code: def entropy(pk, qk=None, base=None, axis=0):
      """Calculate the entropy of a distribution for given probability values.
  
      If only probabilities ``pk`` are given, the entropy is calculated as
      ``S = -sum(pk * log(pk), axis=axis)``.
  
      If ``qk`` is not None, then compute the Kullback-Leibler divergence
      ``S = sum(pk * log(pk / qk), axis=axis)``.
  
      This routine will normalize ``pk`` and ``qk`` if they don't sum to 1.
  
      Args:
          pk (ndarray): Defines the (discrete) distribution. ``pk[i]`` is the
              (possibly unnormalized) probability of event ``i``.
          qk (ndarray, optional): Sequence against which the relative entropy is
              computed. Should be in the same format as ``pk``.
          base (float, optional): The logarithmic base to use, defaults to ``e``
              (natural logarithm).
          axis (int, optional): The axis along which the entropy is calculated.
              Default is 0.
  
      Returns:
          S (cupy.ndarray): The calculated entropy.
  
      """
      if (
          pk.dtype.kind == "c"
          or qk is not None
          and qk.dtype.kind == "c"
      ):
          raise TypeError("complex dtype not supported")
  
      float_type = (
          cupy.float32 if pk.dtype.char in "ef" else cupy.float64
      )
      pk = pk.astype(float_type, copy=False)
      pk = _normalize(pk, axis)
      if qk is None:
          vec = special.entr(pk)
      else:
          if qk.shape != pk.shape:
              raise ValueError("qk and pk must have same shape.")
          qk = qk.astype(float_type, copy=False)
          qk = _normalize(qk, axis)
          vec = special.rel_entr(pk, qk)
      s = cupy.sum(vec, axis=axis)
      if base is not None:
          s /= math.log(base)
      return s
  
imports: import math
  import cupy
  from cupyx.scipy import special
label_desc: Use image node_id 2 for calling a global function with example usage: entropy(pk, qk, base, axis) and returns: s