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Statistics and maths / Statistical inference

Bayesian vs Frequentist Dart Game

13 min read, 14 sections

Contents
  1. Bayesian vs Frequentist Fundamentals
  2. Maximum Likelihood Estimation Implementation
  3. Bayesian Parameter Estimation
  4. Frequentist A/B Testing Implementation
  5. Bayesian A/B Testing
  6. Bayesian Linear Regression
  7. Frequentist Time Series Analysis
  8. Bayesian Time Series Modeling
  9. Bayesian Neural Networks
  10. Frequentist vs Bayesian Model Selection
  11. Implementing Bayesian Optimization
  12. Bayesian Change Point Detection
  13. Frequentist vs Bayesian Clustering
  14. Additional Resources

Bayesian vs Frequentist Fundamentals

In statistical inference, understanding the core differences between Bayesian and Frequentist approaches is crucial. This implementation demonstrates basic probability calculations using both methodologies through a coin flip experiment simulation.

import numpy as np
from scipy import stats

class CoinFlipExperiment:
    def __init__(self, true_probability=0.5):
        self.true_prob = true_probability
        self.data = []
        
    def flip_coin(self, n_flips):
        return np.random.binomial(1, self.true_prob, n_flips)
    
    def frequentist_estimate(self, n_flips):
        flips = self.flip_coin(n_flips)
        self.data.extend(flips)
        return np.mean(self.data)
    
    def bayesian_estimate(self, prior_a=1, prior_b=1):
        # Using Beta distribution as conjugate prior
        successes = sum(self.data)
        failures = len(self.data) - successes
        posterior_a = prior_a + successes
        posterior_b = prior_b + failures
        return posterior_a / (posterior_a + posterior_b)

# Example usage
experiment = CoinFlipExperiment(true_probability=0.7)
freq_estimate = experiment.frequentist_estimate(100)
bayes_estimate = experiment.bayesian_estimate()

print(f"True probability: {experiment.true_prob}")
print(f"Frequentist estimate: {freq_estimate:.3f}")
print(f"Bayesian estimate: {bayes_estimate:.3f}")

Maximum Likelihood Estimation Implementation

Maximum Likelihood Estimation (MLE) represents a cornerstone of frequentist statistics. This implementation demonstrates MLE for estimating parameters of a normal distribution using numerical optimization.

import numpy as np
from scipy.optimize import minimize

def normal_negative_log_likelihood(params, data):
    mu, sigma = params
    return -np.sum(stats.norm.logpdf(data, mu, sigma))

# Generate sample data
true_mu, true_sigma = 2.5, 1.5
np.random.seed(42)
sample_data = np.random.normal(true_mu, true_sigma, 1000)

# MLE estimation
initial_guess = [0, 1]
result = minimize(normal_negative_log_likelihood, 
                 initial_guess,
                 args=(sample_data,),
                 method='Nelder-Mead')

estimated_mu, estimated_sigma = result.x
print(f"True parameters: μ={true_mu}, σ={true_sigma}")
print(f"MLE estimates: μ={estimated_mu:.3f}, σ={estimated_sigma:.3f}")

Bayesian Parameter Estimation

This implementation showcases Bayesian parameter estimation using Markov Chain Monte Carlo (MCMC) with PyMC3, demonstrating how to incorporate prior beliefs and update them with observed data.

import pymc3 as pm
import matplotlib.pyplot as plt

def bayesian_parameter_estimation(data, n_samples=2000):
    with pm.Model() as model:
        # Prior distributions
        mu = pm.Normal('mu', mu=0, sd=10)
        sigma = pm.HalfNormal('sigma', sd=10)
        
        # Likelihood
        observations = pm.Normal('obs', 
                               mu=mu, 
                               sigma=sigma, 
                               observed=data)
        
