mean-field ADVI, we have $\rho$ and $\mu$ (inspired by Bayes by BackProp).
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advi.approx.shared_params
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There are convenient shortcuts to relevant statistics associated with the approximation. This can be useful, for example, when specifying a mass matrix for NUTS sampling:
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advi.approx.mean.eval(), advi.approx.std.eval()
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We can roll these statistics into the Tracker callback.
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tracker = pm.callbacks.Tracker(
mean=advi.approx.mean.eval, # callable that returns mean
std=advi.approx.std.eval # callable that returns std
)
Now, calling advi.fit will record the mean and standard deviation of the approximation as it runs.
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approx = advi.fit(20000, callbacks=[tracker])
We can now plot both the evidence lower bound and parameter traces:
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fig = plt.figure(figsize=(16, 9))
mu_ax = fig.add_subplot(221)
std_ax = fig.add_subplot(222)
hist_ax = fig.add_subplot(212)
mu_ax.plot(tracker['mean'])
mu_ax.set_title('Mean track')
std_ax.plot(tracker['std'])
std_ax.set_title('Std track')
hist_ax.plot(advi.hist)
hist_ax.set_title('Negative ELBO track');
Notice that there are convergence issues with the mean, and that lack of convergence does not seem to change the ELBO trajectory significantly. As we are using the OO API, we can run the approximation longer until convergence is achieved.
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advi.refine(100000)
Let's take a look:
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fig = plt.figure(figsize=(16, 9))
mu_ax = fig.add_subplot(221)
std_ax = fig.add_subplot(222)
hist_ax = fig.add_subplot(212)
mu_ax.plot(tracker['mean'])
mu_ax.set_title('Mean track')
std_ax.plot(tracker['std'])
std_ax.set_title('Std track')
hist_ax.plot(advi.hist)
hist_ax.set_title('Negative ELBO track');
We still see evidence for lack of convergence, as the mean has devolved into a random walk. This could be the result of choosing a poor algorithm for inference. At any rate, it is unstable and can produce very different results even using different random seeds.
Let's compare results with the NUTS output:
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import seaborn as sns
ax = sns.kdeplot(trace['x'], label='NUTS');
sns.kdeplot(approx.sample(10000)['x'], label='ADVI');
Again, we see that ADVI is not able to cope with multimodality; we can instead use SVGD, which generates an approximation based on a large number of particles.
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with model:
svgd_approx = pm.fit(300, method='svgd', inf_kwargs=dict(n_particles=1000),
obj_optimizer=pm.sgd(learning_rate=0.01))
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ax = sns.kdeplot(trace['x'], label='NUTS');
sns.kdeplot(approx.sample(10000)['x'], label='ADVI');
sns.kdeplot(svgd_approx.sample(2000)['x'], label='SVGD');
That did the trick, as we now have a multimodal approximation using SVGD.
With this, it is possible to calculate arbitrary functions of the parameters with this variational approximation. For example we can calculate $x^2$ and $sin(x)$, as with the NUTS model.
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# recall x ~ NormalMixture
a = x**2
b = pm.math.sin(x)
To evaluate these expressions with the approximation, we need approx.sample_node.
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help(svgd_approx.sample_node)
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a_sample = svgd_approx.sample_node(a)
a_sample.eval()
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a_sample.eval()
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a_sample.eval()
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Every call yields a different value from the same theano node. This is because it is stochastic.
By applying replacements, we are now free of the dependence on the PyMC3 model; instead, we now depend on the approximation. Changing it will change the distribution for stochastic nodes:
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sns.kdeplot(np.array([a_sample.eval() for _ in range(2000)]));
plt.title('$x^2$ distribution');
There is a more convinient way to get lots of samples at once: sample_node
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a_samples = svgd_approx.sample_node(a, size=1000)
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sns.kdeplot(a_samples.eval());
plt.title('$x^2$ distribution');
The sample_node function includes an additional dimension, so taking expectations or calculating variance is specified by axis=0.
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a_samples.var(0).eval() # variance
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a_samples.mean(0).eval() # mean
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A symbolic sample size can also be specified:
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i = theano.tensor.iscalar('i')
i.tag.test_value = 1
a_samples_i = svgd_approx.sample_node(a, size=i)
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a_samples_i.eval({i: 100}).shape
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a_samples_i.eval({i: 10000}).shape
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Unfortunately the size must be a scalar value.
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trace_approx = pm.Empirical(trace, model=model)
trace_approx
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We can then draw samples from the Emipirical object:
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pm.plot_posterior(trace_approx.sample(10000));
In [41]:
from sklearn.datasets import load_iris
from sklearn.model_selection import train_test_split
import theano.tensor as tt
import pandas as pd
X, y = load_iris(True)
X_train, X_test, y_train, y_test = train_test_split(X, y)
A relatively simple model will be sufficient here because the classes are roughly linearly separable; we are going to fit multinomial logistic regression.
