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Машинное обучение: Наивный байесовский классификатор. Теория и реализация. С нуля

В этой статье я привел основные сведения о трех основных видах НБК и показал реализацию каждого.

Содержание: формулировка задачи, независимость признаков, теория для мультиномиального, гауссовского и бернулли и их реализация.

Введение

Наивный байесовский классификатор (НБК) является простым, но эффективным алгоритмом в машинном обучении, используемым для задач классификации.

НБК основан на теореме Байеса, которая позволяет рассчитать условную вероятность события на основе предварительной информации. В контексте классификации, он используется для расчета вероятности принадлежности объекта к определенному классу на основе его признаков.

В основе идеи НБК лежит допущение (предположение) о том, что все признаки, описывающие объект, независимы друг от друга. Это предположение часто не выполняется в реальных задачах, но несмотря на это, алгоритм часто показывает хорошие результаты.

Выделяют три типа НБК, которые зависят от распределения набора данных: гауссовский, мультиномиальный и бернулли. Гауссовский используется, когда набор данных распределен нормально, это подходит для непрерывных признаков. Мультиномиальный используется, когда набор данных содержит дискретные значения, используется в текстовой классификации. Бернулли используется при работе над задачами бинарной классификации с бинарными признаками.

НБК используется в таких задачах, как: классификация текста, например, классификация электронных писем как спам или не спам; анализ комментариев - определение тона (положительный, отрицательный, нейтральный); системы рекомендаций - может быть использован для рекомендаций на основе пользовательских предпочтений.

Формулировка задачи

Пусть есть набор данных, который представляет собой наблюдения входных и выходных событий. Входные события — входные признаки, выходные события — выходные признаки. Выходные события зависят от наступления входных событий. Когда на вход подаются входные признаки, то нужно вычислить вероятность наступления каждого выходного события (класса).

На основе набора данных можно рассчитать вероятности для каждого входного и выходного события на основе их частоты. Вероятности входных или выходных событий, рассчитанные на основе набора данных, называются априорными. Вероятности выходных событий, рассчитанные для новых данных, не участвующих в вычислении априорных вероятностей, называются апостериорными. Задача в том, чтобы рассчитать апостериорные вероятности для новых входных данных. Для этого будет использоваться теорема Байеса.

Теорема Байеса. Она позволяет рассчитать апостериорную вероятность (вероятность события при условии выполнения другого события). Формула теоремы Байеса выглядит следующим образом:

P(A|B) = \frac{P(B|A) P(A)}{P(B)}

P(A|B) — апостериорная вероятность события A при условии B (при наличии события B, то есть для новых входных признаков), P(B|A) — априорная вероятность события B при условии A (рассчитывается из набора данных), P(A) — априорная вероятность события A (рассчитывается из набора данных), P(B) — априорная вероятность события B (рассчитывается из набора данных).

Независимость признаков

Вероятностная модель НБК — это условная модель p(c|x_1,...,x_n) над зависимой переменной класса c, зависимая от нескольких переменных (x_1,...,x_n), n — число признаков.

По формуле Байеса:

P(c|x_1,...,x_n) = \frac{P(x_1,...,x_n|c) P(c)}{P(x_1,...,x_n)}

Все признакия x_i являются зависимыми друг от друга, они описывают какой-то объект, который относится к классу c. Так как они зависимы, то вероятность P(x_1,...,x_n|c) вычисляется через правило совместной вероятности:

P(x_1,...,x_n|c) = P(x_1|c) P(x_2|c, x_1) P(x_3|c, x_1, x_2) \cdot ... \cdot P(x_n|c, x_1,...,x_{n-1})

Вычисление такой вероятности вычислительно сложно, поэтому делается допущение, что признаки независимы, тогда вычисление вероятности упрощается.

Допустим, что каждый признак x_i независим от других, тогда вероятность P(x_1,...,x_n|c) будет рассчитываться так:

P(x_1,...,x_n|c) = P(x_1|c) \cdot P(x_2|c) \cdot ... \cdot P(x_n|c) = \prod_{i = 1}^{n} P(x_i|c)

Вероятность P(x_1,...,x_n) вычисляется так же по правилу совместной вероятности:

P(x_1,...,x_n) = P(x_1) \cdot P(x_2|x_1) \cdot ... \cdot P(x_n|x_1,...,x_{n-1})

если признаки независимы, то:

P(x_1,...,x_n) = P(x_1) \cdot P(x_2) \cdot ... \cdot P(x_n)

НБК делает предположение о независимости признаков друг от друга при заданном классе, поэтому он называется наивным, это упрощает вычисления, но не всегда верно в реальных данных (почти всегда не верно), так как признаки определенно зависят друг от друга.

Мультиномиальный НБК

Мультиномиальный НБК является одним из вариантов НБК, специально предназначенный для работы с дискретными признаками, которые распределены мультиномиально. Этот тип НБК является мощным инструментом для классификации текстовых данных и других задач, где признаки имеют мультиномиальное распределение.

Теорема Байеса позволяет рассчитать апостериорную вероятность наступления события c при условии, что событие x произошло, следующим образом:

P(c_k|x) = {{\frac{1}{P(x)} P(x|c_k) \cdot P(c_k)}} = \frac{1}{P(x)} P(c_k) \cdot \prod_{i = 1}^{n} p(x_i|c_k)

P(c_k|x) — апостериорная вероятность класса c_k при условии признаков x (задача в том, чтобы вычислить ее), P(x|c_k) — правдоподобие априорных вероятностей признаков для класса c_k, P(c_k) — априорная вероятность классаc_k, P(x) — априорная вероятность признаков x, n — общее число различных признаков в наборе данных, k — число различных классов в наборе данных.

