Original code copyright (C) 2009-2022 Rudolf Cardinal (

This file is part of cardinal_pythonlib.

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Miscellaneous mathematical functions that use Numpy (which can be slow to load).

cardinal_pythonlib.maths_numpy.inv_logistic(y: Union[float, numpy.ndarray], k: float, theta: float) → Optional[float][source]

Inverse standard logistic function:

x = ( log( \frac {1} {y} - 1) / -k ) + \theta

  • yy
  • kk
  • theta\theta


cardinal_pythonlib.maths_numpy.logistic(x: Union[float, numpy.ndarray], k: float, theta: float) → Optional[float][source]

Standard logistic function.

y = \frac {1} {1 + e^{-k (x - \theta)}}

  • xx
  • kk
  • theta\theta


cardinal_pythonlib.maths_numpy.pick_from_probabilities(probabilities: Union[List[float], numpy.ndarray]) → int[source]

Given a list of probabilities like [0.1, 0.3, 0.6], returns the index of the probabilistically chosen item. In this example, we would return 0 with probability 0.1; 1 with probability 0.3; and 2 with probability 0.6.


probabilities – list of probabilities, which should sum to 1


the index of the chosen option

  • ValueError if a random number in the range [0, 1) is greater
  • than or equal to the cumulative sum of the supplied probabilities (i.e.
  • if you’ve specified probabilities adding up to less than 1)

Does not object if you supply e.g. [1, 1, 1]; it’ll always pick the first in this example.

cardinal_pythonlib.maths_numpy.softmax(x: numpy.ndarray, b: float = 1.0) → numpy.ndarray[source]

Standard softmax function:

P_i = \frac {e ^ {\beta \cdot x_i}} { \sum_{i}{\beta \cdot x_i} }

  • x – vector (numpy.array) of values
  • b – exploration parameter \beta, or inverse temperature [Daw2009], or 1/t; see below

vector of probabilities corresponding to the input values


  • t is temperature (towards infinity: all actions equally likely; towards zero: probability of action with highest value tends to 1)
  • Temperature is not used directly as optimizers may take it to zero, giving an infinity; use inverse temperature instead.
  • [Daw2009] Daw ND, “Trial-by-trial data analysis using computational methods”, 2009/2011; in “Decision Making, Affect, and Learning: Attention and Performance XXIII”; Delgado MR, Phelps EA, Robbins TW (eds), Oxford University Press.