Address Copilot review comments on MAB PR

- Cast rng.integers() results to Python int in EpsilonGreedy and
  RandomStrategy select_arm, fixing doctest flakiness from np.int64
- Fix grammar in module docstring and RandomStrategy docstring
- Add missing doctest for Bandit.__init__
- Split test_mab_strategies into a real deterministic assertion-based
  test and a separate demo_mab_strategies for the stochastic plot
- Revert DIRECTORY.md to upstream (auto-generated, out of scope here)

Author: sephml

DIRECTORY.md

@@ -712,18 +712,14 @@
   * [Loss Functions](machine_learning/loss_functions.py)
   * Lstm
     * [Lstm Prediction](machine_learning/lstm/lstm_prediction.py)
-  * [Mab](machine_learning/mab.py)
-  * [Mean Shift](machine_learning/mean_shift.py)
   * [Mfcc](machine_learning/mfcc.py)
   * [Mini Batch Gradient Descent](machine_learning/mini_batch_gradient_descent.py)
   * [Multilayer Perceptron Classifier](machine_learning/multilayer_perceptron_classifier.py)
-  * [Naive Bayes Text Classification](machine_learning/naive_bayes_text_classification.py)
   * [Polynomial Regression](machine_learning/polynomial_regression.py)
   * [Principle Component Analysis](machine_learning/principle_component_analysis.py)
   * [Q Learning](machine_learning/q_learning.py)
   * [Random Forest Classifier](machine_learning/random_forest_classifier.py)
   * [Random Forest Regressor](machine_learning/random_forest_regressor.py)
-  * [Rmsprop](machine_learning/rmsprop.py)
   * [Scoring Functions](machine_learning/scoring_functions.py)
   * [Self Organizing Map](machine_learning/self_organizing_map.py)
   * [Sequential Minimum Optimization](machine_learning/sequential_minimum_optimization.py)
@@ -740,7 +736,6 @@
   * [Arc Length](maths/arc_length.py)
   * [Area](maths/area.py)
   * [Area Under Curve](maths/area_under_curve.py)
-  * [Autocorrelation](maths/autocorrelation.py)
   * [Average Absolute Deviation](maths/average_absolute_deviation.py)
   * [Average Mean](maths/average_mean.py)
   * [Average Median](maths/average_median.py)
@@ -786,7 +781,6 @@
   * [Fibonacci](maths/fibonacci.py)
   * [Find Max](maths/find_max.py)
   * [Find Min](maths/find_min.py)
-  * [First Fundamental Form](maths/first_fundamental_form.py)
   * [Floor](maths/floor.py)
   * [Gamma](maths/gamma.py)
   * [Gaussian](maths/gaussian.py)
@@ -849,7 +843,6 @@
     * [Square Root](maths/numerical_analysis/square_root.py)
     * [Weierstrass Method](maths/numerical_analysis/weierstrass_method.py)
   * [Odd Sieve](maths/odd_sieve.py)
-  * [Padovan Sequence](maths/padovan_sequence.py)
   * [Pell Number](maths/pell_number.py)
   * [Perfect Cube](maths/perfect_cube.py)
   * [Perfect Number](maths/perfect_number.py)
@@ -879,8 +872,6 @@
   * [Reverse Factorial Recursive](maths/reverse_factorial_recursive.py)
   * [Segmented Sieve](maths/segmented_sieve.py)
   * Series
-    * [Alternate Harmonic Series](maths/series/alternate_harmonic_series.py)
-    * [Alternating Harmonic Series](maths/series/alternating_harmonic_series.py)
     * [Arithmetic](maths/series/arithmetic.py)
     * [Geometric](maths/series/geometric.py)
     * [Geometric Series](maths/series/geometric_series.py)
@@ -920,7 +911,6 @@
     * [Polygonal Numbers](maths/special_numbers/polygonal_numbers.py)
     * [Pronic Number](maths/special_numbers/pronic_number.py)
     * [Proth Number](maths/special_numbers/proth_number.py)
-    * [Spy Number](maths/special_numbers/spy_number.py)
     * [Triangular Numbers](maths/special_numbers/triangular_numbers.py)
     * [Trimorphic Number](maths/special_numbers/trimorphic_number.py)
     * [Ugly Numbers](maths/special_numbers/ugly_numbers.py)

machine_learning/mab.py

@@ -9,7 +9,7 @@
 - There are N arms, each with a different probability of giving a reward.
 - The agent must learn to choose the best arm to pull in order to maximize its reward.
 