Априорные вероятности P(c_k) классов представляют собой вероятность появления каждого класса в наборе данных. Эти вероятности можно оценить как частота появления объектов каждого класса в обучающей выборке. Формула для вычисления:

P(c_k) = \frac{n_c}{n}

n_c — число объектов класса c_k, n — общее число объектов в обучающей выборке.

Априорные вероятности появления каждого признака для каждого класса вычисляются на основе обучающих данных. В случае мультиномиального распределения, это делается по формуле:

P(x_i|c_k) = \frac{n_{ic} + \alpha}{n_с + \alpha n_v}

n_{ic} — количество раз, когда признак x_i встречается в классе c_k (количество появлений данного признака в наборе данных для этого класса), n_c — общее количество всех признаков в классе c_k в наборе данных (общая сумма появлений каждого признака для класса c_k), \alpha — сглаживающий параметр, предотвращает появление нулевых вероятностей, n_v — число уникальных признаков (слов) в словаре (словарь — vocabulary). Если априорная вероятность какого-то признака будет нулевой, тогда вся дробь в формуле Байеса станет нулевой, это нам не нужно (их можно просто не учитывать).

Априорная вероятность P(x) наступления события x в знаменателе формулы, рассчитывается так:

P(x) = p(x_1) \cdot p(x_2) \cdot ... \cdot p(x_n)

Вероятность каждого признака рассчитывается как частота появления этого признака в наборе данных:

p(x_i) = \frac{n_{xi}}{n}

n_{xi} — число появления признака x_i в наборе данных, n — общее число появлений всех признаков в наборе данных.

Так как для всех новых данных знаменатель в формуле один и тот же, то на практике его не учитывают, остается только числитель. Апостериорные вероятности рассчитываются как произведение априорных вероятностей классов и априорных вероятностей признаков:

P(c_k|x) = P(c) \cdot \prod_{i = 1}^{n} P(x_i|c_k)

x — список признаков нового объекта, n — число признаков в списке (или общее число уникальных признаков в наборе данных). Класс с максимальной апостериорной вероятностью выбирается как прогноз, ответ модели.

Реализация мультиномиального НБК

Для проверки мультиномиального и бернулли будем использовать несколько предложений, позитивных, нейтральных и негативных тонов речи, которые записаны в отдельном файле.

Для начала создаем класс, который содержит общие поля и методы для всех трех видов НБК.

class NBC:
    def __init__(self):
        self.labels_probabilities = dict()
        self.unique_labels = list()

    def _unique_labels_(self, labels):
        return list(set(labels))

    def _labels_probabilities_(self, labels):
        label_score = {label: 0 for label in self.unique_labels}

        for label in labels:
            label_score[label] += 1

        return {label: label_score[label] / len(labels) for label in self.unique_labels}

Общие для всех — список уникальных меток классов и их априорные вероятности. Считаем число каждой метки в наборе данных и делим на общее число всех меток.

Далее создаем класс для мультиномиального НБК.

class MultinomialNBC(NBC):
    def __init__(self):
        NBC.__init__(self)

        self.unique_features = list()
        self.features_probabilities = dict()
        self.alpha = 1

    def _unique_features_(self, inputs):
        features = list()

        for inp in inputs:
            for feature in inp:
                if feature not in features:
                    features.append(feature)

        return features

Делаем дополнительные поля: уникальные признаки, их условные вероятности и параметр alpha для предотвращения нулевых вероятностей. В методе unique_features выбираются уникальные признаки feature из каждого объекта inp из списка inputs обучающего набора данных.

Далее делаем метод для расчета условных вероятностей признаков.

def _features_probabilities_(self, inputs, labels):
    probabilities = {label: dict() for label in self.unique_labels}

    for label in self.unique_labels:
        probabilities[label] = {feature: 0 for feature in self.unique_features}
        features_count = 0

        for feature in self.unique_features:
            feature_count = self._feature_count_per_label_(feature, label, inputs, labels)
            probabilities[label][feature] = feature_count
            features_count += feature_count

        for feature in self.unique_features:
            numerator = probabilities[label][feature] + self.alpha
            denominator = features_count + self.alpha * len(self.unique_features)
            probabilities[label][feature] = numerator / denominator

    return probabilities

Создаем словарь probabilities, в котором будут храниться значения условных вероятностей для каждого уникального признака для каждой уникальной метки класса. Для каждого класса label считаем общее число появления всех признаков features_count. Для каждого уникального признака считаем число его появлений feature_count для конкретного класса label. Далее рассчитываем условные вероятности признаков по формуле, которая дана в теоретической части.

Чтобы посчитать число отдельного признака feature_count для данного класса label делаем отдельный метод.

def _feature_count_per_label_(self, feature, label, inputs, labels):
    feature_count = 0

    for i, inp in enumerate(inputs):
        if labels[i] == label:

            for inp_feature in inp:
                if inp_feature == feature:
                    feature_count += 1

    return feature_count

Проходимся по каждому объекту inp в списке всех объектов inputs и считаем число появлений feature_count рассматриваемого признака feature для данного класса label.

Теперь делаем метод fit, в котором будут рассчитываться все данные для модели.

def fit(self, inputs, labels, alpha=1):
    self.alpha = alpha
    self.unique_features = self._unique_features_(inputs)
    self.unique_labels = self._unique_labels_(labels)
    self.features_probabilities = self._features_probabilities_(inputs, labels)
    self.labels_probabilities = self._labels_probabilities_(labels)

Сначала нужно выделить уникальные признаки и метки, потом на их основе вычислять вероятности.

