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Простыми словами о методе максимального правдоподобия и информации Фишера

Всем привет! Сегодня я хотел бы рассказать про метод максимального правдоподобия и информацию Фишера и еще несколько смежных тем, которые активно используются в машинном обучении и анализе данных. Расскажу я об этом просто, понятно и без воды, но с практическими примерами, в том числе на Python.

Сейчас я кратенько и водянисто расскажу о проблематике, а потом перейдем непосредственно к теме. Обещаю всю воду оставить исключительно здесь)

Итак, в свое время, на осознание метода максимального правдоподобия и информации Фишера я потратил достаточно большое количество времени и усилий, так как все, что находилось по этой теме в Интернете, было очень скомкано, намешано, лишено практического смысла и, ну, просто как-то фрагментировано. На данный момент, ситуация не сильно поменялась (если это не так, буду рад увидеть материалы в комментариях). Тем не менее, в конце статьи я оставлю дополнительные материалы, которые раскрывают то, что описано в статье либо глубже, либо компактнее, либо с другой стороны.

В данной статье я постараюсь изложить информацию таким образом, чтобы даже относительно малоподготовленный читатель смог понять как все устроено и работает на практике, и так, как предпочел, чтобы тему объяснили мне, то есть предоставлю инструментарий, объясню как им пользоваться в разных ситуациях и покажу это на практике.

Все же, некий бэкграунд у читателя должен быть:

средний уровень статистики, в целом;
дискретные и непрерывные распределения;
смысл производных первых двух порядков;
простейшие преобразования векторов и матриц;
свойства логарифмов;
понимание смысла пределов и асимптотики.

На этом, вроде бы, все, присаживайтесь поудобнее, заварите кофейку и достаньте печеньки, нам предстоит интересный путь.

P.S. Я достаточно формальный человек, но хотел бы написать эту статью на максимально понятном языке для простоты восприятия и нивелирования эффекта отторжения. Так что не бросайтесь камнями, формальщины в Интернете как раз-таки полно. С другой стороны, после скроллинга, не пугайтесь обилием формул, большинство из них относятся к разделу с примерами.

Содержание

Введение

В современном мире анализа данных и машинного обучения статистические методы занимают центральное место в построении надёжных и интерпретируемых моделей. И здесь как раз крайне важными темами являются метод максимального правдоподобия и информация Фишера, которые вместе образуют теоретическую и практическую базу для оценки параметров, анализа неопределённости и построения доверительных интервалов в широком спектре приложений.

Метод максимального правдоподобия (ММП) является фундаментальным подходом, предназначенным для получения оценок параметров, обладающих оптимальными асимптотическими свойствами (Свойства оценок правдоподобия) при соблюдении регулярных условий (Условия регулярности). Вся мощность данного метода проявляется в задачах, где минимизируются показатели ошибок вроде MSE, MAE, logloss, Huber loss и другие. Также, с помощью ММП мы можем оценить какое распределение лучше описывает наши данные. Для этого необходимо оценить параметры обоих распределений с использованием данного метода и сравнить значения функции правдоподобия.

Информация Фишера позволяет количественно оценить, насколько данные «информативны» относительно неизвестных параметров, то есть насколько много они содержат в себе информации. Данный показатель служит основой для определения асимптотической дисперсии оценок, полученных методом максимального правдоподобия. Этот показатель критически важен при анализе чувствительности моделей, построении доверительных интервалов и проведении гипотез тестирования, что достаточно актуально при работе с алгоритмами классификации и регрессионными задачами.

Хотел бы подчеркнуть, что по ходу статьи я буду ссылаться на условия регулярности, но каждый раз расписывать их не буду, а размещу их, вместе со следствиями и подробным разбором в конце статьи в разделе Условия регулярности статистической модели. Если хотите, можете подробно с ними ознакомиться там, однако понимание всех условий скорее вишенка на торте, а не крайняя необходимость, так что решайте сами. В любом случае, я там все расписываю достаточно понятным языком шаг за шагом.

Метод максимального правдоподобия

Теория

Правдоподобие (likelihood function) — это вероятность получить наблюдаемую выборку при конкретном значении параметра.

Оценка максимального правдоподобия — значение параметра, которое максимизирует правдоподобие.

Интуиция метода заключается в том, что существует некая генеральная совокупность, из которой к нам в руки попала некоторая выборка. Мы же, как исследователи, хотим понять каким образом устроена генеральная совокупность. Для этого мы делаем предположение, что выборка, которая к нам пришла, соответствует определенной модели. В простейшем случае, мы можем сформулировать предположение так: выборка к нам пришла из распределения с плотностью $f(x \ | \ \theta)$ . Параметр $\theta$ является константой, мы его не знаем и пытаемся оценить по выборке.

Поясню, чтобы не путаться в обозначениях, что под $\theta$ может подразумеваться как один, так и несколько параметров, то есть:

Если у нашей модели один параметр, он обозначается как $\theta$ , где $\theta$ — скаляр.
Если у модели несколько параметров, они тоже обозначаются как $\theta$ , где $\theta$ — вектор.

Но причем тут "максимальное" правдоподобие? Дело в том, что главная задача ММП заключается в том, чтобы максимизировать вероятность появления имеющейся выборки по параметру $\theta$ . То есть мы пытаемся оценить $\theta$ так, чтобы вероятность получить текущую выборку из определенной модели была максимальной.

Ключевым понятием данного метода является функция правдоподобия, поэтому давайте выпишем ее математическую постановку. Строго говоря, если мы работаем с дискретным распределением, тогда постановка будет следующей:

$L(\theta \ | \ x_1, ..., x_n) = \mathbb{P}(x_1, ..., x_n \ | \ \theta)$

если же речь идет о непрерывном распределении, тогда:

$L(\theta \ | \ x_1, ..., x_n) = f(x_1, ..., x_n \ | \ \theta)$

При этом, если выполняется условие независимости выборки, тогда мы можем разделить функцию плотности непрерывного распределения на маленькие множители:

$L(\theta \ | \ x_1, ..., x_n) = f(x_1 \ | \ \theta) \ \cdot \ ... \ \cdot \ f(x_n \ | \ \theta) = \prod_{i=1}^nf(x_i \ | \ \theta)$ Хорошо, но какое значение параметра более правдоподобно, а какое менее?

С этим все крайне просто. Говорят, что значение параметра $\theta_1$ более правдоподобно, нежели значения параметра $\theta_2$ , если выполняется условие:

$L(\theta_1 \ | \ x_1, ..., x_n) > L(\theta_2 \ | \ x_1, ..., x_n)$

Таким образом, вспоминая, что ключевой задачей является максимизация правдоподобия приходим к следующей постановке:

$L(\theta \ | \ x_1, ..., x_n) = \prod_{i=1}^nf(x_i \ | \ \theta) \to \max_{\theta}$

Более формально это записывается как:

$\hat\theta^{MLE}=\arg\max_\theta L(\theta \ | \ x_1, ..., x_n)$

Здесь под MLE подразумевается maximum likelihood estimation, то есть оценка максимального правдоподобия.

Теперь, когда у нас имеется постановка, ммы можем приступить к поиску той самой оптимальной оценки. Однако, для начала, давайте поймем, что, на самом деле, максимизация функции $L(\theta)$ равносильна максимизации натурального логарифма от нее. В свою очередь, полученный натуральный логарифм максимизировать гораздо проще, потому что по свойствам логарифма произведения превращаются в суммы, а с суммами работать намного удобнее. Таким образом, давайте откорректируем постановку:

$\ln{L}=\sum_{i=1}^n\ln{(f(x_i \ | \ \theta))} \to \max_{\theta}$

Итак, далее, чтобы решить данную задачу, мы должны взять производную от функции максимального правдоподобия и найти соответствующий экстремум. Математически, мы можем записать это следующим образом:

$\frac{\partial \ln{L}}{\partial \theta}=\sum_{i=1}^n\frac{\partial \ln{f(x_i \ | \ \theta)}}{\partial\theta} = 0$

В целом, первый этап пройден! Возможно, на данный момент, интуиция все еще не совсем ясна, но ниже я опишу по-этапно алгоритм решения и приведу несколько наглядных примеров с дискретным и непрерывным распределениями.

Полная схема

Теперь, когда мы изучили постановку задачи, давайте строго сформулируем алгоритм действий, а затем испытаем его на двух примерах.

Определяем функцию плотности:
$f(x \ | \ \theta) \text{ или } \mathbb{P}(x \ | \ \theta)$
Формулируем функцию правдоподобия:
$L(\theta \ | \ x) = f(x_1 \ | \ \theta) \cdot \ldots \cdot f(x_n \ | \ \theta)$
Сокращаем функцию и устанавливаем параметры для максимизации:
$\prod_{i=1}^nf(x_i \ | \ \theta) \to \max_{\theta}$
Логарифмируем функцию правдоподобия :
$\ln L = \ln\prod_{i=1}^nf(x_i \ | \ \theta)= \sum_{i=1}^n\ln f(x_i \ | \ \theta) \to \max_{\theta}$
Дифференцируем функцию правдоподобия по параметру $\theta$ :
$\frac{\partial \ln{L}}{\partial \theta}=\sum_{i=1}^n\frac{\partial \ln{f(x_i \ | \ \theta)}}{\partial \theta}$
Приравниваем к нулю и ищем решения:
$\sum_{i=1}^n\frac{\partial \ln{f(x_i \ | \ \theta)}}{\partial \theta} = 0$

Пример с непрерывным распределением

Теперь, переходя от теории к практике, давайте посмотрим на несколько примеров. Один из них будет для дискретного случая, а другой — для непрерывного. Суть примеров будет заключатся в том, что мы попытаемся найти оценку параметров распределений методом максимального правдоподобия.

