class_docstr: Metropolis-Hastings [@metropolis1953equation; @hastings1970monte].
  
  #### Notes
  
  In conditional inference, we infer $z$ in $p(z, \beta
  \mid x)$ while fixing inference over $\beta$ using another
  distribution $q(\beta)$.
  To calculate the acceptance ratio, `MetropolisHastings` uses an
  estimate of the marginal density,
  
  $p(x, z) = \mathbb{E}_{q(\beta)} [ p(x, z, \beta) ]
            \approx p(x, z, \beta^*)$
  
  leveraging a single Monte Carlo sample, where $\beta^* \sim
  q(\beta)$. This is unbiased (and therefore asymptotically exact as a
  pseudo-marginal method) if $q(\beta) = p(\beta \mid x)$.
  
  `MetropolisHastings` assumes the proposal distribution has the same
  support as the prior. The `auto_transform` attribute in
  the method `initialize()` is not applicable.
  
  #### Examples
  
  ```python
  mu = Normal(loc=0.0, scale=1.0)
  x = Normal(loc=mu, scale=1.0, sample_shape=10)
  
  qmu = Empirical(tf.Variable(tf.zeros(500)))
  proposal_mu = Normal(loc=mu, scale=0.5)
  inference = ed.MetropolisHastings({mu: qmu}, {mu: proposal_mu},
                                    data={x: np.zeros(10, dtype=np.float32)})
  ```
docstr: Draw sample from proposal conditional on last sample. Then
  accept or reject the sample based on the ratio,
  
  $\text{ratio} =
        \log p(x, z^{\text{new}}) - \log p(x, z^{\text{old}}) -
        \log g(z^{\text{new}} \mid z^{\text{old}}) +
        \log g(z^{\text{old}} \mid z^{\text{new}})$
  
  #### Notes
  
  The updates assume each Empirical random variable is directly
  parameterized by `tf.Variable`s.
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code: def build_update(self):
      """Draw sample from proposal conditional on last sample. Then
      accept or reject the sample based on the ratio,
  
      $\\text{ratio} =
            \log p(x, z^{\\text{new}}) - \log p(x, z^{\\text{old}}) -
            \log g(z^{\\text{new}} \mid z^{\\text{old}}) +
            \log g(z^{\\text{old}} \mid z^{\\text{new}})$
  
      #### Notes
  
      The updates assume each Empirical random variable is directly
      parameterized by `tf.Variable`s.
      """
      old_sample = {
          z: tf.gather(qz.params, tf.maximum(self.t - 1, 0))
          for z, qz in six.iteritems(self.latent_vars)
      }
      old_sample = OrderedDict(old_sample)
  
      # Form dictionary in order to replace conditioning on prior or
      # observed variable with conditioning on a specific value.
      dict_swap = {}
      for x, qx in six.iteritems(self.data):
          if isinstance(x, RandomVariable):
              if isinstance(qx, RandomVariable):
                  qx_copy = copy(qx, scope="conditional")
                  dict_swap[x] = qx_copy.value()
              else:
                  dict_swap[x] = qx
  
      dict_swap_old = dict_swap.copy()
      dict_swap_old.update(old_sample)
      base_scope = tf.get_default_graph().unique_name("inference") + "/"
      scope_old = base_scope + "old"
      scope_new = base_scope + "new"
  
      # Draw proposed sample and calculate acceptance ratio.
      new_sample = old_sample.copy()  # copy to ensure same order
      ratio = 0.0
      for z, proposal_z in six.iteritems(self.proposal_vars):
          # Build proposal g(znew | zold).
          proposal_znew = copy(
              proposal_z, dict_swap_old, scope=scope_old
          )
          # Sample znew ~ g(znew | zold).
          new_sample[z] = proposal_znew.value()
          # Increment ratio.
          ratio -= tf.reduce_sum(proposal_znew.log_prob(new_sample[z]))
  
      dict_swap_new = dict_swap.copy()
      dict_swap_new.update(new_sample)
  
      for z, proposal_z in six.iteritems(self.proposal_vars):
          # Build proposal g(zold | znew).
          proposal_zold = copy(
              proposal_z, dict_swap_new, scope=scope_new
          )
          # Increment ratio.
          ratio += tf.reduce_sum(
              proposal_zold.log_prob(dict_swap_old[z])
          )
  
      for z in six.iterkeys(self.latent_vars):
          # Build priors p(znew) and p(zold).
          znew = copy(z, dict_swap_new, scope=scope_new)
          zold = copy(z, dict_swap_old, scope=scope_old)
          # Increment ratio.
          ratio += tf.reduce_sum(znew.log_prob(dict_swap_new[z]))
          ratio -= tf.reduce_sum(zold.log_prob(dict_swap_old[z]))
  
      for x in six.iterkeys(self.data):
          if isinstance(x, RandomVariable):
              # Build likelihoods p(x | znew) and p(x | zold).
              x_znew = copy(x, dict_swap_new, scope=scope_new)
              x_zold = copy(x, dict_swap_old, scope=scope_old)
              # Increment ratio.
              ratio += tf.reduce_sum(x_znew.log_prob(dict_swap[x]))
              ratio -= tf.reduce_sum(x_zold.log_prob(dict_swap[x]))
  
      # Accept or reject sample.
      u = tf.random_uniform([], dtype=ratio.dtype)
      accept = tf.log(u) < ratio
      sample_values = tf.cond(
          accept,
          lambda: list(six.itervalues(new_sample)),
          lambda: list(six.itervalues(old_sample)),
      )
      if not isinstance(sample_values, list):
          # `tf.cond` returns tf.Tensor if output is a list of size 1.
          sample_values = [sample_values]
  
      sample = {
          z: sample_value
          for z, sample_value in zip(
              six.iterkeys(new_sample), sample_values
          )
      }
  
      # Update Empirical random variables.
      assign_ops = []
      for z, qz in six.iteritems(self.latent_vars):
          variable = qz.get_variables()[0]
          assign_ops.append(
              tf.scatter_update(variable, self.t, sample[z])
          )
  
      # Increment n_accept (if accepted).
      assign_ops.append(
          self.n_accept.assign_add(tf.where(accept, 1, 0))
      )
      return tf.group(*assign_ops)
  
file_path: edward/edward/inferences/metropolis_hastings.py
imports: from __future__ import absolute_import
  from __future__ import division
  from __future__ import print_function
  import six
  import tensorflow
  from collections import OrderedDict
  from edward.inferences.monte_carlo import MonteCarlo
  from edward.models import RandomVariable
  from edward.util import check_latent_vars, copy
label_desc: Use image node_id 3 for calling the MetropolisHastings obj's underlying member method code with example usage: obj.build_update() and returns: tf