# Empirical Approximation overview¶

For most models we use sampling MCMC algorithms like Metropolis or NUTS. In pymc3 we got used to store traces of MC samples and then do analysis using them. As new VI interface was implememted it needed a lot of approximation types.

One of them was so-called Empirical. This type of approximation stores particles for SVGD sampler. But there is no difference between independent SVGD particles and MCMC trace. So the idea was pretty simple to understand and realize: make Empirical be a bridge between MCMC sampling output and full-fledged VI utils like apply_replacements or sample_node. For the interface description, see variational_api_quickstart. Here I will just focus on Emprical and give an overview of specific things for Empirical approximation

In [1]:
%matplotlib inline
import matplotlib.pyplot as plt
import theano
import numpy as np
import pymc3 as pm
np.random.seed(42)
pm.set_tt_rng(42)

## Multimodal density¶

Let’s recall the problem from variational_api_quickstart where we first got a NUTS trace

In [2]:
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)
trace = pm.sample(50000)
WARNING (theano.gof.compilelock): Overriding existing lock by dead process '26463' (I am process '27017')
Auto-assigning NUTS sampler...
100%|██████████| 50500/50500 [00:25<00:00, 2015.98it/s]
In [3]:
pm.traceplot(trace);

Great. First having a trace we can create Empirical approx

In [4]:
print(pm.Empirical.__doc__)
Single Group Full Rank Approximation
In [5]:
with model:
approx = pm.Empirical(trace)
In [6]:
approx
Out[6]:
<pymc3.variational.approximations.Empirical at 0x1193a41d0>

This type of approximation has it’s own underlying storage for samples that is theano.shared itself

In [7]:
approx.histogram
Out[7]:
histogram
In [8]:
approx.histogram.get_value()[:10]
Out[8]:
array([[-0.28495539],
[-0.27002112],
[-0.27578667],
[-0.36169251],
[-0.43738686],
[-0.479884  ],
[-0.3633637 ],
[-0.31469654],
[-0.36427262],
[-0.36479427]])
In [9]:
approx.histogram.get_value().shape
Out[9]:
(50000, 1)

It has exactly the same number of samples that you had in trace before. In our particular case it is 50k. Another thing to notice is thet if you have multitrace with more than one chain you’ll get much more samples stored at once. We flatten all the trace for creating Empirical.

This histogram is about how we store samples. The structure is pretty simple: (n_samples, n_dim) The order of these variables is stored internally in the class and in most cases will not be needed for end user

In [10]:
approx.ordering
Out[10]:
<pymc3.blocking.ArrayOrdering at 0x1190be390>

Sampling from posterior is done uniformly with replacements. Call approx.sample(1000) and you’ll get again the trace but the order is not determined. There is no way now to reconstruct the underlying trace again with approx.sample.

In [11]:
new_trace = approx.sample(50000)
In [12]:
%timeit new_trace = approx.sample(50000)
1.54 s ± 24.3 ms per loop (mean ± std. dev. of 7 runs, 1 loop each)

After sampling function is compiled sampling bacomes really fast

In [13]:
pm.traceplot(new_trace);

You see there is no order any more but reconstructed density is the same.

## 2d density¶

In [14]:
mu = pm.floatX([0., 0.])
cov = pm.floatX([[1, .5], [.5, 1.]])
with pm.Model() as model:
pm.MvNormal('x', mu=mu, cov=cov, shape=2)
trace = pm.sample(1000)
Auto-assigning NUTS sampler...
100%|██████████| 1500/1500 [00:02<00:00, 677.32it/s]
In [15]:
with model:
approx = pm.Empirical(trace)
In [16]:
pm.traceplot(approx.sample(10000))
Out[16]:
array([[<matplotlib.axes._subplots.AxesSubplot object at 0x11839eac8>,
<matplotlib.axes._subplots.AxesSubplot object at 0x118631160>]], dtype=object)
In [17]:
import seaborn as sns
In [18]:
sns.kdeplot(approx.sample(1000)['x'])
Out[18]:
<matplotlib.axes._subplots.AxesSubplot at 0x118a29240>

Previously we had a trace_cov function

In [19]:
with model:
print(pm.trace_cov(trace))
[[ 1.03635135  0.56102585]
[ 0.56102585  1.03247922]]

Now we can estimate the same covariance using Empirical

In [20]:
print(approx.cov)
Elemwise{true_div,no_inplace}.0

That’s a tensor itself

In [21]:
print(approx.cov.eval())
[[ 1.035315    0.56046482]
[ 0.56046482  1.03144674]]

Estimations are very close and differ due to precision error. We can get the mean in the same way

In [22]:
print(approx.mean.eval())
[ 0.03685694 -0.01467962]