Для первого примера возьмем нормальное распределение, которое имеет 2 параметра: $\mu$ и $\sigma^2$ . Убедимся, что $\hat\sigma^2=\cfrac{1}{n} \sum{(x_i - \bar{x})^2}$ , а $\hat\mu=\cfrac{1}{n} \sum{(x_i)}$ . Таким образом, параметр $\theta$ будет представлять из себя вектор.

Определяем функцию плотности:
$N(\mu, \sigma^2) = \frac{1}{\sqrt{2 \pi \sigma^2}} \cdot e^{-\cfrac{(x - \mu)^2}{2\sigma^2}}$
Формулируем функцию правдоподобия:
$L(\mu, \sigma^2 \ | \ x) = \frac{1}{\sqrt{2 \pi \sigma^2}} \cdot e^{-\cfrac{(x_1 - \mu)^2}{2\sigma^2}} \cdot ... \cdot \frac{1}{\sqrt{2 \pi \sigma^2}} \cdot e^{-\cfrac{(x_n - \mu)^2}{2\sigma^2}}$
Сокращаем функцию и устанавливаем параметры для максимизации:
$(2\pi\sigma^2)^{-\cfrac{n}{2}} \cdot e^{-\cfrac{\sum{(x_i - \mu)^2}}{2\sigma^2}} \to \max_{\mu, \sigma^2}$
Логарифмируем функцию правдоподобия:
$\begin{align} \ln{L} &=\ln\Bigg((2\pi\sigma^2)^{-\cfrac{n}{2}} \cdot e^{-\cfrac{\sum{(x_i - \mu)^2}}{2\sigma^2}}\Bigg) = \\ &= -\frac{n}{2} \cdot \ln(2\pi) -\frac{n}{2} \cdot \ln(\sigma^2) - \frac{1}{2\sigma^2} \cdot \sum(x_i-\mu)^2 \to \max_{\mu, \sigma^2} \end{align}$
Дифференцируем логарифмическую функцию правдоподобия по параметру $\theta$ :
- Производная по $\mu$ :
  $\frac{\partial \ln{L}}{\partial \mu} =\frac{1}{\sigma^2} \cdot \sum{(x_i - \mu)}$
- Производная по $\sigma^2$ :
  $\frac{\partial \ln{L}}{\partial \sigma^2} = -\frac{n}{2\sigma^2} + \frac{1}{2\sigma^4} \cdot \sum{(x_i - \mu)^2}$
Приравниваем к нулю и ищем решение:
$\begin{cases} \cfrac{1}{\hat\sigma^2} \cdot \sum{(x_i - \hat\mu)} = 0 \\ -\cfrac{n}{2\hat\sigma^2} + \cfrac{1}{2\hat\sigma^4} \cdot \sum{(x_i - \hat\mu)^2} = 0 \end{cases} \Rightarrow \begin{cases} \sum{(x_i - \hat\mu)} = 0 \\ -n + \cfrac{1}{\hat\sigma^2} \cdot \sum{(x_i - \hat\mu)^2} = 0 \end{cases}$
Получаем решение:
$\begin{cases} \hat\mu_{ML} = \cfrac{1}{n}\sum{x_i} = \bar{x} \\ \hat\sigma^2_{ML}= \cfrac{1}{n} \sum{(x_i - \bar{x})^2}=\hat\sigma^2 \end{cases}$

Таким образом, мы приходим к тому, что привычные нам формулы для нахождения оценочных среднего и дисперсии являются оптимальными с точки зрения ММП.

Пример с дискетным распределением

Следующий пример будет раскрывать принцип работы ММП с дискретным распределением, а именно с распределением Бернулли.

Допустим, что у нас есть некоторая выборка наблюдений о том, любит ли человек кофе или нет. В данном случае, вариант ответа “Люблю кофе” — 1, а “Не люблю кофе” — 0. Респонденты были опрошены независимо. Таким образом, наша выборка имеет следующий вид: x_1 = 1 , x_2 = 0 , x_3 = 1 , , x_n=0 .

Давайте построим таблицу вероятностей, которая будет выглядеть следующим образом:

$\bf{X_i}$	$\bf{0}$	$\bf{1}$
$\mathbb{P}(X_i = k)$

Таким образом, нашей задачей здесь является найти ML-оценку (maximum likelihood) для параметра .

Приступим к простейшим преобразованиям (здесь шаги 1-3 объединены для краткости):
$\begin{align} L(p \ | \ x_1, \ldots, x_n) &= \mathbb{P}(x_1 \ | \ p) \ \times \ldots \times \ \mathbb{P}(x_n \ | \ p) = \\ &= p \times (1-p) \times p \times \ldots \times (1-p) = \\ &= \bf{p^{\Sigma{x_i}} \times (1-p)^{n -{\Sigma{x_i}}} \to \max_p}\end{align}$
Прологарифмируем выражение:
$\begin{align} \ln{(L)} &= \ln{(p^{\Sigma{x_i}} \cdot (1-p)^{n -{\Sigma{x_i}}})} = \\ &= \sum{x_i} \cdot \ln{(p)} + (n - \sum{x_i}) \cdot \ln{(1 - p)} \to \max_p \end{align}$
Теперь нам необходимо найти производную по :
$\frac{\partial \ln{L}}{\partial p} = \frac{\sum{x_i}}{p} - \frac{(n - \sum{x_i})}{(1-p)}$
Далее над появляются колпачки (не забываем про это), так как мы приступаем к оцениванию параметра:
$\frac{\sum{x_i}}{\hat{p}} - \frac{(n - \sum{x_i})}{(1-\hat{p})} = 0$
Пользуясь простейшими преобразованиями получаем следующее решение:
$\hat{p}^{ML} = \frac{1}{n}\sum{x_i}=\bar{x}$

Вот и все! Неплохая получилась история: интересная, весёлая, порой немного грустная, а главное поучительная...
©️ Огузок

А мы двигаемся дальше к информации Фишера)

Информация Фишера

Одним из важнейших аспектов является то, как функция правдоподобия ведет себя вблизи точки максимума, то есть в ее окрестностях. Например, в окрестностях точки максимума функция может быть плоской, тогда, при перемещении вдоль функции в ее окрестностях, правдоподобие не будет меняться. Такая ситуация будет нам говорить о том, что текущая выборка предоставляет мало информации о значениях параметров.

В идеале, нам хочется, чтобы функция правдоподобия была выпуклой и очень острой, чтобы максимум был как можно более ярко выраженным. Считается, что чем более очевиден максимум функции, то есть чем более он выражен, тем больше выборка содержит в себе информации о параметрах модели. При увеличении количества наблюдений (то есть размера выборки), мы накапливаем информацию, что напрямую влияет на пикообразность функции правдоподобия.

За выпуклость функции, как известно из математического анализа, отвечает вторая производная. В статистике, вторая производная от функции правдоподобия называется наблюдаемой информацией Фишера.

Информация Фишера — это способ количественно оценить количество информации, которое одна случайная величина содержит о другой.

Представьте, что у вас есть две связанные между собой величины, например, рост и вес человека. Зная рост, можно примерно предсказать вес, и наоборот. То есть одна величина несет информацию о другой. Информация Фишера позволяет измерить, насколько много информации одна величина дает о другой.

Далее приведенные формулы будут подразумевать выполнение условий регулярности 1-5, которые на практике выполняются почти всегда. Плюс ко всему, на практике проще оценить информацию Фишера вторыми производными. А с канонической постановкой информации Фишера можно ознакомиться в Следствии №2 в Условия регулярности.

Наблюдаемая информация Фишера (Observed Fisher information) — это информация о параметре $\theta$ , которую несет конкретная реализация случайной величины . Она вычисляется по формуле:
$I_{obs}(\theta) = - \frac{\partial^2\ln{L}}{\partial\theta^2}$
Помним, что функция правдоподобия раскрывается и подставляются конкретные . Таким образом, — это конкретное наблюдаемое значение случайной величины .
Если же требуется оценить несколько параметров модели, формально говоря, вектор параметров, тогда информация Фишера будет представлена матрицей Гессе (гессианом).
$I_{obs}(\theta) = - \frac{\partial^2\ln{L}}{\partial\theta_i\partial\theta_j}=-H$
Важно понимать, что употребляя в формулах (гессиан), мы должны интерпретировать это как многомерный случай.
Ожидаемая/Теоретическая информация Фишера (Expected Fisher information) — это среднее количество информации, которое случайная величина дает о неизвестном параметре $\theta$ некоторого распределения. Так, если мы хотим взять не наблюдаемую, а теоретическую информацию, тогда нам необходимо взять математическое ожидание от гессиана, такая матрица будет называться информационной матрицей Фишера:
$I(\theta) = - \mathbb{E}\bigg[\frac{\partial^2\ln{L}}{\partial\theta_i\partial\theta_j}\bigg]=-\mathbb{E}(H)$
Смысл этой формулы в том, что информация тем больше, чем сильнее $\ln{L}$ меняется при изменении параметра $\theta$ . Вторая производная логарифма как раз измеряет эту чувствительность. Соответственно, такую величину следует интерпретировать как крутизну функции правдоподобия для одного наблюдения.