Теперь делаем метод для вычисления ответа модели. Делаем два метода: для расчета вероятности для конкретного класса и для определения класса, у которого максимальная вероятность.

def _predict_label_(self, label, inp):  # calculate posterior probability for each class
    inp_probability = 1

    for feature in inp:
        if feature in self.unique_features:
            inp_probability *= self.features_probabilities[label][feature]

    return inp_probability * self.labels_probabilities[label]

def predict(self, inp):
    probabilities = [self._predict_label_(label, inp) for label in self.unique_labels]
    max_item_idx = argmax(probabilities)

    return self.unique_labels[max_item_idx]  # return class with the highest posterior

Проходимся по каждому классу label в списке уникальных меток и вычисляем для него апостериорную вероятность. В методе predict_label происходит перемножение условных вероятностей всех входных признаков, затем происходит умножение на априорную вероятность данного класса label. После вычисления апостериорной вероятности для каждого класса выбираем метку класса с максимальным значением с помощью функции argmax.

def argmax(items):
    max_item_idx = 0

    for idx, item in enumerate(items):
        if item > items[max_item_idx]:
            max_item_idx = idx

    return max_item_idx

Эта функция возвращает индекс элемента с максимальным значением

Проверяем и тестируем с помощью набора данных.

from nbc_data import positive, negative, neutral

all_messages = positive[:-1] + negative[:-1] + neutral[:-1]
test_messages = [positive[-1], negative[-1], neutral[-1]]

translate_table = str.maketrans({char: '' for char in "!,.?"})

inputs = [inp.translate(translate_table).split() for inp in all_messages]
test_inputs = [inp.translate(translate_table).split() for inp in test_messages]

labels = [0 for _ in positive[:-1]]
labels += [1 for _ in negative[:-1]]
labels += [2 for _ in neutral[:-1]]

test_labels = [0, 1, 2]

mult_nbc = MultinomialNBC()
mult_nbc.fit(inputs, labels)

for i, msg in enumerate(test_inputs):
    print(test_labels[i], mult_nbc.predict(msg))

print("================================")
for i, msg in enumerate(inputs):
    print(labels[i], mult_nbc.predict(msg))
output:
0 0
1 1
2 2
================================
0 0
...
2 2

Из каждого предложения убираем ненужные символы или подстроки с помощью таблицы translate_table. Создаем ее с помощью метода maketrans и словаря соответствий. В этой таблице записывается соответствие определенной подстроки другой подстроке. В этом примере происходит замена каждого из четырех знаков препинания на «пустой символ». Применяем эту таблицу к каждому предложению в наборе данных методом translate и разделяем предложения на слова методом split. В итоге получается список из списков слов.

Далее делаем списки меток для каждого из положительных, нейтральных и отрицательных тонов речи. Создаем объект класса MultinimialNBC и обучаем модель. Затем проверяем результаты.

Перед тем, как передавать текст в модель, его было бы хорошо как-то предварительно обработать, например, привести все слова в начальную форму, чтобы одни и те же слова в разных склонениях рассматривались как одно и то же слово.

Гауссовский НБК

Этот тип НБК предназначен для классификации данных с непрерывными значениями признаков. Априорные вероятности каждого класса вычисляются как и для мультиномиального НБК.

Для каждого признака и каждого класса вычисляются средние значения и стандартные отклонения по обучающим данным.

Среднее значение:

\mu_k = \frac{1}{n} \sum_{j=1}^{n} x_{ij}

n — число всех i-тых признаков для класса c_k (число всех объектов, у которых этот признак не нулевой для класса c_k).

Стандартное отклонение:

\sigma_k = \sqrt{\frac{1}{n}\sum_{j=1}^{n} (x_{ij} - \mu_k)^2}

n — число всех i-тых признаков для класса c_k.

Используя нормальное распределение (гауссовское распределение), рассчитывается вероятностная плотность признаков (априорная вероятность признаков) для каждого класса. Другими словами происходит масштабирование признаков для каждого класса по нормальному распределению. Формула для нормального распределения:

P(x_i|с_k) = \frac{1}{\sqrt{2 \pi \sigma_k^2}} exp(-{\frac{(x_i - \mu_k)^2}{2 \sigma_k^2}})

x_i — значение i-го признака, μ_k — среднее значение i-го признака в классе c_k, \sigma_k — стандартное отклонение i-го признака в классе c_k.

Априорные вероятности для каждого признака рассчитываются аналогично как и для мультиномиального НБК, только с использованием формулы для расчета плотности вероятности. Нужно будет рассчитывать по всем признакам независимо от класса.

Апостериорные вероятности рассчитываются как произведение априорных вероятностей классов и вероятностных плотностей признаков нового объекта. Учитывая предположение, что признаки независимы:

P(с_k|x) = P(с_k) \cdot \prod_{i = 1}^{n} P(x_i|с_k)

x — вектор признаков, n — число признаков.

После того, как все было рассчитано, класс с максимальной апостериорной вероятностью выбирается как прогноз модели.

Реализация гауссовского НБК

Для проверки гауссовского используем генератор набора данных из sclearn.

Для этого НБК понадобятся среднее значение и значение стандартного отклонения. Создаем новый класс, наследуем его от класса NBC, в котором есть общие для всех поля и методы, и инициализируем его в методе init. Добавляем поля для средних значений и стандартного отклонения.

class GaussianNBC(NBC):
    def __init__(self):
        NBC.__init__(self)

        self.mean = dict()
        self.std = dict()

    def _mean_(self, feature_items):
        return sum([item for item in feature_items]) / len(feature_items)

    def _std_(self, feature_items):  # standart deviation
        mean = self._mean_(feature_items)
        squared_sum = sum([(item - mean) ** 2 for item in feature_items])

        return sqrt(squared_sum / len(feature_items))

Делаем методы для вычисления среднего значения для списка чисел и для вычисления стандартного отклонения для списка чисел. Вычисление происходит по формулам, описанным в теоретической части.