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Xt = theano.shared(X_train)
yt = theano.shared(y_train)
with pm.Model() as iris_model:
# Coefficients for features
β = pm.Normal('β', 0, sd=1e2, shape=(4, 3))
# Transoform to unit interval
a = pm.Flat('a', shape=(3,))
p = tt.nnet.softmax(Xt.dot(β) + a)
observed = pm.Categorical('obs', p=p, observed=yt)
PyMC3 models have symbolic inputs for latent variables. To evaluate an espression that requires knowledge of latent variables, one needs to provide fixed values. We can use values approximated by VI for this purpose. The function sample_node removes the symbolic dependenices.
sample_node will use the whole distribution at each step, so we will use it here. We can apply more replacements in single function call using the more_replacements keyword argument in both replacement functions.
HINT: You can use
more_replacementsargument when callingfittoo:
pm.fit(more_replacements={full_data: minibatch_data})inference.fit(more_replacements={full_data: minibatch_data})
In [43]:
with iris_model:
# We'll use SVGD
inference = pm.SVGD(n_particles=500, jitter=1)
# Local reference to approximation
approx = inference.approx
# Here we need `more_replacements` to change train_set to test_set
test_probs = approx.sample_node(p, more_replacements={Xt: X_test}, size=100)
# For train set no more replacements needed
train_probs = approx.sample_node(p)
By applying the code above, we now have 100 sampled probabilities (default number for sample_node is None) for each observation.
Next we create symbolic expressions for sampled accuracy scores:
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test_ok = tt.eq(test_probs.argmax(-1), y_test)
train_ok = tt.eq(train_probs.argmax(-1), y_train)
test_accuracy = test_ok.mean(-1)
train_accuracy = train_ok.mean(-1)
Tracker expects callables so we can pass .eval method of theano node that is function itself.
Calls to this function are cached so they can be reused.
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eval_tracker = pm.callbacks.Tracker(
test_accuracy=test_accuracy.eval,
train_accuracy=train_accuracy.eval
)
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inference.fit(100, callbacks=[eval_tracker]);
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import seaborn as sns
sns.tsplot(np.asarray(eval_tracker['test_accuracy']).T, color='red')
sns.tsplot(np.asarray(eval_tracker['train_accuracy']).T, color='blue')
plt.legend(['test_accuracy', 'train_accuracy'])
plt.title('Training Progress')
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Training does not seem to be working here. Let's use a different optimizer and boost the learning rate.
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inference.fit(400, obj_optimizer=pm.adamax(learning_rate=0.1), callbacks=[eval_tracker]);
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sns.tsplot(np.asarray(eval_tracker['test_accuracy']).T, color='red', alpha=.5)
sns.tsplot(np.asarray(eval_tracker['train_accuracy']).T, color='blue', alpha=.5)
plt.legend(['test_accuracy', 'train_accuracy'])
plt.title('Training Progress');
This is much better!
So, Tracker allows us to monitor our approximation and choose good training schedule.
When dealing with large datasets, using minibatch training can drastically speed up and improve approximation performance. Large datasets impose a hefty cost on the computation of gradients.
There is a nice API in pymc3 to handle these cases, which is avaliable through the pm.Minibatch class. The minibatch is just a highly specialized Theano tensor:
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issubclass(pm.Minibatch, theano.tensor.TensorVariable)
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To demonstrate, let's simulate a large quantity of data:
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# Raw values
data = np.random.rand(40000, 100)
# Scaled values
data *= np.random.randint(1, 10, size=(100,))
# Shifted values
data += np.random.rand(100) * 10
For comparison, let's fit a model without minibatch processing:
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with pm.Model() as model:
mu = pm.Flat('mu', shape=(100,))
sd = pm.HalfNormal('sd', shape=(100,))
lik = pm.Normal('lik', mu, sd, observed=data)
Just for fun, let's create a custom special purpose callback to halt slow optimization. Here we define a callback that causes a hard stop when approximation runs too slowly:
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def stop_after_10(approx, loss_history, i):
if (i > 0) and (i % 10) == 0:
raise StopIteration('I was slow, sorry')
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with model:
advifit = pm.fit(callbacks=[stop_after_10])
Inference is too slow, taking several seconds per iteration; fitting the approximation would have taken hours!
Now let's use minibatches. At every iteration, we will draw 500 random values:
Remember to set
total_sizein observed
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X = pm.Minibatch(data, batch_size=500)
with pm.Model() as model:
mu = pm.Flat('mu', shape=(100,))
sd = pm.HalfNormal('sd', shape=(100,))
likelihood = pm.Normal('likelihood', mu, sd, observed=X, total_size=data.shape)
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with model:
advifit = pm.fit()
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plt.plot(advifit.hist);
Minibatch inference is dramatically faster. Multidimensional minibatches may be needed for some corner cases where you do matrix factorization or model is very wide.
Here is the docstring for Minibatch to illustrate how it can be customized.
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print(pm.Minibatch.__doc__)