code: def gp_inference(
      k_train_train,
      y_train: jnp.ndarray,
      diag_reg: float = 0.0,
      diag_reg_absolute_scale: bool = False,
      trace_axes: Axes = (-1,),
  ):
      r"""Compute the mean and variance of the 'posterior' of NNGP/NTK/NTKGP.
  
      NNGP - the exact posterior of an infinitely wide Bayesian NN. NTK - exact
      distribution of an infinite ensemble of infinitely wide NNs trained with
      gradient flow for infinite time. NTKGP - posterior of a GP (Gaussian process)
      with the NTK covariance (see
      "`Bayesian Deep Ensembles via the Neural Tangent Kernel
      <https://arxiv.org/abs/2007.05864>`_" for how this can correspond to infinite
      ensembles of infinitely wide NNs as well).
  
      Note that first invocation of the returned `predict_fn` will be slow and
      allocate a lot of memory for its whole lifetime, as a Cholesky factorization
      of `k_train_train.nngp` or `k_train_train.ntk` (or both) is performed and
      cached for future invocations.
  
      Args:
        k_train_train:
          train-train kernel. Can be (a) :class:`jax.numpy.ndarray`,
          (b) `Kernel` namedtuple, (c) :class:`~neural_tangents.Kernel` object.
          Must contain the necessary `nngp` and/or `ntk` kernels for arguments
          provided to the returned `predict_fn` function. For example, if you
          request to compute posterior test [only] NTK covariance in future
          `predict_fn` invocations, `k_train_train` must contain both `ntk` and
          `nngp` kernels.
  
        y_train:
          train targets.
  
        diag_reg:
          a scalar representing the strength of the diagonal regularization for
          `k_train_train`, i.e. computing `k_train_train + diag_reg * I` during
          Cholesky factorization.
  
        diag_reg_absolute_scale:
          `True` for `diag_reg` to represent regularization in absolute units,
          `False` to be `diag_reg * jnp.mean(jnp.trace(k_train_train))`.
  
        trace_axes:
          `f(x_train)` axes such that `k_train_train`,
          `k_test_train`[, and `k_test_test`] lack these pairs of dimensions and
          are to be interpreted as :math:`\Theta \otimes I`, i.e. block-diagonal
          along `trace_axes`. These can can be specified either to save space and
          compute, or to even improve approximation accuracy of the infinite-width
          or infinite-samples limit, since in these limits the covariance along
          channel / feature / logit axes indeed converges to a  constant-diagonal
          matrix. However, if you target linearized dynamics of a specific
          finite-width network, `trace_axes=()` will yield most accurate result.
  
      Returns:
        A function of signature `predict_fn(get, k_test_train, k_test_test)`
        computing 'posterior' Gaussian distribution (mean or mean and covariance)
        on a given test set.
      """
      even, odd, first, last = _get_axes(_get_first(k_train_train))
      trace_axes = utils.canonicalize_axis(trace_axes, y_train)
  
      @lru_cache(2)
      def solve(g: str):
          k_dd = _get_attr(k_train_train, g)
          return _get_cho_solve(k_dd, diag_reg, diag_reg_absolute_scale)
  
      @lru_cache(2)
      def k_inv_y(g: str):
          return solve(g)(y_train, trace_axes)
  
      @utils.get_namedtuple("Gaussians")
      def predict_fn(
          get: Optional[Get] = None, k_test_train=None, k_test_test=None
      ) -> dict[str, Union[jnp.ndarray, Gaussian]]:
          """`test`-set posterior given respective covariance matrices.
  
          Args:
            get:
              string, the mode of the Gaussian process, either "nngp", "ntk", "ntkgp",
              (see "`Bayesian Deep Ensembles via the Neural Tangent Kernel
              <https://arxiv.org/abs/2007.05864>`_") or a tuple, or `None`. If `None`
              then both `nngp` and `ntk` predictions are returned.
  
            k_test_train:
              test-train kernel. Can be (a) :class:`jax.numpy.ndarray`,
              (b) `Kernel` namedtuple, (c) :class:`~neural_tangents.Kernel` object.
              Must contain the necessary `nngp` and/or `ntk` kernels for arguments
              provided to the returned `predict_fn` function. For example, if you
              request to compute posterior test [only] NTK covariance, `k_test_train`
              must contain both `ntk` and `nngp` kernels. If `None`, returns
              predictions on the training set. Note that train-set outputs are always
              `N(y_train, 0)` and mostly returned for API consistency.
  