Важно отметить различия:

наблюдаемая информация Фишера зависит от конкретного значения наблюдений;
ожидаемая (теоретическая) информация Фишера зависит исключительно от конкретного вида модели, подобранного для наблюдений, и отражает какую информацию вносит в правдоподобие одно наблюдение в среднем.

Теперь откатимся немного назад и с текущими знаниями проговорим пару моментов.

Следствием выдвинутых условий регулярности является то, что мы можем вывести формулу для информации Фишера, в двух разных вариантах, а именно:
$I_1(\theta) = \mathbb{E}\bigg[\bigg(\frac{\partial \ln{f(x \ | \ \theta)}}{\partial \theta}\bigg)^2\bigg] = - \mathbb{E}\bigg[\frac{\partial^2 \ln{f(x \ | \ \theta)}}{\partial \theta^2}\bigg]$
Эта формула используется для одного теоретического наблюдения, при расчете наблюдаемой информации Фишера, знаки математического ожидания сгорают и меняется на $x_1, \ldots, x_n$ .
Важно понимать, что теоретическая информация Фишера по всем наблюдениям это $n \cdot I_1(\theta)$ в то время, как при подсчете эмпирической информации по всей выборке придется по-честному посчитать вторую производную от при каждом и сложить их все, так как разные наблюдаемые значения, несут в себе разную информацию.
Таким образом, если к нам приходят наблюдения из хвостов выборки, они будут нести больше информации Фишера, чем те, что приходят к нам из средиземья. Это происходит по той причине, что наблюдения пришедшие из крайнеземья дают нам больше информации о вариативности.

На этом с информацией Фишера aka дисперсией ML-оценки у нас все) Осталось только обсудить эффективность

Неравенство Рао-Крамера

Далее, важным фактом является то, что, если функция плотности распределения удовлетворяет условиям регулярности №1-4 и $\hat\theta$ является несмещенной оценкой параметра $\theta$ , тогда выполняется так называемое неравенство Рао-Крамера:

$Var(\hat\theta) \ge I_n(\theta)^{-1}$

И на этом моменте давайте договоримся записывать $I(\theta)$ в подобных случаях как $I(\theta)^{-1}$ , а не в виде $\frac{1}{I(\theta)}$ , подразумевая многомерный случай.

Этим неравенством подразумевается, что если модель регулярная, тогда мы не сможем получить оценку со "слишком быстро" стремящейся к нулю дисперсией. В свою очередь, дисперсия оценки является некоторой мерой ее эффективности, говорящая о том, как быстро она сходится к необходимому качеству.

Важно отметить, что вне условий, выдвинутых для формулирования неравенства Рао-Крамера, существуют "сверхэффективные" оценки, чья дисперсии стремится к нулю быстрее, нежели для тех оценок, которые удовлетворяют поставленным условиям. В частности, для $\mathbb{U}(0, \theta)$ (равномерного распределения с неизвестным правым концом) оценка $Var(\hat\theta_{ML})$ стремится к нулю как $\frac{1}{n^2}$ , в то время, как для регулярных моделей $Var(\hat\theta_{ML})$ стремится к нулю как $\frac{1}{n}$ , то есть медленнее, по той причине, что $I_n(\hat\theta)=n \cdot I_1(\hat\theta)$ , где — размер выборки.

Довольно часто можно перепутать смысл неравенства Рао-Крамера и информации Фишера. Неравенство Рао-Крамера говорит о дисперсии $\hat\theta_{ML}$ (ML-оценки), а информация Фишера — о дисперсии $\frac{\partial\ln{L}}{\partial\theta}$ (функции счетчика), которая обеспечивает информативность данных.

Свойства оценок правдоподобия

Если выполнены Условия регулярности №1-6, тогда решение уравнения $\frac{\partial \ln{L}}{\partial \theta}=0$ действительно является оценкой максимального правдоподобия. То есть, если приведенные условия выполнены, тогда наша оценка гарантировано будет максимумом, а не просто экстремумом и проверять ничего не нужно.

Непосредственно сами свойства:

состоятельность — $\lim_{n \to \infty}\hat\theta \stackrel{p}{\to} \theta$ ;
асимптотическая нормальность — $\hat\theta \stackrel{asy}{\sim} N(\theta, I_1(\theta)^{-1})$ — среди всех асимптотически нормальных оценок эта дисперсия глобально самая маленькая.
асимптотическая эффективность — $\lim_{n \to \infty}Var(\hat\theta)=I_n(\theta)^{-1}$ — это означает, что при увеличении числа наблюдений в выборке, в пределе, дисперсия ML-оценки ложится на свою нижнюю границу, которую задает неравенство Рао-Крамера.
инвариантность — если — непрерывная функция, тогда $g(\hat\theta)$ оценка для $g(\theta)$ .

Тест отношения правдоподобий

Метод тестирования отношения правдоподобий позволяет сравнить, насколько хорошо данные объясняются моделью при выполнении ограничений (нулевая гипотеза H_0 ) по сравнению с моделью без этих ограничений (альтернативная гипотеза H_1 ).

Тест отношения правдоподобий заключается в том, что мы считаем $\hat\theta_{ML}$ по логарифмированной функции правдоподобия без ограничений и $\hat\theta_0$ с ограничениями. А дальше вычитаем их друг из друга и смотрим насколько большая получилась разность.

Но почему тогда "отношение", если мы вычитаем? Напомню, что работать проще с логарифмами функций правдоподобия, а после простейших преобразований с логарифмами, отношение превращается в вычитание.

Под упомянутыми в начале ограничениями здесь подразумевается фиксация одного из параметров и оптимизация по другому. Для нормального распределения мы можем зафиксировать $\mu$ и оптимизировать логарифмированную функцию правдоподобия по $\sigma^2$ .

Итак, переходим к алгоритм:

Формулируем гипотезы. В нулевой гипотезе задается определенное ограничение на параметры модели — $H_0: \ \theta = \theta_0$ . В альтернативной параметр(ы) не ограничены, то есть принимается, что данные могут иметь иное значение параметра — $H_a: \ \theta \ne \theta_0$ .
Формулируем функции правдоподобия для каждой из гипотез. Для гипотезы наша функция правдоподобия будет обозначаться как $\ln{L_{R}}$ (restricted), а для — $\ln{L_{UR}}$ (unrestricted).
Подставляем значения и вычисляем результат по каждой из функций правдоподобия.
Вычисляем статистику. После того, как мы получили некоторые значения для $\ln{L_{R}}$ и $\ln{L_{UR}}$ , нам необходимо вычислить интересующую нас статистик по следующей формуле $2(\ln{L_{UR}} - \ln{L_{R}}) \sim \chi^2_q$ . Утверждается, что такая статистика будет иметь распределение хи-квадрат с степенями свободы, где — количество наложенных ограничений.
Находим критическое значение статистики на интересующим нас уровне значимости $\alpha$ как делаем это всегда.
Сравниваем критическое значение с наблюдаемым. Если наблюдаемое значение слева от критического — не отвергается, что означает то, что расстояние очень мало, если наоборот, нулевая гипотеза отвергается и принимается .

Теперь, чтобы немного попрактиковаться давайте поведем тестирование отношения правдоподобий на Python. Мы будем работать со случайной выборкой из 100 наблюдений, которая пришла к нам из нормального распределения со следующими параметрами — N(0, 1) . Гипотеза H_0 будет заключаться в том, что $\mu=0.15$ , а H_a будет утверждать, что $\mu=\hat\mu$ , то есть обычному оценочному среднему. Приступим к решению:

import numpy as np
from scipy.stats import chi2, norm


np.random.seed(42)

n = 100          # размер выборки
mu_true = 0      # истинное среднее
sigma = 1        # известное стандартное отклонение
alpha = 0.05     # уровень значимости

# Генерируем выборку из нормального распределения
data = np.random.normal(loc=mu_true, scale=sigma, size=n)

# Оценка параметра при H1 (альтернативная гипотеза)
# MLE для параметра `mu` при нормальном распределении с известной дисперсией – выборочное среднее
mu_hat = np.mean(data)

# Вычисление логарифмов правдоподобия
logL_R = np.sum(norm.logpdf(data, loc=0.15, scale=sigma))     # при mu = 0.15 (H0)
logL_UR = np.sum(norm.logpdf(data, loc=mu_hat, scale=sigma))  # при mu = mu_hat (H1)

# Вычисление статистики теста: -2 * (logL_R - logL_UR)
test_statistic = -2 * (logL_R - logL_UR)
print("Статистика теста: {:.4f}".format(test_statistic))

# Вычисление критического значения для уровня значимости alpha = 0.05
critical_value = chi2.ppf(1 - alpha, df=1)
print("Критическое значение: {:.4f}".format(critical_value))

# Вычисление p-value c 1 степенью свободы, ибо мы ограничили один параметр
p_value = 1 - chi2.cdf(test_statistic, df=1)
print("p-value: {:.4f}".format(p_value))

# Принятие решения
if p_value < alpha:
    print("Отвергаем H0: данные не согласуются с предположением mu = 0.15")
else:
    print("Нет оснований отвергать H0: данные согласуются с предположением mu = 0.15")

Таким образом, если мы запустим приведенный выше код, то в выводе получим следующее:

Статистика теста: 6.4438
Критическое значение: 3.8415
p-value: 0.0111
Отвергаем H0: данные не согласуются с предположением mu = 0.15

Все отработало корректно. Понятно, что это крайне наивная имплементация, но для иллюстрации думаю, что более чем предостаточно.