        # Inference
        trace = pm.sample(n_samples, 
                         tune=1000, 
                         return_inferencedata=False)
    
    return trace

# Generate and analyze data
np.random.seed(42)
data = np.random.normal(2.5, 1.5, 100)
trace = bayesian_parameter_estimation(data)

print("Posterior estimates:")
print(f"μ: {trace['mu'].mean():.3f} ± {trace['mu'].std():.3f}")
print(f"σ: {trace['sigma'].mean():.3f} ± {trace['sigma'].std():.3f}")

Frequentist A/B Testing Implementation

A practical implementation of frequentist A/B testing using hypothesis testing and confidence intervals, demonstrating how to analyze conversion rates between two different webpage versions.

import scipy.stats as stats
import numpy as np

class FrequentistABTest:
    def __init__(self, control_data, treatment_data):
        self.control = control_data
        self.treatment = treatment_data
        
    def calculate_statistics(self):
        n1, n2 = len(self.control), len(self.treatment)
        p1, p2 = np.mean(self.control), np.mean(self.treatment)
        
        # Calculate pooled standard error
        p_pooled = (sum(self.control) + sum(self.treatment)) / (n1 + n2)
        se = np.sqrt(p_pooled * (1 - p_pooled) * (1/n1 + 1/n2))
        
        # Calculate z-score
        z_score = (p2 - p1) / se
        p_value = 2 * (1 - stats.norm.cdf(abs(z_score)))
        
        return {
            'control_conv': p1,
            'treatment_conv': p2,
            'difference': p2 - p1,
            'z_score': z_score,
            'p_value': p_value
        }

# Example usage
np.random.seed(42)
control = np.random.binomial(1, 0.10, 1000)    # 10% conversion
treatment = np.random.binomial(1, 0.12, 1000)  # 12% conversion

ab_test = FrequentistABTest(control, treatment)
results = ab_test.calculate_statistics()

for key, value in results.items():
    print(f"{key}: {value:.4f}")

Bayesian A/B Testing

Implementing Bayesian A/B testing using Beta distributions as conjugate priors for conversion rates, providing a more nuanced view of uncertainty in the testing process.

class BayesianABTest:
    def __init__(self, alpha_prior=1, beta_prior=1):
        self.alpha_prior = alpha_prior
        self.beta_prior = beta_prior
        
    def analyze(self, control_successes, control_trials,
               treatment_successes, treatment_trials,
               n_samples=100000):
        
        # Posterior distributions
        control_post = stats.beta(
            self.alpha_prior + control_successes,
            self.beta_prior + control_trials - control_successes
        )
        
        treatment_post = stats.beta(
            self.alpha_prior + treatment_successes,
            self.beta_prior + treatment_trials - treatment_successes
        )
        
        # Generate samples
        control_samples = control_post.rvs(n_samples)
        treatment_samples = treatment_post.rvs(n_samples)
        
        # Calculate probability of improvement
        prob_improvement = np.mean(treatment_samples > control_samples)
        
        # Expected lift
        expected_lift = np.mean((treatment_samples - control_samples) / 
                              control_samples)
        
        return {
            'prob_improvement': prob_improvement,
            'expected_lift': expected_lift,
            'control_mean': np.mean(control_samples),
            'treatment_mean': np.mean(treatment_samples)
        }

# Example usage
bayes_test = BayesianABTest(alpha_prior=1, beta_prior=1)
results = bayes_test.analyze(
    control_successes=100,
    control_trials=1000,
    treatment_successes=120,
    treatment_trials=1000
)

for key, value in results.items():
    print(f"{key}: {value:.4f}")

Bayesian Linear Regression

A comprehensive implementation of Bayesian Linear Regression using probabilistic programming, demonstrating parameter uncertainty estimation and posterior predictive distributions.

import pymc3 as pm
import numpy as np
import matplotlib.pyplot as plt

class BayesianLinearRegression:
    def __init__(self):
        self.trace = None
        self.model = None
    
    def fit(self, X, y, n_samples=2000):
        with pm.Model() as self.model:
            # Priors for unknown model parameters
            alpha = pm.Normal('alpha', mu=0, sd=10)
            beta = pm.Normal('beta', mu=0, sd=10)
            sigma = pm.HalfNormal('sigma', sd=1)
            
