code: def filter_and_predict(self, data, smoothing=False):
      (
          trans_logits,
          trans_probs,
          trans_mvn,
          obs_mvn,
          x_trans_dist,
          y_dist,
      ) = self.get_tensors_and_dists()
  
      log_prob = funsor.Number(0.0)
  
      s_vars = {
          -1: funsor.Tensor(torch.tensor(0), dtype=self.num_components)
      }
      x_vars = {-1: None}
  
      predictive_x_dists, predictive_y_dists, filtering_dists = (
          [],
          [],
          [],
      )
      test_LLs = []
  
      for t, y in enumerate(data):
          s_vars[t] = funsor.Variable(
              f"s_{t}", funsor.Bint[self.num_components]
          )
          x_vars[t] = funsor.Variable(
              f"x_{t}", funsor.Reals[self.hidden_dim]
          )
  
          log_prob += dist.Categorical(
              trans_probs(s=s_vars[t - 1]), value=s_vars[t]
          )
  
          if t == 0:
              log_prob += self.x_init_mvn(value=x_vars[t])
          else:
              log_prob += x_trans_dist(
                  s=s_vars[t], x=x_vars[t - 1], y=x_vars[t]
              )
  
          if t > 0:
              log_prob = log_prob.reduce(
                  ops.logaddexp, {s_vars[t - 1], x_vars[t - 1]}
              )
  
          # do 1-step prediction and compute test LL
          if t > 0:
              predictive_x_dists.append(log_prob)
              _log_prob = log_prob - log_prob.reduce(ops.logaddexp)
              predictive_y_dist = (
                  y_dist(s=s_vars[t], x=x_vars[t]) + _log_prob
              )
              test_LLs.append(
                  predictive_y_dist(y=y)
                  .reduce(ops.logaddexp)
                  .data.item()
              )
              predictive_y_dist = predictive_y_dist.reduce(
                  ops.logaddexp, {f"x_{t}", f"s_{t}"}
              )
              predictive_y_dists.append(
                  funsor_to_mvn(predictive_y_dist, 0, ())
              )
  
          log_prob += y_dist(s=s_vars[t], x=x_vars[t], y=y)
  
          # save filtering dists for forward-backward smoothing
          if smoothing:
              filtering_dists.append(log_prob)
  
      # do the backward recursion using previously computed ingredients
      if smoothing:
          # seed the backward recursion with the filtering distribution at t=T
          smoothing_dists = [filtering_dists[-1]]
          T = data.size(0)
  
          s_vars = {
              t: funsor.Variable(
                  f"s_{t}", funsor.Bint[self.num_components]
              )
              for t in range(T)
          }
          x_vars = {
              t: funsor.Variable(
                  f"x_{t}", funsor.Reals[self.hidden_dim]
              )
              for t in range(T)
          }
  
          # do the backward recursion.
          # let p[t|t-1] be the predictive distribution at time step t.
          # let p[t|t] be the filtering distribution at time step t.
          # let f[t] denote the prior (transition) density at time step t.
          # then the smoothing distribution p[t|T] at time step t is
          # given by the following recursion.
          # p[t-1|T] = p[t-1|t-1] <p[t|T] f[t] / p[t|t-1]>
          # where <...> denotes integration of the latent variables at time step t.
          for t in reversed(range(T - 1)):
              integral = smoothing_dists[-1] - predictive_x_dists[t]
              integral += dist.Categorical(
                  trans_probs(s=s_vars[t]), value=s_vars[t + 1]
              )
              integral += x_trans_dist(
                  s=s_vars[t], x=x_vars[t], y=x_vars[t + 1]
              )
              integral = integral.reduce(
                  ops.logaddexp, {s_vars[t + 1], x_vars[t + 1]}
              )
              smoothing_dists.append(filtering_dists[t] + integral)
  
      # compute predictive test MSE and predictive variances
      predictive_means = torch.stack(
          [d.mean for d in predictive_y_dists]
      )  # T-1 ydim
      predictive_vars = torch.stack(
          [
              d.covariance_matrix.diagonal(dim1=-1, dim2=-2)
              for d in predictive_y_dists
          ]
      )
      predictive_mse = (
          (predictive_means - data[1:, :]).pow(2.0).mean(-1)
      )
  
      if smoothing:
          # compute smoothed mean function
          smoothing_dists = [
              funsor_to_cat_and_mvn(d, 0, (f"s_{t}",))
              for t, d in enumerate(reversed(smoothing_dists))
          ]
          means = torch.stack(
              [d[1].mean for d in smoothing_dists]
          )  # T 2 xdim
          means = torch.matmul(
              means.unsqueeze(-2), self.observation_matrix
          ).squeeze(
              -2
          )  # T 2 ydim
  
          probs = torch.stack(
              [d[0].logits for d in smoothing_dists]
          ).exp()
          probs = probs / probs.sum(-1, keepdim=True)  # T 2
  
          smoothing_means = (probs.unsqueeze(-1) * means).sum(
              -2
          )  # T ydim
          smoothing_probs = probs[:, 1]
  
          return (
              predictive_mse,
              torch.tensor(np.array(test_LLs)),
              predictive_means,
              predictive_vars,
              smoothing_means,
              smoothing_probs,
          )
      else:
          return predictive_mse, torch.tensor(np.array(test_LLs))
  
imports: import argparse
  import time
  from collections import OrderedDict
  from os.path import exists
  from urllib.request import urlopen
  import numpy
  import pyro
  import torch
  import torch.nn
  import funsor
  import funsor.ops
  import funsor.torch.distributions
  from funsor.pyro.convert import funsor_to_cat_and_mvn, funsor_to_mvn, matrix_and_mvn_to_funsor, mvn_to_funsor
label_desc: Use image node_id 4 for calling the SLDS obj's underlying member method code with example usage: obj.filter_and_predict(data, smoothing) and returns: predictive_mse,  torch,  predictive_means,  predictive_vars,  smoothing_means,  smoothing_probs, predictive_mse,  torch