Подытоживая, хотелось бы упомянуть, что тест на отношение правдоподобий еще крайне удобен тем, что в системе мы можем тестировать большое количество гипотез. Обычно в таких ситуациях пользуются поправкой Бонферрони или Холма для того, чтобы устранить эффект множественного сравнения. Однако в случае с тестом отношения правдоподобий корректировать уровень значимости не нужно.

Условия регулярности статистической модели

Условия

Условия регулярности не то, что бы обязательны к пониманию, однако очень желательны. В этом разделе я буду писать несколько суше и преимущественно в математических терминах, но постараюсь сглаживать некоторые углы и приводить максимально простые объяснения.

Важно отметить, что основных условий регулярности 4, однако есть дополнительные, которые необходимы, например, для выведения неравенства Рао-Крамера или определенных свойств ML-оценки.

Условие №1

$\mathbb{A} := \{x: f(x \ | \ \theta) > 0\} \text{ не зависит от } \theta$ Пояснение

То есть здесь требуется, чтобы область определения наших иксов не зависела от параметра $\theta$ . Например, данное условие нарушается для $\mathbb{U}(0, \theta)$ (равномерного распределения с неизвестным концом).

Условие №2

$f(x \ | \ \theta) \text{ непрерывно дифференцируема по } \theta \text{ на всем множестве } \mathbb{A}$ Пояснение

Тут все очевидно и без комментариев, могу лишь сказать, что данное условие нарушается, например, для $\mathbb{L}(\alpha, \theta)$ (распределение Лапласа с неизвестным параметром сдвига), так как внутри функции плотности $f(x)=\frac{\alpha}{2}e^{-\alpha|x-\beta|}$ в степени экспоненты содержится модуль, который мы не сможем продифференцировать.

Условие №3

Производная для $\int_\mathbb{A}f(x \ | \ \theta)dx$ в непрерывном случае и для $\Sigma_\mathbb{A}f(x \ | \ \theta)$ в дискретном
случае может быть вычислена путем изменения порядка дифференцирования и интегрирования/суммирования.

Пояснение

Это условие самое важное из всех, потому что позволяет с помощью некоторых математических преобразований получить математическое ожидание и дисперсию для оценки максимального правдоподобия, а также помогает вывести неравенство Рао-Крамера (о нем позже).

Условие №4

Вот такая величина $\mathbb{E}\bigg[\bigg(\cfrac{\partial \ln{f(x \ | \ \theta)}}{\partial \theta}\bigg)^2\bigg]$ положительна и конечна. То есть, с точки зрения математической постановки это выглядит так:

$0 < \mathbb{E}\bigg[\bigg(\frac{\partial \ln{f(x \ | \ \theta)}}{\partial \theta}\bigg)^2\bigg] < \infty$ Поянение

Позволю себе, немного забегая вперед (до Следствия №2), сказать, что эта величина является дисперсией для $\frac{\partial \ln{f(x \ | \ \theta)}}{\partial \theta}$ и по совместительству информацией Фишера. И здесь становится очевидно, что дисперсия должна быть конечной и положительной, в том числе потому, что мы будем использовать ее в знаменателе при расчете $Var(\hat\theta)$ (дисперсии оценки максимального правдоподобия).

И буквально самую малость про нулевую и "бесконечную" дисперсии, в общем:

Равная нулю дисперсия означает, что все значения в наборе данных одинаковы, то есть нет вариации или разброса данных. Причиной тому могут быть то, что все наблюдения в выборки действительно имеют одинаковое значение, или выборка слишком мала и не отражает вариативность в данных, или могла произойти ошибка при сборе или записи данных.

Типичный пример распределения с бесконечной дисперсией — распределение Коши. Сама же по себе бесконечная дисперсия означает, что теоретически разброс данных настолько велик, что дисперсия не может быть определена. И как раз для распределения Коши соответствующий интеграл расходится, что и является причиной бесконечной дисперсии. На самом деле, с точки зрения математики, выражение “бесконечная дисперсия” неверно, в таком случае уместнее говорить, что дисперсия не определена.

Условие №5 (дополнительное)

$f(x \ | \ \theta) \text{ дважды дифференцируема по } \theta \text{ на всем множестве } \mathbb{A}$ Пояснение

Это условие необходимо для того, чтобы мы могли посчитать информацию Фишера. Подробнее об этом можно прочитать в Следствии №2.

Условие №6 (дополнительное)

$f(x \ | \ \theta) \text{ трижды дифференцируема по } \theta \text{ на всем множестве } \mathbb{A}$ Пояснение

Данное условие, позволяет нам говорить о том, что оценка является асимптотически эффективной. Подробнее об этом можно прочитать в Следствии №3.

Следствия

А теперь давайте рассмотрим следствия выдвинутых условий. Самые важные из них заключаются в определении математического ожидания и дисперсии для $\frac{\partial \ln{f(x \ | \ \theta)}}{\partial \theta}$ , а также информации Фишера.

Следствие №1

Математическое ожидание формулируется следующим образом:

$\mathbb{E}\bigg[\frac{\partial \ln{f(x \ | \ \theta)}}{\partial \theta}\bigg]=0, \forall\theta \in \Theta$ Подробное объяснение

Итак, для самых любознательных постараюсь разложить на атомы как мы вообще приходим к этому.

Подготовка.

В начале было слово... Ой, не то. В начале было определение функции плотности распределения $1 = \int f(x)dx$ . Потом мы применили некоторые математические преобразования к $\frac{\partial}{\partial\theta}f(x \ | \ \theta)$ для того, чтобы перейти к работе математическим ожиданием:

$\frac{\partial f(x \ | \ \theta)}{\partial\theta} = \frac{\frac{\partial f(x \ | \ \theta)}{\partial\theta}}{f(x \ | \ \theta)} \cdot f(x \ | \ \theta) = \frac{\partial\ln{f(x \ | \ \theta)}}{\partial\theta} \cdot f(x \ | \ \theta)$

Поясняю, что мы сделали:

Первым шагом, мы поделили и умножили производную на одну и ту же величину.
На втором шаге, воспользовались следующим правилом дифференцирования логарифма:

$\frac{d}{dx}\ln{f(x)} = \frac{f'(x)}{f(x)}$

Основная часть

Теперь, давайте продифференцируем функцию плотности с обеих сторон, тогда получится:

$0=\int \frac{\partial f(x \ | \ \theta)}{\partial\theta}dx$

А дальше сделаем подстановку и получим:

$0=\int \frac{\partial\ln{f(x \ | \ \theta)}}{\partial\theta} \cdot f(x \ | \ \theta)dx=\mathbb{E}\bigg[\frac{\partial \ln{f(x \ | \ \theta)}}{\partial \theta}\bigg]$

Таким образом, мы и получаем нулевое математическое ожидание.

Следствие №2

Дисперсия функции счетчика формулируется следующим образом:

$Var\bigg[\frac{\partial \ln{f(x \ | \ \theta)}}{\partial \theta}\bigg]=\mathbb{E}\bigg[\bigg(\frac{\partial \ln{f(x \ | \ \theta)}}{\partial \theta}\bigg)^2\bigg]$ Подробное объяснение

Здесь все проще, так как у нас есть базовая формула на все случаи жизни:

$Var[X] = \mathbb{E}[X^2] + \mathbb{E}[X]^2$

Теперь давайте, произведем подстановку, тогда получится, что:

$\begin{align} Var\bigg[\frac{\partial \ln{f(x \ | \ \theta)}}{\partial \theta}\bigg] &= \mathbb{E}\bigg[\bigg(\frac{\partial \ln{f(x \ | \ \theta)}}{\partial \theta}\bigg)^2\bigg] + \mathbb{E}\bigg[\frac{\partial \ln{f(x \ | \ \theta)}}{\partial \theta}\bigg]^2 = \\ &= \mathbb{E}\bigg[\bigg(\frac{\partial \ln{f(x \ | \ \theta)}}{\partial \theta}\bigg)^2\bigg] + 0^2 = \\ &= \mathbb{E}\bigg[\bigg(\frac{\partial \ln{f(x \ | \ \theta)}}{\partial \theta}\bigg)^2\bigg] \end{align}$

Поясню, раз, как уже выяснилось из предыдущего пункта, математическое ожидание равно нулю, то есть $\mathbb{E}\bigg[\frac{\partial \ln{f(x \ | \ \theta)}}{\partial \theta}\bigg]=0$ , тогда возводя ноль в степень мы получаем ноль и, соответственно, остается только первое слагаемое.