            # Expected value of outcome
            mu = alpha + beta * X
            
            # Likelihood (sampling distribution) of observations
            y_obs = pm.Normal('y_obs', mu=mu, sd=sigma, observed=y)
            
            # Inference
            self.trace = pm.sample(n_samples, 
                                 tune=1000, 
                                 return_inferencedata=False)
        
        return self
    
    def predict(self, X_new):
        with self.model:
            pm.set_data({'X': X_new})
            posterior_pred = pm.sample_posterior_predictive(
                self.trace, 
                vars=[self.model.alpha, self.model.beta]
            )
        return posterior_pred

# Example usage
np.random.seed(42)
X = np.linspace(0, 10, 100)
true_alpha, true_beta = 2, 0.5
y = true_alpha + true_beta * X + np.random.normal(0, 0.5, size=100)

model = BayesianLinearRegression()
model.fit(X, y)

# Print parameter estimates
print(f"True alpha: {true_alpha}, Estimated: {model.trace['alpha'].mean():.3f}")
print(f"True beta: {true_beta}, Estimated: {model.trace['beta'].mean():.3f}")

Frequentist Time Series Analysis

Implementation of classical time series decomposition and forecasting using frequentist methods, demonstrating seasonal decomposition and ARIMA modeling for prediction.

from statsmodels.tsa.seasonal import seasonal_decompose
from statsmodels.tsa.arima.model import ARIMA
import numpy as np
import pandas as pd

class FrequentistTimeSeriesAnalysis:
    def __init__(self, data, frequency=12):
        self.data = data
        self.frequency = frequency
        
    def decompose(self):
        result = seasonal_decompose(self.data, 
                                  period=self.frequency,
                                  model='multiplicative')
        return {
            'trend': result.trend,
            'seasonal': result.seasonal,
            'residual': result.resid
        }
    
    def fit_arima(self, order=(1,1,1)):
        model = ARIMA(self.data, order=order)
        fitted = model.fit()
        return fitted
    
    def forecast(self, model, steps=12):
        forecast = model.forecast(steps=steps)
        conf_int = model.get_forecast(steps=steps).conf_int()
        return forecast, conf_int

# Example usage
np.random.seed(42)
dates = pd.date_range(start='2020-01-01', periods=100, freq='M')
trend = np.linspace(0, 10, 100)
seasonal = np.sin(np.linspace(0, 8*np.pi, 100))
noise = np.random.normal(0, 0.5, 100)
data = trend + seasonal + noise

ts_analysis = FrequentistTimeSeriesAnalysis(data)
decomposition = ts_analysis.decompose()
model = ts_analysis.fit_arima()
forecast, conf_intervals = ts_analysis.forecast(model)

print(f"AIC: {model.aic:.2f}")
print(f"Forecast next 3 steps: {forecast[:3]}")

Bayesian Time Series Modeling

Implementation of a Bayesian structural time series model using PyMC3, incorporating prior knowledge and uncertainty in trend and seasonal components.

import pymc3 as pm
import numpy as np

class BayesianTimeSeriesModel:
    def __init__(self, data, seasonal_periods=12):
        self.data = data
        self.seasonal_periods = seasonal_periods
        self.trace = None
        
    def build_and_fit(self, samples=2000):
        n = len(self.data)
        
        with pm.Model() as model:
            # Level and trend components
            level_sd = pm.HalfNormal('level_sd', sd=0.5)
            trend_sd = pm.HalfNormal('trend_sd', sd=0.1)
            
            level = pm.GaussianRandomWalk('level', 
                                        sd=level_sd,
                                        shape=n)
            trend = pm.GaussianRandomWalk('trend',
                                        sd=trend_sd,
                                        shape=n)
            