Метод для того, чтобы выбирать i-й признак для определенного класса.

def _feature_items_per_label_(self, feature_idx, label, inputs, labels):
    feature_items = list()

    for i, inp in enumerate(inputs):
        if labels[i] == label:
            feature_items.append(inp[feature_idx])

    return feature_items

Создаем список для каждого значения i-го признака (feature_idx), который относится к данному класса label. Проходимся по каждому объекту в обучающих данных. Если метка объекта соответствует рассматриваемой метке label, то выбираем из этого объекта inp признак с данным feature_idx индексом. Таким образом мы отбираем нужные нам признаки из объектов, которые относятся к определенному классу. Далее на основе этого списка признаков будет рассчитываться среднее значение и стандартное отклонение.

Теперь делаем метод для отбора каждого i-го признака для каждого класса.

def _features_items_(self, inputs, labels):
    features_num = len(inputs[0])
    features_items = {label: dict() for label in self.unique_labels}

    for label in self.unique_labels:
        features_items[label] = {i: list() for i in range(features_num)}

        for i in range(features_num):
            features_items[label][i] = self._feature_items_per_label_(i, label, inputs, labels)

    return features_items

Создаем переменную с числом признаков для каждого объекта. Создаем словарь, индексами которого являются уникальные значения меток классов. Для каждого класса создается словарь, в котором хранятся списки i-ых признаков.

Проходимся по каждой уникальной метке класса и создаем для нее словарь, в котором будут храниться списки значений для каждого i-го признака. Для каждого i-го признака создаем список признаков с помощью метода feature_items_per_label.

Теперь делаем метод, в котором будут создаваться словари для хранения значений среднего и стандартного отклонения для каждого признака для каждого класса.

def _get_features_parameters_(self, inputs, labels):
    features_items = self._features_items_(inputs, labels)
    mean = {label: dict() for label in self.unique_labels}
    std = {label: dict() for label in self.unique_labels}

    for label, feature_items in features_items.items():
        mean[label] = {i: 0 for i in range(len(inputs[0]))}
        std[label] = {i: 0 for i in range(len(inputs[0]))}

        for feature_idx, items in feature_items.items():
            mean[label][feature_idx] = self._mean_(feature_items=items)
            std[label][feature_idx] = self._std_(feature_items=items)

    return mean, std

В нем создается словарь features_items, в котором хранятся списки для каждого признака для каждого класса. Например, список всех значений первых признаков, которые состоят в объектах первого класса, для объектов другого класса значения первых признаков учитываться не будет.

В словаре mean хранятся средние значения для каждого i-го признака для каждого из классов, в словаре std то же самое, только для стандартных отклонений.

Рассчитываем значения mean и std для каждой группы признаков и сохраняем их для модели.

Вызываем все эти методы в методе fit.

def fit(self, inputs, labels):
    self.unique_labels = self._unique_labels_(labels)
    self.labels_probabilities = self._labels_probabilities_(labels)
    self.mean, self.std = self._get_features_parameters_(inputs, labels)

Делаем метод, в котором будет происходить нормировка каждого признака для нового входного объекта. Нормировка производится на основе рассчитанных mean и std в методе fit.

def _gaus_scaling_(self, inp, label):
    scaled_inp = list()

    for idx, value in enumerate(inp):
        numerator = e ** (-(value - self.mean[label][idx]) ** 2 / (2 * self.std[label][idx] ** 2))
        denominator = sqrt(2 * pi * self.std[label][idx] ** 2)
        scaled_inp.append(numerator / denominator)

    return scaled_inp

Используем формулу нормального распределения, которая дана в теоретической части. Преобразуем каждый входной признак и получаем объект с отнормированными признаками.

Теперь делаем метод для получения ответа модели.

def _predict_label_(self, inp, label):  # calculate posterior probability for each class
    return self.labels_probabilities[label] * product(self._gaus_scaling_(inp, label))

def predict(self, inp):
    probabilities = [self._predict_label_(inp, label) for label in self.unique_labels]
    max_item_idx = argmax(probabilities)

    return self.unique_labels[max_item_idx]  # return class with the highest posterior

Проходимся по каждой уникальной метке класса и вычисляем апостериорную вероятность для этих классов. Выбираем метку класса, у которого максимальная вероятность.

Функция умножения product реализуется так.

def product(items):
    res = 1

    for item in items:
        res *= item

    return res

Проверяем работу модели.

from sklearn.model_selection import train_test_split
from sklearn.datasets import make_blobs

x_dataset, y_dataset = make_blobs(n_samples=150, n_features=4, centers=3, random_state=123)
x_train, x_test, y_train, y_test = train_test_split(x_dataset, y_dataset, test_size=0.2, random_state=123)

gauss_nbc = GaussianNBC()
gauss_nbc.fit(inputs=x_train, labels=y_train)

for i in range(len(x_test)):
    print(y_test[i], gauss_nbc.predict(x_test[i]))
output:
1 1
1 1
...
2 2

Создаем искусственный набор данных с помощью функции make_blobs. Указываем число признаков, классов и число объектов. Делим их на тестовые и обучающие с помощью функции train_test_split в отношении 20% к 80%. Обучаем модель и проверяем.