Эта дисперсия является информацией Фишера, которую можно выразить в двух разных вариантах:

$I(\theta) = \mathbb{E}\bigg[\bigg(\frac{\partial \ln{f(x \ | \ \theta)}}{\partial \theta}\bigg)^2\bigg] = - \mathbb{E}\bigg[\frac{\partial^2 \ln{f(x \ | \ \theta)}}{\partial \theta^2}\bigg]$ Подробное объяснение

В начале было $1 = \int f(x)dx$ и Условие №4, которое позволяет нам менять операции интегрирования и дифференцирования местами.
Теперь давайте продифференцируем обе части $1 = \int f(x)dx$ , получим:
$0=\int \frac{\partial f(x \ | \ \theta)}{\partial\theta}dx$
Далее, поймем, что $\frac{\partial f(x \ | \ \theta)}{\partial\theta}$ мы можем выразить иначе:
$\frac{\partial f(x \ | \ \theta)}{\partial\theta} = \frac{\frac{\partial f(x \ | \ \theta)}{\partial\theta}}{f(x \ | \ \theta)} \cdot f(x \ | \ \theta) = \frac{\partial\ln{f(x \ | \ \theta)}}{\partial\theta} \cdot f(x \ | \ \theta)$
Сделаем подстановку:
$0=\int \frac{\partial\ln{f(x \ | \ \theta)}}{\partial\theta} \cdot f(x \ | \ \theta)dx$
Продифференцируем еще раз (работает Условие №5):
$0=\int \frac{\partial}{\partial \theta} \bigg[\frac{\partial\ln{f(x \ | \ \theta)}}{\partial\theta} \cdot f(x \ | \ \theta) \bigg]dx$
Получаем тождество из применения правила дифференцирования логарифма дважды:
$\begin{align} \frac{\partial^2f(x \ | \ \theta)}{\partial\theta^2} &= \frac{\partial}{\partial \theta} \bigg[\frac{\partial\ln{f(x \ | \ \theta)}}{\partial\theta} \cdot f(x \ | \ \theta) \bigg] = \\ &= \bigg[\frac{\partial^2\ln f(x \ | \ \theta)}{\partial\theta^2} \bigg] \cdot f(x \ | \ \theta) + \bigg[\frac{\partial\ln{f(x \ | \ \theta)}}{\partial\theta}\bigg] \bigg[\frac{\partial f(x \ | \ \theta)}{\partial\theta}\bigg] = \\ &= \bigg[\frac{\partial^2\ln f(x \ | \ \theta)}{\partial\theta^2} \bigg] \cdot f(x \ | \ \theta) + \bigg[\frac{\partial\ln{f(x \ | \ \theta)}}{\partial\theta}\bigg] \bigg[\frac{\partial\ln{f(x \ | \ \theta)}}{\partial\theta}\bigg] \cdot f(x \ | \ \theta) = \\ &= \bigg[\frac{\partial^2\ln f(x \ | \ \theta)}{\partial\theta^2} + \bigg(\frac{\partial\ln f(x \ | \ \theta)}{\partial\theta}\bigg)^2 \bigg] \cdot f(x \ | \ \theta) \end{align}$
Применяем тождество к интегралу:
$\begin{align} 0 &= \int \bigg[\frac{\partial^2\ln f(x \ | \ \theta)}{\partial\theta^2} + \bigg(\frac{\partial\ln f(x \ | \ \theta)}{\partial\theta}\bigg)^2 \bigg] \cdot f(x \ | \ \theta) = \\ &= \int \frac{\partial^2\ln f(x \ | \ \theta)}{\partial\theta^2} \cdot f(x \ | \ \theta) + \int \bigg(\frac{\partial\ln f(x \ | \ \theta)}{\partial\theta}\bigg)^2 \cdot f(x \ | \ \theta) \end{align}$
Отсюда следует, что:
$-\mathbb{E}\bigg[\frac{\partial^2\ln f(x \ | \ \theta)}{\partial\theta^2}\bigg] = \mathbb{E}\bigg[\bigg(\frac{\partial\ln f(x \ | \ \theta)}{\partial\theta}\bigg)^2\bigg]$

Соответственно, правое выражение у нас положительное, а левое — отрицательное, поэтому мы его делаем положительным домножая на -1. Итого, мы можем сформулировать информацию Фишера в двух выражениях:

$I(\theta) = \mathbb{E}\bigg[\bigg(\frac{\partial\ln f(x \ | \ \theta)}{\partial\theta}\bigg)^2\bigg] =-\mathbb{E}\bigg[\frac{\partial^2\ln f(x \ | \ \theta)}{\partial\theta^2}\bigg]$

Следствие №3

Последнее следствие касается свойств оценки максимального правдоподобия, а именно асимптотической эффективности. Для этого нам снова нужно дополнительное условие, а именно: $f(x \ | \ \theta)$ трижды дифференцируема по $\theta$ на всем множестве $\mathbb{A}$ .

В совокупности выполнение условий №3 и №4 гарантирует, что статистическая модель будет вести себя "хорошо" и позволит получать надежные оценки параметров и доверительные интервалы.

Заключение

Давайте подведем итоги, сформулировав ключевые тезисы:

Метод максимального правдоподобия позволяет оценить неизвестный параметр по имеющейся выборке.
Информация Фишера позволяет измерить объем информации, которую несет в себе одно наблюдение или выборка наблюдений в целом.
Тест отношения правдоподобий позволяет легко проверить гипотезу о том, что параметр или вектор параметров равны какому-то значению.

Таким образом, мы научились:

находить MLE (maximum likelihood estimation) интересующих нас параметров;
определять какая из двух оценок наиболее правдоподобна;
оценивать информативность выборки или конкретного наблюдения;
проводить тест отношения правдоподобий.

В целом, я постарался достаточно подробно разобрать данную тему, но данный материал может не покрывать всю ее глубину конкретно для вас, так что рекомендую как-нибудь на досуге ознакомиться с дополнительными материалами. Все же, к изучению сложных тем стоит подходить с разных сторон)

...

Буду крайне признателен фидбеку в комментариях. Например, вы можете предложить как улучшить материал/подачу. Также, если я где-то ошибся или опечатался, поправьте меня, так как БуКаВ действительно много)

Как и обещал, вот список хороших источников:

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17 февр. 2025 г.

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30.07.26 00:04 Ahmed

A really cool analysis, thank you. I was especially struck by how strictly the order is defined: data processing and formatting come first, with color only at the very end. This really saves you from the typical mistake of "first a pretty palette, then we figure out what the chart is for." And the palette validator with OKLCH + colorblindness check is absolutely fantastic; you almost never see that in regular tools. By the way, while reading about rank-trajectory and the logarithmic scale, I immediately remembered how convenient it is to analyze dynamics on charts in ExpertOption—I've been trading there for quite some time now. When the data is well visualized, decisions are noticeably easier and more relaxed. I also liked the point about "no more than eight colors" and the dual-axis ban. Strict restrictions sometimes actually produce better results than complete freedom. I'll try this approach myself.
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03.08.26 20:05 Philip

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03.08.26 20:06 Philip

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04.08.26 11:05 Kisnoles

Excellent analysis, thank you. I was particularly struck by how rigidly the skill sets the order: data processing and form come first, with color coming last. This really cures the habit of "first a pretty palette, then we'll figure it out." A palette validator using OKLCH + color blindness + WCAG is something most chart generators lack. And regarding the boundaries of competence, it's very clear. Where there's a self-checking loop (color), you delegate freely. Where there's a heuristic (form, data interpretation), you remain the final filter. This is a universal principle, not just for /dataviz. By the way, when you look at trading dashboards (including those of decent platforms like ExpertOption), it's immediately clear who thought about readability and who just threw in rainbow lines. Tools like this skill could greatly improve the quality of analytics. Thanks again for the detailed analysis – saved.
04.08.26 12:37 kimberlyhebertt6877

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04.08.26 12:37 kimberlyhebertt6877

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20.08.26 11:32 michaeldavenport218

I was recently scammed out of $53,000 by a fraudulent Bitcoin investment scheme, which added significant stress to my already difficult health issues, as I was also facing cancer surgery expenses. Desperate to recover my funds, I spent hours researching and consulting other victims, which led me to discover the excellent reputation of Capital Crypto Recover, I came across a Google post It was only after spending many hours researching and asking other victims for advice that I discovered Capital Crypto Recovery’s stellar reputation. I decided to contact them because of their successful recovery record and encouraging client testimonials. I had no idea that this would be the pivotal moment in my fight against cryptocurrency theft. Thanks to their expert team, I was able to recover my lost cryptocurrency back. The process was intricate, but Capital Crypto Recovery's commitment to utilizing the latest technology ensured a successful outcome. I highly recommend their services to anyone who has fallen victim to cryptocurrency fraud. For assistance contact [email protected] and on Telegram OR WhatsApp Number +1 (336)390-6684 via email: [email protected] you can visit his website: https://recovercapital.wixsite.com/capital-crypto-rec-1
23.08.26 20:02 leslieyee

Excellent analysis by /dataviz! I was particularly struck by how strictly the order is defined: "data meaning first, color last" and how the validator actually adjusts the palette according to OKLCH, color blindness, and WCAG standards. These rules greatly improve the quality of charts. By the way, a similar approach to clean and legible charts is highly valued in trading—for example, on the ExpertOption platform, the interface and price visualization are designed to ensure information is read instantly and without unnecessary noise.
25.08.26 13:44 lydiassmith567