            # Seasonal component
            season_sd = pm.HalfNormal('season_sd', sd=0.1)
            season_raw = pm.Normal('season_raw', 
                                 mu=0,
                                 sd=season_sd,
                                 shape=self.seasonal_periods)
            
            season = pm.Deterministic('season',
                                    season_raw - pm.math.mean(season_raw))
            
            # Final model
            sigma = pm.HalfNormal('sigma', sd=1)
            mu = level + trend + season[np.arange(n) % self.seasonal_periods]
            
            y = pm.Normal('y', mu=mu, sd=sigma, observed=self.data)
            
            # Inference
            self.trace = pm.sample(samples, 
                                 tune=1000,
                                 return_inferencedata=False)
        
        return self.trace

# Example usage
np.random.seed(42)
n_points = 100
trend = np.linspace(0, 5, n_points)
seasonal = np.tile(np.sin(np.linspace(0, 2*np.pi, 12)), n_points//12 + 1)[:n_points]
data = trend + seasonal + np.random.normal(0, 0.2, n_points)

model = BayesianTimeSeriesModel(data)
trace = model.build_and_fit()

print("Posterior estimates:")
print(f"Noise σ: {trace['sigma'].mean():.3f} ± {trace['sigma'].std():.3f}")
print(f"Level σ: {trace['level_sd'].mean():.3f} ± {trace['level_sd'].std():.3f}")

Bayesian Neural Networks

Implementation of a Bayesian Neural Network using PyMC3, demonstrating how to incorporate uncertainty in weight parameters and make probabilistic predictions.

import pymc3 as pm
import numpy as np
import theano.tensor as tt

class BayesianNeuralNetwork:
    def __init__(self, input_dim, hidden_dim=50):
        self.input_dim = input_dim
        self.hidden_dim = hidden_dim
        self.trace = None
        
    def build_and_fit(self, X, y, samples=2000):
        with pm.Model() as model:
            # Prior distributions for weights
            w1 = pm.Normal('w1', 
                         mu=0, 
                         sd=1, 
                         shape=(self.input_dim, self.hidden_dim))
            w2 = pm.Normal('w2', 
                         mu=0, 
                         sd=1, 
                         shape=(self.hidden_dim, 1))
            
            # Prior for bias
            b1 = pm.Normal('b1', mu=0, sd=1, shape=(self.hidden_dim,))
            b2 = pm.Normal('b2', mu=0, sd=1, shape=(1,))
            
            # Neural network architecture
            act_1 = tt.tanh(tt.dot(X, w1) + b1)
            mu = tt.dot(act_1, w2) + b2
            
            # Likelihood
            sigma = pm.HalfNormal('sigma', sd=1)
            y_obs = pm.Normal('y_obs', mu=mu, sd=sigma, observed=y)
            
            # Inference
            self.trace = pm.sample(samples, 
                                 tune=1000,
                                 return_inferencedata=False)
        
        return self.trace
    
    def predict(self, X_new, samples=1000):
        preds = np.zeros((samples, len(X_new)))
        for i in range(samples):
            idx = np.random.randint(0, len(self.trace['w1']))
            w1_samp = self.trace['w1'][idx]
            w2_samp = self.trace['w2'][idx]
            b1_samp = self.trace['b1'][idx]
            b2_samp = self.trace['b2'][idx]
            
            # Forward pass
            act_1 = np.tanh(np.dot(X_new, w1_samp) + b1_samp)
            preds[i] = np.dot(act_1, w2_samp) + b2_samp.reshape(-1)
            
        return preds.mean(axis=0), preds.std(axis=0)

# Example usage
np.random.seed(42)
X = np.linspace(-3, 3, 100).reshape(-1, 1)
y = np.sin(X) + np.random.normal(0, 0.1, size=X.shape)

model = BayesianNeuralNetwork(input_dim=1)
trace = model.build_and_fit(X, y)
y_pred_mean, y_pred_std = model.predict(X)

print("Model trained. Example predictions:")
print(f"X: {X[0]}, Pred: {y_pred_mean[0]:.3f} ± {y_pred_std[0]:.3f}")