НБК Бернулли

НБК Бернулли - вариант НБК для работы с бинарными признаками, которые имеют бернуллиевское распределение и с двумя классами (0 или 1). В данном случае признаки представляют собой бинарные индикаторы наличия или отсутствия определённых характеристик у объекта, а ответ представляет собой вероятность одного из классов. Например, в таких задачах, как бинарная классификация текста, классификация электронных писем как спам или не спам, где каждое слово в тексте представлено бинарным признаком (присутствует или отсутствует).

Аналогичным образом вычисляются априорные вероятности каждого класса на основе обучающих данных, только теперь будет всего 2 класса.

Для НБК Бернулли, каждый признак является бинарным (0 или 1), соответственно, вероятности признаков P(x_i|c_k) представляют собой вероятность появления каждого бинарного признака x_i в объекте с классом c_k, то есть наличие или отсутствие в объекте. Эти вероятности можно вычислить так:

P(x_i = 1|c_k) = p_i, P(x_i = 0|c_k) = 1 - p_i

p_i — вероятность появления признака x_i в классе c_k, которая оценивается как частота появления этого признака в классе c_k в обучающей выборке,

p_i = \frac{n_c(x_i = 1)}{n}

n_c — число появлений i-го признака для класса c_k, n — число всех объектов класса c. То есть нужно посчитать число объектов с данным признаком и с данным классом и поделить на число всех объектов данного класса.

Для каждого признака вычисляются априорная вероятность по этой формуле:

P(x_i|c_k) = p_i x_i + (1 - p_i)(1 - x_i)

x_i — значение i-го признака (0 или 1) в рассматриваемом объекте, p_i — вероятность появления i-го признака в классе c. Если признак x_i присутствует в объекте (x_i= 1), то получится вероятность появления признака p_i, если признак отсутствует (x_i= 0), то получится вероятность не появления признака (1 - p_i). НБК Бернулли учитывает, что в ответе модели признаки не появляются во входном объекте.

Всего 2 класса, но как прогноз выбирается так же класс с максимальной апостериорной вероятностью. Можно вычислить сначала вероятность для одного класса и затем, чтобы вычесть вероятность второго класса, можно вычесть из единицы значение вероятности первого класса. Но для этого понадобится учесть знаменатель в формуле Байеса. Поэтому будет проще просто вычислить апостериорную вероятность для обоих классов без учета знаменателя.

Реализация НБК Бернулли

Создаем класс BernoullyNBC и наследуем его от MultinimialNBC, чтобы не переписывать готовый код. Нужно будет всего лишь изменить пару методов.

class BernoullyNBC(MultinomialNBC):
    def __init__(self):
        MultinomialNBC.__init__(self)

Переопределяем методы features_probabilities и feature_count_per_label, так как в этом НБК они рассчитываются по-другому.

def _feature_count_per_label_(self, feature, label, inputs, labels):
    feature_count, label_count = 0, 0

    for i, inp in enumerate(inputs):
        if labels[i] == label:
            label_count += 1

            for inp_feature in inp:
                if inp_feature == feature:
                    feature_count += 1
                    break  # до первого появления в примере

    return feature_count, label_count

  
def _features_probabilities_(self, inputs, labels):
    probabilities = {label: dict() for label in self.unique_labels}

    for label in self.unique_labels:
        probabilities[label] = {feature: 0 for feature in self.unique_features}

        for feature in self.unique_features:
            feature_count, label_count = self._feature_count_per_label_(feature, label, inputs, labels)
            probabilities[label][feature] = feature_count / label_count

    return probabilities

Нужно посчитать число появления (присутствия) каждого признака для каждого класса. В методе feature_count_per_label происходит подсчет числа определенного признака для определенного класса, учитывается лишь присутствие (0 или 1) этого признака, поэтому только до первого появления.

Потом в методе features_probabilities рассчитывается вероятность появления каждого признака. Нужно поделить число появления данного признака на число объектов, в которых присутствует этот признак. Формулы даны в теоретической части.

Так как для этого НБК важно только наличие признаков, то нужно сделать метод, в котором выделяются только уникальные признаки для нового входного объекта.

def _unique_features_per_input_(self, inp):
    features = list()

    for feature in inp:
        if feature not in features:
            features.append(feature)

    return features

Осталось только изменить метод для расчета числителя.

def _predict_label_(self, label, inp):
    inp_probability = 1
    unique_inp_features = self._unique_features_per_input_(inp)

    for feature in inp:
        if feature in self.unique_features:
            if feature in unique_inp_features:
                inp_probability *= self.features_probabilities[label][feature]

            else:
                inp_probability *= 1 - self.features_probabilities[label][feature]

    return inp_probability * self.labels_probabilities[label]

Числитель вычисляется тоже по-другому, формула дана в теоретической части. Если признак присутствует, то учитывается его условная вероятность присутствия, если отсутствует — учитывается его условная вероятность отсутствия.

Проверяем и тестируем.

from nbc_data import positive, negative

train_pos, train_neg = positive[:-2], negative[:-2]
test_pos, test_neg = positive[-2:], negative[-2:]

translate_table = str.maketrans({char: '' for char in "!,.?"})

inputs = [inp.translate(translate_table).split() for inp in train_pos + train_neg]
test_inputs = [inp.translate(translate_table).split() for inp in test_pos + test_neg]

labels = [0 for _ in train_pos]
labels += [1 for _ in train_neg]
test_labels = [0 for _ in test_pos]
test_labels += [1 for _ in test_neg]

bernoully_nbc = BernoullyNBC()
bernoully_nbc.fit(inputs, labels)

for i, msg in enumerate(inputs):
    print(labels[i], bernoully_nbc.predict(msg))

print("==========================")

for i, msg in enumerate(test_inputs):
    print(test_labels[i], bernoully_nbc.predict(msg))
output:
0 0
...
1 1
==========================
0 0
...
1 1

Преимущества и недостатки

НБК относительно прост в реализации.