HIRE A HACKER YOUR STOLEN CRYPTO RECOVERY / BTC / USDT / ETH WITH THE HELP OF CAPITAL CRYPTO RECOVER. I want to share my experience publicly regarding cryptocurrency wallet recovery, I highly recommend you to contact CAPITAL CRYPTO RECOVER, a professional private investigator in the Bitcoin world. They have a certified expert security team specializing in Bitcoin Recovery Services and have helped many people worldwide recover their lost funds. My wife and I were defrauded by an online manipulator posing as an experienced crypto investment professional. We lost $9.2 Million Stolen BTC in cryptocurrency and were left feeling homeless. After spending hours searching for a reliable crypto recovery service, I discovered CAPITAL CRYPTO RECOVER online. By patiently explaining my situation to their team, I was able to recover all my funds. Remarkably, my money was returned to my wallet in less than 24 hours. I am extremely grateful to CAPITAL CRYPTO RECOVER for their excellent assistance—they truly were a godsend in my difficult situation. If you have fallen victim to a cryptocurrency scam, you can reach CAPITAL CRYPTO RECOVER through the following channels Email: [email protected] OR Call/WhatsApp: +1 (336) 390-6684 Contact: [email protected] Website: https://recovercapital.wixsite.com/capital-crypto-rec-1
25.08.26 13:44 lydiassmith567

HIRE A HACKER YOUR STOLEN CRYPTO RECOVERY / BTC / USDT / ETH WITH THE HELP OF CAPITAL CRYPTO RECOVER. I want to share my experience publicly regarding cryptocurrency wallet recovery, I highly recommend you to contact CAPITAL CRYPTO RECOVER, a professional private investigator in the Bitcoin world. They have a certified expert security team specializing in Bitcoin Recovery Services and have helped many people worldwide recover their lost funds. My wife and I were defrauded by an online manipulator posing as an experienced crypto investment professional. We lost $9.2 Million Stolen BTC in cryptocurrency and were left feeling homeless. After spending hours searching for a reliable crypto recovery service, I discovered CAPITAL CRYPTO RECOVER online. By patiently explaining my situation to their team, I was able to recover all my funds. Remarkably, my money was returned to my wallet in less than 24 hours. I am extremely grateful to CAPITAL CRYPTO RECOVER for their excellent assistance—they truly were a godsend in my difficult situation. If you have fallen victim to a cryptocurrency scam, you can reach CAPITAL CRYPTO RECOVER through the following channels Email: [email protected] OR Call/WhatsApp: +1 (336) 390-6684 Contact: [email protected] Website: https://recovercapital.wixsite.com/capital-crypto-rec-1
30.08.26 15:37 [email protected]

Discovering I had been defrauded, I searched online for services claiming to recover stolen ETH. I contacted several companies, but none succeeded. Although some claimed they could trace assets or recover funds, I could not verify their success or recover my money. I later found SYLVESTER BRYANT Recovery through Google. After contacting him, he recovered 450,000,00. He can be reached at [email protected] or WhatsApp at +1 512 577 7957.
30.08.26 15:37 [email protected]

Discovering I had been defrauded, I searched online for services claiming to recover stolen ETH. I contacted several companies, but none succeeded. Although some claimed they could trace assets or recover funds, I could not verify their success or recover my money. I later found SYLVESTER BRYANT Recovery through Google. After contacting him, he recovered 450,000,00. He can be reached at [email protected] or WhatsApp at +1 512 577 7957.
01.09.26 11:16 lisawerth897

My Experience With Cryptocurrency — CAPITAL CRYPTO RECOVER I lost over $82,000 in Bitcoin after falling victim to a fraudulent online investment scheme. After realizing I had been scammed, I spent considerable time researching possible ways to recover my funds. During my research, I came across CAPITAL CRYPTO RECOVER and decided to contact them after reading their reported success record and encouraging client positive reviews and testimonials. Their team guided me through the recovery process, and I was grateful to successfully recover my lost cryptocurrency. The experience was challenging, but I appreciated the assistance and support I received throughout the process. I’m sharing my experience in the hope that it may encourage other victims to carefully research their options and verify any recovery service before proceeding. I will forever be thankful to you CAPITAL CRYPTO RECOVER 📧 [email protected] 🌐 Website: recovercapital.wixsite.com/capital-crypto-rec-1 📧 [email protected] 📞 Call/WhatsApp: +1 (336) 390-6684
01.09.26 11:16 lisawerth897

My Experience With Cryptocurrency — CAPITAL CRYPTO RECOVER I lost over $82,000 in Bitcoin after falling victim to a fraudulent online investment scheme. After realizing I had been scammed, I spent considerable time researching possible ways to recover my funds. During my research, I came across CAPITAL CRYPTO RECOVER and decided to contact them after reading their reported success record and encouraging client positive reviews and testimonials. Their team guided me through the recovery process, and I was grateful to successfully recover my lost cryptocurrency. The experience was challenging, but I appreciated the assistance and support I received throughout the process. I’m sharing my experience in the hope that it may encourage other victims to carefully research their options and verify any recovery service before proceeding. I will forever be thankful to you CAPITAL CRYPTO RECOVER 📧 [email protected] 🌐 Website: recovercapital.wixsite.com/capital-crypto-rec-1 📧 [email protected] 📞 Call/WhatsApp: +1 (336) 390-6684
01.09.26 17:34 Garry42

Really crazy world. These fraudsters go at any length to steal your hard earned funds. I have been a victim of a bitcoin scam about 7 months back. A Con artist gained access to my cashapp account through a phishing scam. They stole $409,000. I was really devastated. I did everything to get back my funds by contacting the FBI but they claimed there was nothing they could do. A friend told me about a recovery expert. He helps fight against various phishing and investment scams and they were able to help trace and recover my funds even though it took over 2 days. you can reach out to him through recoverydarek@gmail. com . I can guarantee his services are still active.
01.09.26 17:34 Garry42

Really crazy world. These fraudsters go at any length to steal your hard earned funds. I have been a victim of a bitcoin scam about 7 months back. A Con artist gained access to my cashapp account through a phishing scam. They stole $409,000. I was really devastated. I did everything to get back my funds by contacting the FBI but they claimed there was nothing they could do. A friend told me about a recovery expert. He helps fight against various phishing and investment scams and they were able to help trace and recover my funds even though it took over 2 days. you can reach out to him through recoverydarek@gmail. com . I can guarantee his services are still active.
02.09.26 03:43 kimberlyhebertt6877

I invested in bitcoin trading After losing $78.4 USDT) linked to a romance fraud scam worth of cryptocurrency through an online investment platform and later discovered it was a scam. After extensive research for recovery options, I contacted CAPITAL CRYPTO RECOVER based on positive client reviews and recommendations. Their professional security team guided me through the recovery process using advanced technology, and I was able to recover my lost cryptocurrency successfully. I am truly grateful for their support and assistance during such a difficult experience. I will advise you to contact CAPITAL CRYPTO RECOVER helped me recover my funds. For anyone facing similar issues, Website: https://recovercapital.wixsite.com/capital-crypto-rec-1 Email: [email protected] Telegram: @Capitalcryptorecover Contact: [email protected] WhatsApp/Text Number: +1 (336) 390-6684
02.09.26 03:43 kimberlyhebertt6877

I invested in bitcoin trading After losing $78.4 USDT) linked to a romance fraud scam worth of cryptocurrency through an online investment platform and later discovered it was a scam. After extensive research for recovery options, I contacted CAPITAL CRYPTO RECOVER based on positive client reviews and recommendations. Their professional security team guided me through the recovery process using advanced technology, and I was able to recover my lost cryptocurrency successfully. I am truly grateful for their support and assistance during such a difficult experience. I will advise you to contact CAPITAL CRYPTO RECOVER helped me recover my funds. For anyone facing similar issues, Website: https://recovercapital.wixsite.com/capital-crypto-rec-1 Email: [email protected] Telegram: @Capitalcryptorecover Contact: [email protected] WhatsApp/Text Number: +1 (336) 390-6684
04.09.26 21:51 Kovengray

Recovering your lost investment funds as the case might be, is not what you can do alone, you’d require the service of a trained recovery specialist. A recovery specialist is a person or a group of people who are well equipped to work around the brokerage network. They have vast knowledge about the whole network and have the right software and private keys to follow any transaction. I was ripped off trading online to an investment broker, good thing I got every penny back through the help of Gavin ray he’s a genius Contact : Gavinray78 at gmail com or WhatsApp +1 352 322 2096 It is also important to be patient and really calm during the process.
05.09.26 21:38 [email protected]

I invested 45,000 Euro, and later aggreviated to 198,000 Euro. I requested to place ‎Withdrawal of my funds to be paid to my Bank account. But nothing happened I was subjected to pay more fee until I can get my funds on my account, i got in touch with Theodore ryan here on this platform who had helped a lot of people, I followed all instructions and he legally got back my withheld funds, I got threatened by the company that if I don’t pay they will get my account frozen, all thanks to Theodore ryan and I highly recommend him to anyone dealing with an unregulated broker company… contact him on his Gmail - theodoreryan318@ gmail . com
09.09.26 21:31 lisawerth897

