code: def model_5(data, history, vectorized):
      x_dim, y_dim = 3, 2
      x_init = pyro.param(
          "x_init",
          lambda: torch.rand(x_dim),
          constraint=constraints.simplex,
      )
      x_init_2 = pyro.param(
          "x_init_2",
          lambda: torch.rand(x_dim, x_dim),
          constraint=constraints.simplex,
      )
      x_trans = pyro.param(
          "x_trans",
          lambda: torch.rand((x_dim, x_dim, x_dim)),
          constraint=constraints.simplex,
      )
      y_probs = pyro.param(
          "y_probs",
          lambda: torch.rand(x_dim, y_dim),
          constraint=constraints.simplex,
      )
  
      x_prev = x_prev_2 = None
      markov_loop = (
          pyro.vectorized_markov(
              name="time", size=len(data), dim=-2, history=history
          )
          if vectorized
          else pyro.markov(range(len(data)), history=history)
      )
      for i in markov_loop:
          if isinstance(i, int) and i == 0:
              x_probs = x_init
          elif isinstance(i, int) and i == 1:
              x_probs = Vindex(x_init_2)[x_prev]
          else:
              x_probs = Vindex(x_trans)[x_prev_2, x_prev]
  
          x_curr = pyro.sample(
              "x_{}".format(i), dist.Categorical(x_probs)
          )
          with pyro.plate("tones", data.shape[-1], dim=-1):
              pyro.sample(
                  "y_{}".format(i),
                  dist.Categorical(Vindex(y_probs)[x_curr]),
                  obs=data[i],
              )
          x_prev_2, x_prev = x_prev, x_curr
  
imports: import pytest
  import torch
  from pyroapi import pyro_backend
  from torch.distributions import constraints
  from pyro.ops.indexing import Vindex
label_desc: Use image node_id 6 for calling a global function with example usage: model_5(data, history, vectorized) without return types