class_docstr: Probability of Improvement
  
  :math:`P(f_{t+1}(\mathbf{X})\geq f(\mathbf{X^+}))` :math:`:= \Phi(\\frac{ \mu(\mathbf{X})-f(\mathbf{X^+}) }
  { \sigma(\mathbf{X}) })` with :math:`f(X^+)` as the incumbent and :math:`\Phi` the cdf of the standard normal.
  
  Parameters
  ----------
  xi : float, defaults to 0.0
      Controls the balance between exploration and exploitation of the acquisition function.
class_docstr_tok: ["Probability","of","Improvement",":","math",":","`","P","(","f_","{","t+1","}","(","\\mathbf","{","X","}",")","\\geq","f","(","\\mathbf","{","X^+","}",")",")","`",":","math",":","`",":","=","\\Phi","(","\\\\frac","{","\\mu","(","\\mathbf","{","X","}",")","-f","(","\\mathbf","{","X^+","}",")","}","{","\\sigma","(","\\mathbf","{","X","}",")","}",")","`","with",":","math",":","`","f","(","X^+",")","`","as","the","incumbent","and",":","math",":","`","\\Phi","`","the","cdf","of","the","standard","normal",".","Parameters","--","--","--","--","--","xi",":","float",",","defaults","to","0.0","Controls","the","balance","between","exploration","and","exploitation","of","the","acquisition","function","."]
code: def meta(self) -> dict[str, Any]:  # noqa: D102
      meta = super().meta
      meta.update({"xi": self._xi})
  
      return meta
  
file_path: SMAC3/smac/acquisition/function/probability_improvement.py
imports: from __future__ import annotations
  from typing import Any
  import numpy
  from scipy.stats import norm
  from smac.acquisition.function.abstract_acquisition_function import AbstractAcquisitionFunction
  from smac.utils.logging import get_logger
label_desc: Use image node_id 3 for calling the PI obj's underlying member method code with example usage: obj.meta() and returns: meta