My Experience With Cryptocurrency — CAPITAL CRYPTO RECOVER I lost over $82,000 in Bitcoin after falling victim to a fraudulent online investment scheme. After realizing I had been scammed, I spent considerable time researching possible ways to recover my funds. During my research, I came across CAPITAL CRYPTO RECOVER and decided to contact them after reading their reported success record and encouraging client positive reviews and testimonials. Their team guided me through the recovery process, and I was grateful to successfully recover my lost cryptocurrency. The experience was challenging, but I appreciated the assistance and support I received throughout the process. I’m sharing my experience in the hope that it may encourage other victims to carefully research their options and verify any recovery service before proceeding. I will forever be thankful to you CAPITAL CRYPTO RECOVER 📧 [email protected] 🌐 Website: recovercapital.wixsite.com/capital-crypto-rec-1 📧 [email protected] 📞 Call/WhatsApp: +1 (336) 390-6684
09.09.26 21:31 lisawerth897

My Experience With Cryptocurrency — CAPITAL CRYPTO RECOVER I lost over $82,000 in Bitcoin after falling victim to a fraudulent online investment scheme. After realizing I had been scammed, I spent considerable time researching possible ways to recover my funds. During my research, I came across CAPITAL CRYPTO RECOVER and decided to contact them after reading their reported success record and encouraging client positive reviews and testimonials. Their team guided me through the recovery process, and I was grateful to successfully recover my lost cryptocurrency. The experience was challenging, but I appreciated the assistance and support I received throughout the process. I’m sharing my experience in the hope that it may encourage other victims to carefully research their options and verify any recovery service before proceeding. I will forever be thankful to you CAPITAL CRYPTO RECOVER 📧 [email protected] 🌐 Website: recovercapital.wixsite.com/capital-crypto-rec-1 📧 [email protected] 📞 Call/WhatsApp: +1 (336) 390-6684
09.09.26 23:22 Fraddy Pual

Hearing that these individuals are focusing on other people makes me very sad. I had a similar situation with them and lost a lot of money, but after learning about (Cruxcipherteam @ proton DoT me), whataqq:+168160-15021, telegram: @Cruxcipherteam I was able to get my money back. It's critical that we all be watchful and keep reporting these occurrences.
11.09.26 03:23 kimberlyhebertt6877

I invested in bitcoin trading After losing $78.4 USDT) linked to a romance fraud scam worth of cryptocurrency through an online investment platform and later discovered it was a scam. After extensive research for recovery options, I contacted CAPITAL CRYPTO RECOVER based on positive client reviews and recommendations. Their professional security team guided me through the recovery process using advanced technology, and I was able to recover my lost cryptocurrency successfully. I am truly grateful for their support and assistance during such a difficult experience. I will advise you to contact CAPITAL CRYPTO RECOVER helped me recover my funds. For anyone facing similar issues, Website: https://recovercapital.wixsite.com/capital-crypto-rec-1 Email: [email protected] Telegram: @Capitalcryptorecover Contact: [email protected] WhatsApp/Text Number: +1 (336) 390-6684
11.09.26 03:23 kimberlyhebertt6877

I invested in bitcoin trading After losing $78.4 USDT) linked to a romance fraud scam worth of cryptocurrency through an online investment platform and later discovered it was a scam. After extensive research for recovery options, I contacted CAPITAL CRYPTO RECOVER based on positive client reviews and recommendations. Their professional security team guided me through the recovery process using advanced technology, and I was able to recover my lost cryptocurrency successfully. I am truly grateful for their support and assistance during such a difficult experience. I will advise you to contact CAPITAL CRYPTO RECOVER helped me recover my funds. For anyone facing similar issues, Website: https://recovercapital.wixsite.com/capital-crypto-rec-1 Email: [email protected] Telegram: @Capitalcryptorecover Contact: [email protected] WhatsApp/Text Number: +1 (336) 390-6684
15.09.26 03:35 elioduncan

I fell victim to a freelance web development contract scam that ultimately cost me CAD 4,000. It all started when I came across a job posting on Indeed Canada, a popular online job portal. The position seemed perfect: a freelance web developer role for an established company looking for someone to design and build a fully functional e-commerce website. The job promised a high payment of CAD 20,000 upon successful completion, which sounded like a great opportunity for me to gain experience and earn decent pay.The employer, who introduced himself as a project manager from a "well-known" tech company, was very persuasive and professional in our initial communication. He explained that the project involved developing a user-friendly, responsive online store for their client and that they had a strict timeline to meet. He assured me that the payment would be made promptly after completing the tasks. However, before starting the work, he told me that I would need to pay an upfront fee of CAD 4,000 to cover certain software tools and licensing fees required for the project. This was supposedly a part of their company policy for freelance contractors, ensuring access to their premium resources. The idea of working on a professional project, coupled with the promise of a substantial payout, convinced me to pay the upfront fee.As soon as I made the payment, the project manager became increasingly difficult to reach. He initially responded to my emails and provided some vague instructions on what the project would entail, but as time went on, communication slowed to a complete halt. I never received the required tools or any proper project details. My emails went unanswered, and any attempts to contact the company were met with silence. After waiting for weeks, I realized that I had been scammed.Desperate to recover my money, I turned to TechY Force Cyber Retrieval, a service that specializes in helping victims of online scams. They helped me track the payment and took legal steps to pursue the fraudsters. Through their guidance, I was able to recover the full CAD 4,000. TechY Force Cyber Retrieval worked with my bank to reverse the transaction and liaised with the authorities to trace the scammer's details.This was a hard lesson, and I now know to be highly cautious about online job offers that require upfront payments. It is crucial to research companies thoroughly and avoid any job that seems too good to be true, especially when it involves paying money upfront. WhatsApp https://wa.link/2x6ktp Mail. [email protected]
15.09.26 15:45 lydiassmith567

HIRE A HACKER YOUR STOLEN CRYPTO RECOVERY / BTC / USDT / ETH WITH THE HELP OF CAPITAL CRYPTO RECOVER. I want to share my experience publicly regarding cryptocurrency wallet recovery, I highly recommend you to contact CAPITAL CRYPTO RECOVER, a professional private investigator in the Bitcoin world. They have a certified expert security team specializing in Bitcoin Recovery Services and have helped many people worldwide recover their lost funds. My wife and I were defrauded by an online manipulator posing as an experienced crypto investment professional. We lost $9.2 Million Stolen BTC in cryptocurrency and were left feeling homeless. After spending hours searching for a reliable crypto recovery service, I discovered CAPITAL CRYPTO RECOVER online. By patiently explaining my situation to their team, I was able to recover all my funds. Remarkably, my money was returned to my wallet in less than 24 hours. I am extremely grateful to CAPITAL CRYPTO RECOVER for their excellent assistance—they truly were a godsend in my difficult situation. If you have fallen victim to a cryptocurrency scam, you can reach CAPITAL CRYPTO RECOVER through the following channels Email: [email protected] OR Call/WhatsApp: +1 (336) 390-6684 Contact: [email protected] Website: https://recovercapital.wixsite.com/capital-crypto-rec-1
15.09.26 15:45 lydiassmith567

HIRE A HACKER YOUR STOLEN CRYPTO RECOVERY / BTC / USDT / ETH WITH THE HELP OF CAPITAL CRYPTO RECOVER. I want to share my experience publicly regarding cryptocurrency wallet recovery, I highly recommend you to contact CAPITAL CRYPTO RECOVER, a professional private investigator in the Bitcoin world. They have a certified expert security team specializing in Bitcoin Recovery Services and have helped many people worldwide recover their lost funds. My wife and I were defrauded by an online manipulator posing as an experienced crypto investment professional. We lost $9.2 Million Stolen BTC in cryptocurrency and were left feeling homeless. After spending hours searching for a reliable crypto recovery service, I discovered CAPITAL CRYPTO RECOVER online. By patiently explaining my situation to their team, I was able to recover all my funds. Remarkably, my money was returned to my wallet in less than 24 hours. I am extremely grateful to CAPITAL CRYPTO RECOVER for their excellent assistance—they truly were a godsend in my difficult situation. If you have fallen victim to a cryptocurrency scam, you can reach CAPITAL CRYPTO RECOVER through the following channels Email: [email protected] OR Call/WhatsApp: +1 (336) 390-6684 Contact: [email protected] Website: https://recovercapital.wixsite.com/capital-crypto-rec-1
05.10.26 21:26 [email protected]

I Invested 45,000 Euro, and later aggreviated to 198,000 Euro. I requested to place Withdrawal of my funds to be paid to my Bank account. But nothing happened I was subjected to pay more fee until I can get my funds on my account, i got in touch with Theodore ryan here on this platform who had helped a lot of people, I followed all instructions and he legally got back my withheld funds, I got threatened by the company that if I don’t pay they will get my account frozen, all thanks to Theodore ryan and I highly recommend him to anyone dealing with an unregulated broker company… contact him on his Gmail - theodoreryan318@ gmail . com
06.10.26 06:13 melissathomason573

Professional Crypto Recovery Testimonial After losing a significant amount of cryptocurrency through a scam on Twitter, I was unsure whether my Bitcoin funds could ever be recovered. The experience was extremely stressful, particularly because I encountered several individuals and services claiming they could help recover my assets, only to discover that some were fraudulent themselves. A friend eventually referred me to CAPITAL CRYPTO RECOVER. Their team handled my case professionally, discreetly, and efficiently. With their assistance, I successfully recovered approximately $130,000 worth of USDT in just over a week. My experience taught me the importance of being extremely cautious when seeking cryptocurrency recovery services, as scammers often target victims a second time. I am grateful to CAPITAL CRYPTO RECOVER for the assistance I received and wanted to share my experience with others who may be facing a similar situation. If you’ve been compromised, don’t lose hope—and be careful of fraudsters posing as saviors. CAPITAL CRYPTO RECOVER are real professionals in crypto recovery. I am a living proof of their effectiveness, you can reach them through Phone (Call/WhatsApp): +1 (336) 390-6684 Email: [email protected] Alternate Email: [email protected] Website: https://recovercapital.wixsite.com/capital-crypto-rec-1
06.10.26 06:13 melissathomason573

