使用变分推断拟合广义线性混合效应模型

%matplotlib inline
%config InlineBackend.figure_format = 'retina'

import os
from six.moves import urllib

import matplotlib.pyplot as plt; plt.style.use('ggplot')
import numpy as np
import pandas as pd
import seaborn as sns; sns.set_context('notebook')
import tensorflow_datasets as tfds

import tensorflow.compat.v2 as tf
tf.enable_v2_behavior()

import tensorflow_probability as tfp
tfd = tfp.distributions
tfb = tfp.bijectors

if tf.test.gpu_device_name() != '/device:GPU:0':
  print("We'll just use the CPU for this run.")
else:
  print('Huzzah! Found GPU: {}'.format(tf.test.gpu_device_name()))

We'll just use the CPU for this run.

def load_and_preprocess_radon_dataset(state='MN'):
  """Load the Radon dataset from TensorFlow Datasets and preprocess it.

  Following the examples in "Bayesian Data Analysis" (Gelman, 2007), we filter
  to Minnesota data and preprocess to obtain the following features:
  - `county`: Name of county in which the measurement was taken.
  - `floor`: Floor of house (0 for basement, 1 for first floor) on which the
    measurement was taken.

  The target variable is `log_radon`, the log of the Radon measurement in the
  house.
  """
  ds = tfds.load('radon', split='train')
  radon_data = tfds.as_dataframe(ds)
  radon_data.rename(lambda s: s[9:] if s.startswith('feat') else s, axis=1, inplace=True)
  df = radon_data[radon_data.state==state.encode()].copy()

  df['radon'] = df.activity.apply(lambda x: x if x > 0. else 0.1)
  # Make county names look nice. 
  df['county'] = df.county.apply(lambda s: s.decode()).str.strip().str.title()
  # Remap categories to start from 0 and end at max(category).
  df['county'] = df.county.astype(pd.api.types.CategoricalDtype())
  df['county_code'] = df.county.cat.codes
  # Radon levels are all positive, but log levels are unconstrained
  df['log_radon'] = df['radon'].apply(np.log)

  # Drop columns we won't use and tidy the index 
  columns_to_keep = ['log_radon', 'floor', 'county', 'county_code']
  df = df[columns_to_keep].reset_index(drop=True)

  return df

df = load_and_preprocess_radon_dataset()
df.head()

def estimate_log_radon(floor):
    return intercept + floor_effect[floor]

fig, (ax1, ax2) = plt.subplots(ncols=2, figsize=(12, 4))
df.groupby('floor')['log_radon'].plot(kind='density', ax=ax1);
ax1.set_xlabel('Measured log(radon)')
ax1.legend(title='Floor')

df['floor'].value_counts().plot(kind='bar', ax=ax2)
ax2.set_xlabel('Floor where radon was measured')
ax2.set_ylabel('Count')
fig.suptitle("Distribution of log radon and floors in the dataset");

fig, ax = plt.subplots(figsize=(22, 5));
county_freq = df['county'].value_counts()
county_freq.plot(kind='bar', ax=ax)
ax.set_xlabel('County')
ax.set_ylabel('Number of readings');

features = df[['county_code', 'floor']].astype(int)
labels = df[['log_radon']].astype(np.float32).values.flatten()

def make_joint_distribution_coroutine(floor, county, n_counties, n_floors):

  def model():
    county_scale = yield tfd.HalfNormal(scale=1., name='scale_prior')
    intercept = yield tfd.Normal(loc=0., scale=1., name='intercept')
    floor_weight = yield tfd.Normal(loc=0., scale=1., name='floor_weight')
    county_prior = yield tfd.Normal(loc=tf.zeros(n_counties),
                                    scale=county_scale,
                                    name='county_prior')
    random_effect = tf.gather(county_prior, county, axis=-1)

    fixed_effect = intercept + floor_weight * floor
    linear_response = fixed_effect + random_effect
    yield tfd.Normal(loc=linear_response, scale=1., name='likelihood')
  return tfd.JointDistributionCoroutineAutoBatched(model)

joint = make_joint_distribution_coroutine(
    features.floor.values, features.county_code.values, df.county.nunique(),
    df.floor.nunique())

# Define a closure over the joint distribution 
# to condition on the observed labels.
def target_log_prob_fn(*args):
  return joint.log_prob(*args, likelihood=labels)

