EFE_learning_novelty_term.ipynb

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      " \n",
      "Novelty term for small concentration parameter values:\n",
      "0.505\n",
      " \n",
      "Novelty term for intermediate concentration parameter values:\n",
      "0.05050000000000002\n",
      " \n",
      "Novelty term for large concentration parameter values:\n",
      "0.005050000000000001\n",
      " \n"
     ]
    }
   ],
   "source": [
    "# Calculating novelty term in expected free energy when learning 'A' matrix concentration parameters\n",
    "# (which drives parameter exploration)\n",
    "\n",
    "# Supplementary Code for: A Step-by-Step Tutorial on Active Inference Modelling and its \n",
    "# Application to Empirical Data\n",
    "\n",
    "# By: Ryan Smith, Karl J. Friston, Christopher J. Whyte\n",
    "###############################################################################\n",
    "\n",
    "# 参考《主动推理》“新颖性”：P213 式7-11，P374\n",
    "\n",
    "import numpy as np\n",
    "\n",
    "def col_norm(A_norm):\n",
    "    norm_constant = np.sum(A_norm, axis=0)  # create normalizing constant from sum of columns\n",
    "    A_normed = A_norm / norm_constant  # divide columns by constant\n",
    "    return A_normed\n",
    "\n",
    "# small concentration parameter values \n",
    "a1 = np.array([[0.25, 1],  \n",
    "               [0.75, 1]])\n",
    "\n",
    "# intermediate concentration parameter values \n",
    "a2 = np.array([[2.5, 10],\n",
    "               [7.5, 10]])\n",
    "\n",
    "# large concentration parameter values  \n",
    "a3 = np.array([[25, 100],\n",
    "               [75, 100]])\n",
    "\n",
    "# normalize columns in 'a' to get likelihood matrix 'A'\n",
    "A1 = col_norm(a1)\n",
    "A2 = col_norm(a2)\n",
    "A3 = col_norm(a3)\n",
    "\n",
    "# calculate 'a_sum' \n",
    "a1_sum = np.array([[a1[0,0]+a1[1,0], a1[0,1]+a1[1,1]],\n",
    "                   [a1[0,0]+a1[1,0], a1[0,1]+a1[1,1]]])\n",
    "\n",
    "a2_sum = np.array([[a2[0,0]+a2[1,0], a2[0,1]+a2[1,1]],\n",
    "                   [a2[0,0]+a2[1,0], a2[0,1]+a2[1,1]]])\n",
    "\n",
    "a3_sum = np.array([[a3[0,0]+a3[1,0], a3[0,1]+a3[1,1]],\n",
    "                   [a3[0,0]+a3[1,0], a3[0,1]+a3[1,1]]])\n",
    "\n",
    "# element wise inverse for 'a' and 'a_sum'\n",
    "inv_a1 = 1 / a1\n",
    "inv_a2 = 1 / a2\n",
    "inv_a3 = 1 / a3\n",
    "\n",
    "inv_a1_sum = 1 / a1_sum\n",
    "inv_a2_sum = 1 / a2_sum\n",
    "inv_a3_sum = 1 / a3_sum\n",
    "\n",
    "# 'W' term for 'a' matrix\n",
    "W1 = 0.5 * (inv_a1 - inv_a1_sum)\n",
    "W2 = 0.5 * (inv_a2 - inv_a2_sum)\n",
    "W3 = 0.5 * (inv_a3 - inv_a3_sum)\n",
    "\n",
    "# beliefs over states under a policy at a time point\n",
    "s_pi_tau = np.array([0.9, 0.1])\n",
    "\n",
    "# predictive posterior over outcomes (A*s_pi_tau = predicted o_pi_tau)\n",
    "A1s = np.dot(A1, s_pi_tau)\n",
    "A2s = np.dot(A2, s_pi_tau)\n",
    "A3s = np.dot(A3, s_pi_tau)\n",
    "\n",
    "# W term multiplied by beliefs over states under a policy at a time point\n",
    "W1s = np.dot(W1, s_pi_tau)\n",
    "W2s = np.dot(W2, s_pi_tau)\n",
    "W3s = np.dot(W3, s_pi_tau)\n",
    "\n",
    "# compute novelty using dot product function\n",
    "Novelty_smallCP = np.dot(A1s, W1s)\n",
    "Novelty_intermediateCP = np.dot(A2s, W2s)\n",
    "Novelty_largeCP = np.dot(A3s, W3s)\n",
    "\n",
    "# show results\n",
    "print(' ')\n",
    "print('Novelty term for small concentration parameter values:')\n",
    "print(Novelty_smallCP)\n",
    "print(' ')\n",
    "print('Novelty term for intermediate concentration parameter values:')\n",
    "print(Novelty_intermediateCP)\n",
    "print(' ')\n",
    "print('Novelty term for large concentration parameter values:')\n",
    "print(Novelty_largeCP)\n",
    "print(' ')"
   ]
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