Professional Crypto Recovery Testimonial After losing a significant amount of cryptocurrency through a scam on Twitter, I was unsure whether my Bitcoin funds could ever be recovered. The experience was extremely stressful, particularly because I encountered several individuals and services claiming they could help recover my assets, only to discover that some were fraudulent themselves. A friend eventually referred me to CAPITAL CRYPTO RECOVER. Their team handled my case professionally, discreetly, and efficiently. With their assistance, I successfully recovered approximately $130,000 worth of USDT in just over a week. My experience taught me the importance of being extremely cautious when seeking cryptocurrency recovery services, as scammers often target victims a second time. I am grateful to CAPITAL CRYPTO RECOVER for the assistance I received and wanted to share my experience with others who may be facing a similar situation. If you’ve been compromised, don’t lose hope—and be careful of fraudsters posing as saviors. CAPITAL CRYPTO RECOVER are real professionals in crypto recovery. I am a living proof of their effectiveness, you can reach them through Phone (Call/WhatsApp): +1 (336) 390-6684 Email: [email protected] Alternate Email: [email protected] Website: https://recovercapital.wixsite.com/capital-crypto-rec-1
06.10.26 17:44 Darrell1717

I fell victim of a scam I invested to total of 5thousand into. I know it could be risky but after I trusted this man I was dating online he lured me into investing with him. I know hard I worked of my money so I wanted every way possible to get it back. I really appreciate Blockrecoveryexpert for successfully recovered my funds form this scammer. This is all I can say. You can reachout if you ever need help contact blockrecoveryexpert (at) gmail (dot)com
11.10.26 03:04 Patricia014

Most recovery companies will take your money again. I personally think they’re the same set of people running all of them. It’s a sad, unending cycle, and so many people fall for cloned websites. The only recovery agent I know is Agent Rottgen Raphael. I’ve visited their physical company address, and they’ll tell you if they can handle your case instead of taking upfront payment when they know they can’t help. I highly appreciate their support in recovering my money from the scammers’ wallet within 24 hours. RottgenRaphael@Gmail. com
12.10.26 19:34 lydiassmith567

HIRE A HACKER YOUR STOLEN CRYPTO RECOVERY / BTC / USDT / ETH WITH THE HELP OF CAPITAL CRYPTO RECOVER. I want to share my experience publicly regarding cryptocurrency wallet recovery, I highly recommend you to contact CAPITAL CRYPTO RECOVER, a professional private investigator in the Bitcoin world. They have a certified expert security team specializing in Bitcoin Recovery Services and have helped many people worldwide recover their lost funds. My wife and I were defrauded by an online manipulator posing as an experienced crypto investment professional. We lost $9.2 Million Stolen BTC in cryptocurrency and were left feeling homeless. After spending hours searching for a reliable crypto recovery service, I discovered CAPITAL CRYPTO RECOVER online. By patiently explaining my situation to their team, I was able to recover all my funds. Remarkably, my money was returned to my wallet in less than 24 hours. I am extremely grateful to CAPITAL CRYPTO RECOVER for their excellent assistance—they truly were a godsend in my difficult situation. If you have fallen victim to a cryptocurrency scam, you can reach CAPITAL CRYPTO RECOVER through the following channels Email: [email protected] OR Call/WhatsApp: +1 (336) 390-6684 Contact: [email protected] Website: https://recovercapital.wixsite.com/capital-crypto-rec-1
12.10.26 19:34 lydiassmith567

HIRE A HACKER YOUR STOLEN CRYPTO RECOVERY / BTC / USDT / ETH WITH THE HELP OF CAPITAL CRYPTO RECOVER. I want to share my experience publicly regarding cryptocurrency wallet recovery, I highly recommend you to contact CAPITAL CRYPTO RECOVER, a professional private investigator in the Bitcoin world. They have a certified expert security team specializing in Bitcoin Recovery Services and have helped many people worldwide recover their lost funds. My wife and I were defrauded by an online manipulator posing as an experienced crypto investment professional. We lost $9.2 Million Stolen BTC in cryptocurrency and were left feeling homeless. After spending hours searching for a reliable crypto recovery service, I discovered CAPITAL CRYPTO RECOVER online. By patiently explaining my situation to their team, I was able to recover all my funds. Remarkably, my money was returned to my wallet in less than 24 hours. I am extremely grateful to CAPITAL CRYPTO RECOVER for their excellent assistance—they truly were a godsend in my difficult situation. If you have fallen victim to a cryptocurrency scam, you can reach CAPITAL CRYPTO RECOVER through the following channels Email: [email protected] OR Call/WhatsApp: +1 (336) 390-6684 Contact: [email protected] Website: https://recovercapital.wixsite.com/capital-crypto-rec-1
13.10.26 14:14 lisalisa22

Legitimate Cryptocurrency Recovery Services: Consult ZEUS CRYPTO RECOVERY SERVICES Zeus Crypto Recovery Services supports individuals exploring options after cryptocurrency losses from fraud, unauthorised transfers, or wallet access issues. Services may include blockchain analysis, wallet tracing, and evidence gathering for investigations. Each case is reviewed. Zeus Crypto Recovery Services is a trusted company with the highest success rate among cryptocurrency recovery services in 2025/2026 and is highly recommended if you have been scammed. ZEUS CRYPTO RECOVERY SERVICES handles case such as Scammed USDT Lost BTC, Coinbase Wallet Forgotten password, ETH and coins, etc. Anything that has to do with cryptocurrency or lost digital assets, ZEUS can help you Do you Need Help? Website https://zeusrecoveryservices.com Mailbox [email protected] WhatsApp number +447798816938 Telegram https://t.me/Zeuscryptorecoveryservices
13.10.26 14:14 lisalisa22

Legitimate Cryptocurrency Recovery Services: Consult ZEUS CRYPTO RECOVERY SERVICES Zeus Crypto Recovery Services supports individuals exploring options after cryptocurrency losses from fraud, unauthorised transfers, or wallet access issues. Services may include blockchain analysis, wallet tracing, and evidence gathering for investigations. Each case is reviewed. Zeus Crypto Recovery Services is a trusted company with the highest success rate among cryptocurrency recovery services in 2025/2026 and is highly recommended if you have been scammed. ZEUS CRYPTO RECOVERY SERVICES handles case such as Scammed USDT Lost BTC, Coinbase Wallet Forgotten password, ETH and coins, etc. Anything that has to do with cryptocurrency or lost digital assets, ZEUS can help you Do you Need Help? Website https://zeusrecoveryservices.com Mailbox [email protected] WhatsApp number +447798816938 Telegram https://t.me/Zeuscryptorecoveryservices
17.10.26 12:58 melissathomason573

Professional Crypto Recovery Testimonial After losing a significant amount of cryptocurrency through a scam on Twitter, I was unsure whether my Bitcoin funds could ever be recovered. The experience was extremely stressful, particularly because I encountered several individuals and services claiming they could help recover my assets, only to discover that some were fraudulent themselves. A friend eventually referred me to CAPITAL CRYPTO RECOVER. Their team handled my case professionally, discreetly, and efficiently. With their assistance, I successfully recovered approximately $130,000 worth of USDT in just over a week. My experience taught me the importance of being extremely cautious when seeking cryptocurrency recovery services, as scammers often target victims a second time. I am grateful to CAPITAL CRYPTO RECOVER for the assistance I received and wanted to share my experience with others who may be facing a similar situation. If you’ve been compromised, don’t lose hope—and be careful of fraudsters posing as saviors. CAPITAL CRYPTO RECOVER are real professionals in crypto recovery. I am a living proof of their effectiveness, you can reach them through Phone (Call/WhatsApp): +1 (336) 390-6684 Email: [email protected] Alternate Email: [email protected] Website: https://recovercapital.wixsite.com/capital-crypto-rec-1
17.10.26 12:58 melissathomason573

Professional Crypto Recovery Testimonial After losing a significant amount of cryptocurrency through a scam on Twitter, I was unsure whether my Bitcoin funds could ever be recovered. The experience was extremely stressful, particularly because I encountered several individuals and services claiming they could help recover my assets, only to discover that some were fraudulent themselves. A friend eventually referred me to CAPITAL CRYPTO RECOVER. Their team handled my case professionally, discreetly, and efficiently. With their assistance, I successfully recovered approximately $130,000 worth of USDT in just over a week. My experience taught me the importance of being extremely cautious when seeking cryptocurrency recovery services, as scammers often target victims a second time. I am grateful to CAPITAL CRYPTO RECOVER for the assistance I received and wanted to share my experience with others who may be facing a similar situation. If you’ve been compromised, don’t lose hope—and be careful of fraudsters posing as saviors. CAPITAL CRYPTO RECOVER are real professionals in crypto recovery. I am a living proof of their effectiveness, you can reach them through Phone (Call/WhatsApp): +1 (336) 390-6684 Email: [email protected] Alternate Email: [email protected] Website: https://recovercapital.wixsite.com/capital-crypto-rec-1
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