Модель не нуждается в обучении, нужно лишь посчитать частоту, что делает его полезным для больших наборов данных.

Он хорошо работает с разреженными данными (с небольшим набором данных) и не требует масштабирования признаков.

НБК достаточно устойчив к шуму и отсутствующим значениям признаков.

Основным ограничением является предположение о независимости признаков, которое почти всегда не выполняется. Это может привести к плохим результатам, когда между признаками есть сильные корреляции.

Алгоритм не учитывает порядок и вес признаков, что может быть важным в некоторых задачах (возможно получится сделать дополнительную модель с весами, которые будут учитываться в НБК).

Если количество примеров каждого класса в наборе данных сильно отличается, то вероятность будет больше у классов с большим числом примеров, то есть, если набор данных несбалансирован, то вероятность будет больше у классов с большим числом примеров.

В целом, НБК является мощным и простым инструментом для решения задач классификации в машинном обучении, несмотря на его ограничения.

Заключение

Здесь была разобрана теоретическая часть работы основных трех видов НБК. Возможно есть какие-то неточности в описании моделей этих трех видов НБК, но основная идея каждого из них показана. Также дана формулировка задачи и рассказано, почему НБК называется наивным (наивное предположение о независимости признаков).

Я показал, как написать каждый из трех основных видов НБК с нуля. Получилось много кода и много замудренных объяснений, но это не проблема. Если есть желание, то очень легко можно в этом всем разобраться, нужно лишь провернуть эту всю логику в уме пару раз, тем более код достаточно понятный. Есть теоретическая часть, в которой есть все используемые здесь формулы и более подробное описание логики работы НБК.

Ссылка на папку с кодом из статьи на ГитХабе.

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  • 07.09.23 16:24 CherryTeam

    Cherry Team atlyginimų skaičiavimo programa yra labai naudingas įrankis įmonėms, kai reikia efektyviai valdyti ir skaičiuoti darbuotojų atlyginimus. Ši programinė įranga, turinti išsamias funkcijas ir patogią naudotojo sąsają, suteikia daug privalumų, kurie padeda supaprastinti darbo užmokesčio skaičiavimo procesus ir pagerinti finansų valdymą. Štai keletas pagrindinių priežasčių, kodėl Cherry Team atlyginimų skaičiavimo programa yra naudinga įmonėms: Automatizuoti ir tikslūs skaičiavimai: Atlyginimų skaičiavimai rankiniu būdu gali būti klaidingi ir reikalauti daug laiko. Programinė įranga Cherry Team automatizuoja visą atlyginimų skaičiavimo procesą, todėl nebereikia atlikti skaičiavimų rankiniu būdu ir sumažėja klaidų rizika. Tiksliai apskaičiuodama atlyginimus, įskaitant tokius veiksnius, kaip pagrindinis atlyginimas, viršvalandžiai, premijos, išskaitos ir mokesčiai, programa užtikrina tikslius ir be klaidų darbo užmokesčio skaičiavimo rezultatus. Sutaupoma laiko ir išlaidų: Darbo užmokesčio valdymas gali būti daug darbo jėgos reikalaujanti užduotis, reikalaujanti daug laiko ir išteklių. Programa Cherry Team supaprastina ir pagreitina darbo užmokesčio skaičiavimo procesą, nes automatizuoja skaičiavimus, generuoja darbo užmokesčio žiniaraščius ir tvarko išskaičiuojamus mokesčius. Šis automatizavimas padeda įmonėms sutaupyti daug laiko ir pastangų, todėl žmogiškųjų išteklių ir finansų komandos gali sutelkti dėmesį į strategiškai svarbesnę veiklą. Be to, racionalizuodamos darbo užmokesčio operacijas, įmonės gali sumažinti administracines išlaidas, susijusias su rankiniu darbo užmokesčio tvarkymu. Mokesčių ir darbo teisės aktų laikymasis: Įmonėms labai svarbu laikytis mokesčių ir darbo teisės aktų, kad išvengtų baudų ir teisinių problemų. Programinė įranga Cherry Team seka besikeičiančius mokesčių įstatymus ir darbo reglamentus, užtikrindama tikslius skaičiavimus ir teisinių reikalavimų laikymąsi. Programa gali dirbti su sudėtingais mokesčių scenarijais, pavyzdžiui, keliomis mokesčių grupėmis ir įvairių rūšių atskaitymais, todėl užtikrina atitiktį reikalavimams ir kartu sumažina klaidų riziką. Ataskaitų rengimas ir analizė: Programa Cherry Team siūlo patikimas ataskaitų teikimo ir analizės galimybes, suteikiančias įmonėms vertingų įžvalgų apie darbo užmokesčio duomenis. Ji gali generuoti ataskaitas apie įvairius aspektus, pavyzdžiui, darbo užmokesčio paskirstymą, išskaičiuojamus mokesčius ir darbo sąnaudas. Šios ataskaitos leidžia įmonėms analizuoti darbo užmokesčio tendencijas, nustatyti tobulintinas sritis ir priimti pagrįstus finansinius sprendimus. Pasinaudodamos duomenimis pagrįstomis įžvalgomis, įmonės gali optimizuoti savo darbo užmokesčio strategijas ir veiksmingai kontroliuoti išlaidas. Integracija su kitomis sistemomis: Cherry Team programinė įranga dažnai sklandžiai integruojama su kitomis personalo ir apskaitos sistemomis. Tokia integracija leidžia automatiškai perkelti atitinkamus duomenis, pavyzdžiui, informaciją apie darbuotojus ir finansinius įrašus, todėl nebereikia dubliuoti duomenų. Supaprastintas duomenų srautas tarp sistemų padidina bendrą efektyvumą ir sumažina duomenų klaidų ar neatitikimų riziką. Cherry Team atlyginimų apskaičiavimo programa įmonėms teikia didelę naudą - automatiniai ir tikslūs skaičiavimai, laiko ir sąnaudų taupymas, atitiktis mokesčių ir darbo teisės aktų reikalavimams, ataskaitų teikimo ir analizės galimybės bei integracija su kitomis sistemomis. Naudodamos šią programinę įrangą įmonės gali supaprastinti darbo užmokesčio skaičiavimo procesus, užtikrinti tikslumą ir atitiktį reikalavimams, padidinti darbuotojų pasitenkinimą ir gauti vertingų įžvalgų apie savo finansinius duomenis. Programa Cherry Team pasirodo esanti nepakeičiamas įrankis įmonėms, siekiančioms efektyviai ir veiksmingai valdyti darbo užmokestį. https://cherryteam.lt/lt/

