code: def kalman_filter_step(
      F,
      target: Tensor,
      prior_mean: Tensor,
      prior_cov: Tensor,
      emission_coeff: Tensor,
      residual: Tensor,
      noise_std: Tensor,
      latent_dim: int,
      output_dim: int,
  ):
      """
      One step of the Kalman filter.
  
      This function computes the filtered state (mean and covariance) given the
      linear system coefficients the prior state (mean and variance),
      as well as observations.
  
      Parameters
      ----------
      F
      target
          Observations of the system output, shape (batch_size, output_dim)
      prior_mean
          Prior mean of the latent state, shape (batch_size, latent_dim)
      prior_cov
          Prior covariance of the latent state, shape
          (batch_size, latent_dim, latent_dim)
      emission_coeff
          Emission coefficient, shape (batch_size, output_dim, latent_dim)
      residual
          Residual component, shape (batch_size, output_dim)
      noise_std
          Standard deviation of the output noise, shape (batch_size, output_dim)
      latent_dim
          Dimension of the latent state vector
      Returns
      -------
      Tensor
          Filtered_mean, shape (batch_size, latent_dim)
      Tensor
          Filtered_covariance, shape (batch_size, latent_dim, latent_dim)
      Tensor
          Log probability, shape (batch_size, )
      """
      # output_mean: mean of the target (batch_size, obs_dim)
      output_mean = F.linalg_gemm2(
          emission_coeff, prior_mean.expand_dims(axis=-1)
      ).squeeze(axis=-1)
  
      # noise covariance
      noise_cov = make_nd_diag(
          F=F, x=noise_std * noise_std, d=output_dim
      )
  
      S_hh_x_A_tr = F.linalg_gemm2(
          prior_cov, emission_coeff, transpose_b=True
      )
  
      # covariance of the target
      output_cov = (
          F.linalg_gemm2(emission_coeff, S_hh_x_A_tr) + noise_cov
      )
  
      # compute the Cholesky decomposition output_cov = LL^T
      L_output_cov = F.linalg_potrf(output_cov)
  
      # Compute Kalman gain matrix K:
      # K = S_hh X with X = A^T output_cov^{-1}
      # We have X = A^T output_cov^{-1} => X output_cov = A^T => X LL^T = A^T
      # We can thus obtain X by solving two linear systems involving L
      kalman_gain = F.linalg_trsm(
          L_output_cov,
          F.linalg_trsm(
              L_output_cov, S_hh_x_A_tr, rightside=True, transpose=True
          ),
          rightside=True,
      )
  
      # compute the error
      target_minus_residual = target - residual
      delta = target_minus_residual - output_mean
  
      # filtered estimates
      filtered_mean = prior_mean.expand_dims(axis=-1) + F.linalg_gemm2(
          kalman_gain, delta.expand_dims(axis=-1)
      )
      filtered_mean = filtered_mean.squeeze(axis=-1)
  
      # Joseph's symmetrized update for covariance:
      ImKA = F.broadcast_sub(
          F.eye(latent_dim), F.linalg_gemm2(kalman_gain, emission_coeff)
      )
  
      filtered_cov = F.linalg_gemm2(
          ImKA, F.linalg_gemm2(prior_cov, ImKA, transpose_b=True)
      ) + F.linalg_gemm2(
          kalman_gain,
          F.linalg_gemm2(noise_cov, kalman_gain, transpose_b=True),
      )
  
      # likelihood term: (batch_size,)
      log_p = MultivariateGaussian(output_mean, L_output_cov).log_prob(
          target_minus_residual
      )
  
      return filtered_mean, filtered_cov, log_p
  
imports: from typing import Optional, Tuple
  import mxnet
  import numpy
  from gluonts.core.component import validated
  from gluonts.mx import Tensor
  from gluonts.mx.linalg_util import jitter_cholesky
  from gluonts.mx.util import _broadcast_param, make_nd_diag
  from .None import Distribution, Gaussian, MultivariateGaussian
  from .distribution import getF
label_desc: Use image node_id 2 for calling a global function with example usage: kalman_filter_step(F, target, prior_mean, prior_cov, emission_coeff, residual, noise_std, latent_dim, output_dim) and returns: filtered_mean,  filtered_cov,  log_p