            k_test_test:
              test-test kernel. Can be (a) :class:`jax.numpy.ndarray`,
              (b) `Kernel` namedtuple, (c) :class:`~neural_tangents.Kernel` object.
              Must contain the necessary `nngp` and/or `ntk` kernels for arguments
              provided to the returned `predict_fn` function. Provide if you want to
              compute test-test posterior covariance. `k_test_test=None` means to not
              compute it. If `k_test_train is None`, pass any non-`None` value (e.g.
              `True`) if you want to get non-regularized (`diag_reg=0`) train-train
              posterior covariance. Note that non-regularized train-set outputs will
              always be the zero-variance Gaussian `N(y_train, 0)` and mostly
              returned for API consistency. For regularized train-set posterior
              outputs according to a positive `diag_reg`, pass
              `k_test_train=k_train_train`, and, optionally,
              `k_test_test=nngp_train_train`.
  
          Returns:
            Either a :class:`Gaussian` `(mean, variance)` namedtuple or `mean` of the
            GP posterior on the `test` set.
          """
          if get is None:
              get = ("nngp", "ntk")
  
          out = {}
  
          for g in get:
              k = g if g != "ntkgp" else "ntk"
              k_dd = _get_attr(k_train_train, k)
              k_td = (
                  None
                  if k_test_train is None
                  else _get_attr(k_test_train, k)
              )
  
              if k_td is None:
                  # Train set predictions.
                  y = y_train.astype(k_dd.dtype)
              else:
                  # Test set predictions.
                  y = jnp.tensordot(k_td, k_inv_y(k), (odd, first))
                  y = jnp.moveaxis(
                      y, range(-len(trace_axes), 0), trace_axes
                  )
  
              if k_test_test is not None:
                  if k_td is None:
                      out[g] = Gaussian(
                          y, jnp.zeros_like(k_dd, k_dd.dtype)
                      )
                  else:
                      if g == "ntk" and (
                          not hasattr(k_train_train, "nngp")
                          or not hasattr(k_test_train, "nngp")
                      ):
                          raise ValueError(
                              'If `"ntk" in get`, and `k_test_test is not None`, '
                              "and `k_test_train is not None`, i.e. you request the "
                              "NTK posterior covariance on the test set, you need "
                              "both NTK and NNGP train-train and test-train matrices "
                              "contained in `k_test_train` and `k_train_train`. "
                              "Hence they must be `namedtuple`s with `nngp` and "
                              "`ntk` attributes."
                          )
  
                      #  kernel of wide NN at initialization
                      g_init = "nngp" if g != "ntkgp" else "ntk"
  
                      k_td_g_inv_y = solve(k)(
                          _get_attr(k_test_train, g_init), even
                      )
                      k_tt = _get_attr(k_test_test, g_init)
  
                      if g == "nngp" or g == "ntkgp":
                          cov = jnp.tensordot(
                              k_td, k_td_g_inv_y, (odd, first)
                          )
                          cov = k_tt - utils.zip_axes(cov)
                          out[g] = Gaussian(y, cov)
  
                      elif g == "ntk":
                          term_1 = solve(g)(k_td, even)
                          cov = jnp.tensordot(
                              _get_attr(k_train_train, "nngp"),
                              term_1,
                              (odd, first),
                          )
                          cov = jnp.tensordot(
                              term_1, cov, (first, first)
                          )
  
                          term_2 = jnp.tensordot(
                              k_td, k_td_g_inv_y, (odd, first)
                          )
                          term_2 += jnp.moveaxis(term_2, first, last)
                          cov = utils.zip_axes(cov - term_2) + k_tt
                          out[g] = Gaussian(y, cov)
  
                      else:
                          raise ValueError(g)
  
              else:
                  out[g] = y
  
          return out
  
      return predict_fn
  
imports: import collections
  from functools import lru_cache
  from typing import Any, Callable, Generator, Iterable, NamedTuple, Optional, Protocol, Union
  import jax
  from jax import grad
  from jax.experimental import ode
  import jax.numpy
  import jax.scipy
  from jax.tree_util import tree_all
  from jax.tree_util import tree_map
  import numpy
  import scipy
  from .utils import dataclasses
  from .utils import utils
  from .utils.typing import Axes
  from .utils.typing import Get
  from .utils.typing import KernelFn
label_desc: Use image node_id 3 for calling a global function with example usage: gp_inference(k_train_train, y_train, diag_reg, diag_reg_absolute_scale, trace_axes) and returns: predict_fn, _get_cho_solve, solve, out