  • 08.10.23 01:30 davec8080

    The "Shibarium for this confirmed rug pull is a BEP-20 project not related at all to Shibarium, SHIB, BONE or LEASH. The Plot Thickens. Someone posted the actual transactions!!!! https://bscscan.com/tx/0xa846ea0367c89c3f0bbfcc221cceea4c90d8f56ead2eb479d4cee41c75e02c97 It seems the article is true!!!! And it's also FUD. Let me explain. Check this link: https://bscscan.com/token/0x5a752c9fe3520522ea88f37a41c3ddd97c022c2f So there really is a "Shibarium" token. And somebody did a rug pull with it. CONFIRMED. But the "Shibarium" token for this confirmed rug pull is a BEP-20 project not related at all to Shibarium, SHIB, BONE or LEASH.

  • 24.06.24 04:31 tashandiarisha

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  • 26.06.24 18:46 Jacobethannn098

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  • 17.07.24 02:26 thompsonrickey

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  • 06.09.24 01:35 Celinagarcia

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  • 06.09.24 01:44 Celinagarcia

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  • 23.09.24 18:56 matthewshimself

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  • 01.10.24 14:54 Sinewclaudia

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  • 02.10.24 22:27 Emily Hunter

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  • 31.10.24 00:13 ytre89

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  • 02.11.24 14:44 diannamendoza732

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  • 12.11.24 00:50 TERESA

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  • 21.11.24 04:14 ronaldandre617

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  • 21.11.24 08:02 Emily Hunter

    If I hadn't found a review online and filed a complaint via email to support@deftrecoup. com , the people behind this unregulated scheme would have gotten away with leaving me in financial ruins. It was truly the most difficult period of my life.

  • 22.11.24 04:41 [email protected]

    I never could have imagined the nightmare of losing access to my digital wallet. All of my cryptocurrency holdings were abruptly imprisoned, inaccessible, and appeared to be lost forever following a catastrophic hardware breakdown. Years of meticulous investment and careful saving were reduced to nothing more than strings of code that I could no longer control, and I could feel the dread and sorrow that swept through me at that very instant. Thankfully, during my worst moment, I came into (TRUST GEEKS HACK EXPERT), a professional service devoted to recovering lost or inaccessible digital data. With optimism, I went out to their team of skilled technologists, laying bare the full nature of my issue. What followed was a laborious, multi-step process that required an almost surgical level of digital forensics and Bitcoin skill. In order to create a thorough profile of my wallet's contents and activities, the (TRUST GEEKS HACK EXPERT) team first thoroughly examined the transaction history and metadata connected to it. Next, they implemented a series of advanced recovery techniques, using cutting-edge software tools to bypass the access barriers that had left me locked out. The entire process was shrouded in secrecy and discretion, with the (TRUST GEEKS HACK EXPERT) team working tirelessly to protect the confidentiality of my sensitive financial information. After what felt like an eternity of nervous anticipation, the day finally arrived when I received the triumphant notification – my wallet had been successfully restored, and all of my precious digital assets had been returned to my control. The sense of relief was indescribable, as I could finally breathe easy knowing that the fruits of my financial discipline had been safeguarded. While the experience of losing access to my wallet was undoubtedly traumatic, (TRUST GEEKS HACK EXPERT) intervention allowed me to emerge from the ordeal with my cryptocurrency holdings intact, and a renewed appreciation for the importance of proactive digital asset management. You can contact Them through EMAIL: [email protected] - TELEGRAM: TRUSTGEEKSHACKEXPERT

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    Your Lost Bitcoins Are Not Gone Forever? Enquire From iBolt Cyber Hacker iBolt Cyber Hacker is a cybersecurity service that specializes in Bitcoin and cryptocurrency recovery. Even if your Bitcoin is locked away in a scammer inaccessible wallet, they have the tools and expertise to retrieve it. Many people, including seasoned cryptocurrency investors, face the daunting possibility of never seeing their lost funds again. iBolt cyber hacker service is a potential lifeline in these situations. I understand the concerns many people might have about trusting a third-party service to recover their Bitcoin. iBolt Cyber Hacker takes security seriously, implementing encryption and stringent privacy protocols. I was assured that no sensitive data would be compromised during the recovery process. Furthermore, their reputation in the cryptocurrency community, based on positive feedback from previous clients, gave me confidence that I was in good hands. Whtp +39, 351..105, 3619 Em.ail: ibolt @ cyber- wizard. co m

