code: def update_posterior_states(A, obs, prior=None, **kwargs):
      """
      Update marginal posterior over hidden states using mean-field fixed point iteration
      FPI or Fixed point iteration.
  
      See the following links for details:
      http://www.cs.cmu.edu/~guestrin/Class/10708/recitations/r9/VI-view.pdf, slides 13- 18, and http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.137.221&rep=rep1&type=pdf, slides 24 - 38.
  
      Parameters
      ----------
      A: ``numpy.ndarray`` of dtype object
          Sensory likelihood mapping or 'observation model', mapping from hidden states to observations. Each element ``A[m]`` of
          stores an ``np.ndarray`` multidimensional array for observation modality ``m``, whose entries ``A[m][i, j, k, ...]`` store
          the probability of observation level ``i`` given hidden state levels ``j, k, ...``
      obs: 1D ``numpy.ndarray``, ``numpy.ndarray`` of dtype object, int or tuple
          The observation (generated by the environment). If single modality, this can be a 1D ``np.ndarray``
          (one-hot vector representation) or an ``int`` (observation index)
          If multi-modality, this can be ``np.ndarray`` of dtype object whose entries are 1D one-hot vectors,
          or a tuple (of ``int``)
      prior: 1D ``numpy.ndarray`` or ``numpy.ndarray`` of dtype object, default None
          Prior beliefs about hidden states, to be integrated with the marginal likelihood to obtain
          a posterior distribution. If not provided, prior is set to be equal to a flat categorical distribution (at the level of
          the individual inference functions).
      **kwargs: keyword arguments
          List of keyword/parameter arguments corresponding to parameter values for the fixed-point iteration
          algorithm ``algos.fpi.run_vanilla_fpi.py``
  
      Returns
      ----------
      qs: 1D ``numpy.ndarray`` or ``numpy.ndarray`` of dtype object
          Marginal posterior beliefs over hidden states at current timepoint
      """
  
      (
          num_obs,
          num_states,
          num_modalities,
          num_factors,
      ) = utils.get_model_dimensions(A=A)
  
      obs = utils.process_observation(obs, num_modalities, num_obs)
  
      if prior is not None:
          prior = utils.to_obj_array(prior)
  
      return run_vanilla_fpi(
          A, obs, num_obs, num_states, prior, **kwargs
      )
  
file_path: pymdp/pymdp/inference.py
imports: import numpy
  from pymdp import utils
  from pymdp.maths import get_joint_likelihood_seq
  from pymdp.algos import run_vanilla_fpi, run_mmp, _run_mmp_testing
label_desc: Use image node_id 4 for calling a global function with example usage: update_posterior_states(A, obs, prior) and returns: run_vanilla_fpi