-Here there are 3 optimising strategies have been implemented:
+Here 3 optimising strategies have been implemented:
 - Epsilon-Greedy
 - Upper Confidence Bound (UCB)
 - Thompson Sampling
@@ -41,6 +41,11 @@ def __init__(self, probabilities: list[float]) -> None:
 
         Args:
             probabilities: List of probabilities for each arm.
+
+        Example:
+            >>> bandit = Bandit([0.1, 0.5, 0.9])
+            >>> bandit.num_arms
+            3
         """
         self.probabilities = probabilities
         self.num_arms = len(probabilities)
@@ -127,7 +132,7 @@ def select_arm(self) -> int:
         rng = np.random.default_rng()
 
         if rng.random() < self.epsilon:
-            return rng.integers(self.num_arms)
+            return int(rng.integers(self.num_arms))
         else:
             return int(np.argmax(self.values))
 
@@ -274,7 +279,7 @@ def update(self, arm_index: int, reward: int) -> None:
 # Random strategy (full exploration)
 class RandomStrategy(Strategy):
     """
-    A class for choosing totally random at each round to give
+    A class for choosing an arm uniformly at random at each round to give
     a better comparison with the other optimised strategies.
     """
 
@@ -297,10 +302,10 @@ def select_arm(self) -> int:
         Example:
             >>> strategy = RandomStrategy(num_arms=3)
             >>> 0 <= strategy.select_arm() < 3
-            np.True_
+            True
         """
         rng = np.random.default_rng()
-        return rng.integers(self.num_arms)
+        return int(rng.integers(self.num_arms))
 
     def update(self, arm_index: int, reward: int) -> None:
         """
@@ -371,7 +376,62 @@ def update(self, arm_index: int, reward: int) -> None:
 
 def test_mab_strategies() -> None:
     """
-    Test the MAB strategies.
+    Deterministic behavioural tests for the MAB strategies.
+
+    These checks feed each strategy a fixed sequence of rewards and assert
+    on the resulting internal state and arm selection, so a regression in
+    the update/select logic will fail the suite instead of only being
+    visible in the (stochastic) plotted demo.
+    """
+    num_arms = 3
+
+    # After repeatedly rewarding arm 2, a purely greedy strategy must
+    # settle on arm 2.
+    greedy = GreedyStrategy(num_arms=num_arms)
+    for _ in range(10):
+        greedy.update(2, 1)
+    greedy.update(0, 0)
+    greedy.update(1, 0)
+    assert greedy.select_arm() == 2
+
+    # Epsilon-Greedy with epsilon=0 behaves like the greedy strategy.
+    epsilon_greedy = EpsilonGreedy(epsilon=0.0, num_arms=num_arms)
+    for _ in range(10):
+        epsilon_greedy.update(1, 1)
+    epsilon_greedy.update(0, 0)
+    epsilon_greedy.update(2, 0)
+    assert epsilon_greedy.select_arm() == 1
+
+    # UCB must exhaustively try every arm once before repeating any of them.
+    ucb = UCB(num_arms=num_arms)
+    first_round_arms = set()
+    for _ in range(num_arms):
+        arm = ucb.select_arm()
+        first_round_arms.add(arm)
+        ucb.update(arm, 1)
+    assert first_round_arms == set(range(num_arms))
+
+    # Thompson Sampling should heavily favor an arm with only successes
+    # over arms with only failures.
+    thompson = ThompsonSampling(num_arms=num_arms)
+    for _ in range(20):
+        thompson.update(0, 1)
+        thompson.update(1, 0)
+        thompson.update(2, 0)
+    selections = [thompson.select_arm() for _ in range(50)]
+    assert selections.count(0) > len(selections) // 2
+
+    # RandomStrategy.update is a no-op and select_arm always returns a
+    # valid arm index.
+    random_strategy = RandomStrategy(num_arms=num_arms)
+    random_strategy.update(0, 1)
+    assert 0 <= random_strategy.select_arm() < num_arms
+
+
+def demo_mab_strategies() -> None:
+    """
+    Run a stochastic simulation of the MAB strategies and plot their
+    cumulative reward over time for visual comparison.
     """
     # Simulation
     num_arms = 4
@@ -418,3 +478,4 @@ def test_mab_strategies() -> None:
 
     doctest.testmod()
     test_mab_strategies()
+    demo_mab_strategies()