# Initialize locations and scales randomly with `tf.Variable`s and 
# `tfp.util.TransformedVariable`s.
_init_loc = lambda shape=(): tf.Variable(
    tf.random.uniform(shape, minval=-2., maxval=2.))
_init_scale = lambda shape=(): tfp.util.TransformedVariable(
    initial_value=tf.random.uniform(shape, minval=0.01, maxval=1.),
    bijector=tfb.Softplus())
n_counties = df.county.nunique()

surrogate_posterior = tfd.JointDistributionSequentialAutoBatched([
  tfb.Softplus()(tfd.Normal(_init_loc(), _init_scale())),           # scale_prior
  tfd.Normal(_init_loc(), _init_scale()),                           # intercept
  tfd.Normal(_init_loc(), _init_scale()),                           # floor_weight
  tfd.Normal(_init_loc([n_counties]), _init_scale([n_counties]))])  # county_prior

surrogate_posterior = tfp.experimental.vi.build_factored_surrogate_posterior(
  event_shape=joint.event_shape_tensor()[:-1],
  constraining_bijectors=[tfb.Softplus(), None, None, None])

optimizer = tf.optimizers.Adam(learning_rate=1e-2)

losses = tfp.vi.fit_surrogate_posterior(
    target_log_prob_fn, 
    surrogate_posterior,
    optimizer=optimizer,
    num_steps=3000, 
    seed=42,
    sample_size=2)

(scale_prior_, 
 intercept_, 
 floor_weight_, 
 county_weights_), _ = surrogate_posterior.sample_distributions()

print('        intercept (mean): ', intercept_.mean())
print('     floor_weight (mean): ', floor_weight_.mean())
print(' scale_prior (approx. mean): ', tf.reduce_mean(scale_prior_.sample(10000)))

intercept (mean):  tf.Tensor(1.4352839, shape=(), dtype=float32)
     floor_weight (mean):  tf.Tensor(-0.6701997, shape=(), dtype=float32)
 scale_prior (approx. mean):  tf.Tensor(0.28682157, shape=(), dtype=float32)

fig, ax = plt.subplots(figsize=(10, 3))
ax.plot(losses, 'k-')
ax.set(xlabel="Iteration",
       ylabel="Loss (ELBO)",
       title="Loss during training",
       ylim=0);

county_counts = (df.groupby(by=['county', 'county_code'], observed=True)
                   .agg('size')
                   .sort_values(ascending=False)
                   .reset_index(name='count'))

means = county_weights_.mean()
stds = county_weights_.stddev()

fig, ax = plt.subplots(figsize=(20, 5))

for idx, row in county_counts.iterrows():
  mid = means[row.county_code]
  std = stds[row.county_code]
  ax.vlines(idx, mid - std, mid + std, linewidth=3)
  ax.plot(idx, means[row.county_code], 'ko', mfc='w', mew=2, ms=7)

ax.set(
    xticks=np.arange(len(county_counts)),
    xlim=(-1, len(county_counts)),
    ylabel="County effect",
    title=r"Estimates of county effects on log radon levels. (mean $\pm$ 1 std. dev.)",
)
ax.set_xticklabels(county_counts.county, rotation=90);

fig, ax = plt.subplots(figsize=(10, 7))
ax.plot(np.log1p(county_counts['count']), stds.numpy()[county_counts.county_code], 'o')
ax.set(
    ylabel='Posterior std. deviation',
    xlabel='County log-count',
    title='Having more observations generally\nlowers estimation uncertainty'
);

%%shell
exit  # Trick to make this block not execute.

radon = read.csv('srrs2.dat', header = TRUE)
radon = radon[radon$state=='MN',]
radon$radon = ifelse(radon$activity==0., 0.1, radon$activity)
radon$log_radon = log(radon$radon)

# install.packages('lme4')
library(lme4)
fit <- lmer(log_radon ~ 1 + floor + (1 | county), data=radon)
fit

# Linear mixed model fit by REML ['lmerMod']
# Formula: log_radon ~ 1 + floor + (1 | county)
#    Data: radon
# REML criterion at convergence: 2171.305
# Random effects:
#  Groups   Name        Std.Dev.
#  county   (Intercept) 0.3282
#  Residual             0.7556
# Number of obs: 919, groups:  county, 85
# Fixed Effects:
# (Intercept)        floor
#       1.462       -0.693

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print(pd.DataFrame(data=dict(intercept=[1.462, tf.reduce_mean(intercept_.mean()).numpy()],
                             floor=[-0.693, tf.reduce_mean(floor_weight_.mean()).numpy()],
                             scale=[0.3282, tf.reduce_mean(scale_prior_.sample(10000)).numpy()]),
                   index=['lme4', 'vi']))

intercept   floor     scale
lme4   1.462000 -0.6930  0.328200
vi     1.435284 -0.6702  0.287251