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    all thanks to Tech Cyber Force Recovery expert assistance. As a novice in cryptocurrency, I had been carefully accumulating a modest amount of Bitcoin, meticulously safeguarding my digital wallet and private keys. However, as the adage goes, the best-laid plans can often go awry, and that's precisely what happened to me. Due to a series of technical mishaps and human errors, I found myself locked out of my Bitcoin wallet, unable to access the fruits of my digital labors. Panic set in as I frantically searched for a solution, scouring the internet for any glimmer of hope. That's when I stumbled upon the Tech Cyber Force Recovery team, a group of seasoned cryptocurrency specialists who had built a reputation for their ability to recover lost or inaccessible digital assets. Skeptical at first, I reached out, desperate for a miracle. To my utter amazement, the Tech Cyber Force Recovery experts quickly assessed my situation and devised a meticulous plan of attack. Through their deep technical knowledge, unwavering determination, and a keen eye for detail, they were able to navigate the complex labyrinth of blockchain technology, ultimately recovering my entire Bitcoin portfolio. What had once seemed like a hopeless endeavor was now a reality, and I found myself once again in possession of my digital wealth, all thanks to the incredible efforts of the Tech Cyber Force Recovery team. This experience has not only restored my faith in the cryptocurrency ecosystem. Still, it has also instilled in me a profound appreciation for the critical role that expert recovery services can play in safeguarding one's digital assets.   ENAIL < Tech cybers force recovery @ cyber services. com >   WEBSITE < ht tps : // tech cyber force recovery. info  >   TEXT < +1. 561. 726. 3697 >

  • 24.11.24 02:21 [email protected]

    I never could have imagined the nightmare of losing access to my digital wallet. All of my cryptocurrency holdings were abruptly imprisoned, inaccessible, and appeared to be lost forever following a catastrophic hardware breakdown. Years of meticulous investment and careful saving were reduced to nothing more than strings of code that I could no longer control, and I could feel the dread and sorrow that swept through me at that very instant. Thankfully, during my worst moment, I came into (TRUST GEEKS HACK EXPERT), a professional service devoted to recovering lost or inaccessible digital data. With optimism, I went out to their team of skilled technologists, laying bare the full nature of my issue. What followed was a laborious, multi-step process that required an almost surgical level of digital forensics and Bitcoin skill. In order to create a thorough profile of my wallet's contents and activities, the (TRUST GEEKS HACK EXPERT) team first thoroughly examined the transaction history and metadata connected to it. Next, they implemented a series of advanced recovery techniques, using cutting-edge software tools to bypass the access barriers that had left me locked out. The entire process was shrouded in secrecy and discretion, with the (TRUST GEEKS HACK EXPERT) team working tirelessly to protect the confidentiality of my sensitive financial information. After what felt like an eternity of nervous anticipation, the day finally arrived when I received the triumphant notification – my wallet had been successfully restored, and all of my precious digital assets had been returned to my control. The sense of relief was indescribable, as I could finally breathe easy knowing that the fruits of my financial discipline had been safeguarded. While the experience of losing access to my wallet was undoubtedly traumatic, (TRUST GEEKS HACK EXPERT) intervention allowed me to emerge from the ordeal with my cryptocurrency holdings intact, and a renewed appreciation for the importance of proactive digital asset management. You can contact Them through EMAIL: [email protected] - TELEGRAM: TRUSTGEEKSHACKEXPERT

  • 25.11.24 02:19 briankennedy

    COMMENT ON I NEED A HACKER TO RECOVER MONEY FROM BINARY TRADING. HIRE FASTFUND RECOVERY

  • 25.11.24 02:20 briankennedy

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  • 26.11.24 21:59 [email protected]

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  • 26.11.24 23:12 rickrobinson8

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  • 26.11.24 23:12 rickrobinson8

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  • 27.11.24 00:39 [email protected]

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  • 27.11.24 09:10 Michal Novotny

    The biggest issue with cryptocurrency is that it is unregulated, wh ich is why different people can come up with different fake stories all the time, and it is unfortunate that platforms like Facebook and others only care about the money they make from them through ads. I saw an ad on Facebook for Cointiger and fell into the scam, losing over $30,000. I reported it to Facebook, but they did nothing until I discovered deftrecoup . c o m from a crypto community; they retrieved approximately 95% of the total amount I lost.

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  • 04.12.24 22:24 andreygagloev

    When I first heard about Bitcoin back in 2018, I was skeptical. The idea of a decentralized, digital currency seemed too good to be true. But I was intrigued as I learned more about the technology behind it and its potential. I started small, investing just a few hundred dollars, dipping my toes into the cryptocurrency waters. At first, it was exhilarating to watch the value of my investment grow exponentially. I felt like I was part of the future, an early adopter of this revolutionary new asset. But that euphoria was short-lived. One day, I logged into my digital wallet only to find it empty - my Bitcoin had vanished without a trace. It turned out that the online exchange I had trusted had been hacked, and my funds were stolen. I was devastated, both financially and emotionally. All the potential I had seen in Bitcoin was tainted by the harsh reality that with decentralization came a lack of regulation and oversight. My hard-earned money was gone, lost to the ether of the digital world. This experience taught me a painful lesson about the price of trust in the uncharted territory of cryptocurrency. While the technology holds incredible promise, the risks can be catastrophic if you don't approach it with extreme caution. My Bitcoin investment gamble had failed, and I was left to pick up the pieces, wiser but poorer for having placed my faith in the wrong hands. My sincere appreciation goes to MUYERN TRUST HACKER. You are my hero in recovering my lost funds. Send a direct m a i l ( muyerntrusted ( @ ) mail-me ( . )c o m ) or message on whats app : + 1 ( 4-4-0 ) ( 3 -3 -5 ) ( 0-2-0-5 )

  • 12.12.24 00:35 amandagregory

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  • 27.12.24 00:21 swiftdream

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