Frequentist vs Bayesian Model Selection

Implementation comparing frequentist AIC/BIC criteria with Bayesian model selection using Bayes factors for polynomial regression models.

import numpy as np
from scipy import stats
from sklearn.linear_model import LinearRegression
from sklearn.metrics import make_scorer
from sklearn.model_selection import cross_val_score

class ModelSelection:
    def __init__(self, X, y):
        self.X = X
        self.y = y
        self.n = len(y)
        
    def frequentist_selection(self, max_degree=5):
        results = {}
        for degree in range(1, max_degree + 1):
            X_poly = np.vander(self.X, degree + 1)
            model = LinearRegression()
            model.fit(X_poly, self.y)
            
            # Calculate AIC and BIC
            y_pred = model.predict(X_poly)
            mse = np.mean((self.y - y_pred) ** 2)
            k = degree + 1
            
            aic = self.n * np.log(mse) + 2 * k
            bic = self.n * np.log(mse) + np.log(self.n) * k
            
            results[degree] = {
                'aic': aic,
                'bic': bic,
                'r2': model.score(X_poly, self.y)
            }
        return results
    
    def bayesian_selection(self, max_degree=5):
        results = {}
        for degree in range(1, max_degree + 1):
            X_poly = np.vander(self.X, degree + 1)
            
            # Calculate marginal likelihood using BIC approximation
            model = LinearRegression()
            model.fit(X_poly, self.y)
            
            y_pred = model.predict(X_poly)
            mse = np.mean((self.y - y_pred) ** 2)
            k = degree + 1
            
            # Log marginal likelihood approximation
            log_ml = -0.5 * (self.n * np.log(2 * np.pi * mse) + 
                           k * np.log(self.n))
            
            results[degree] = {
                'log_marginal_likelihood': log_ml,
                'bayes_factor': np.exp(log_ml) if degree > 1 
                               else None
            }
        return results

# Example usage
np.random.seed(42)
X = np.linspace(-3, 3, 100)
y = 1 + 2*X + 0.5*X**2 + np.random.normal(0, 0.1, size=X.shape)

selector = ModelSelection(X, y)
freq_results = selector.frequentist_selection()
bayes_results = selector.bayesian_selection()

print("Frequentist Selection Results:")
for degree, metrics in freq_results.items():
    print(f"Degree {degree}: AIC={metrics['aic']:.2f}, BIC={metrics['bic']:.2f}")

print("\nBayesian Selection Results:")
for degree, metrics in bayes_results.items():
    print(f"Degree {degree}: Log ML={metrics['log_marginal_likelihood']:.2f}")

Implementing Bayesian Optimization

A practical implementation of Bayesian optimization using Gaussian Processes for hyperparameter tuning, demonstrating the balance between exploration and exploitation.

import numpy as np
from scipy.stats import norm
from scipy.optimize import minimize

class BayesianOptimizer:
    def __init__(self, bounds, n_init=5):
        self.bounds = np.array(bounds)
        self.n_init = n_init
        self.X = []
        self.y = []
        
    def acquisition_function(self, X, model):
        mu, sigma = model.predict(X.reshape(-1, 1))
        sigma = np.maximum(sigma, 1e-4)
        
        # Expected Improvement
        imp = mu - np.max(self.y) - 0.01
        Z = imp / sigma
        ei = imp * norm.cdf(Z) + sigma * norm.pdf(Z)
        return -ei
    
    def optimize(self, objective_func, n_iter=20):
        # Initial random sampling
        X_init = np.random.uniform(
            self.bounds[0], 
            self.bounds[1], 
            size=self.n_init
        )
        y_init = [objective_func(x) for x in X_init]
        
        self.X = list(X_init)
        self.y = list(y_init)
        
        for i in range(n_iter):
            # Fit GP model
            kernel = RBF(length_scale=1.0)
            model = GaussianProcessRegressor(
                kernel=kernel,
                alpha=1e-6,
                normalize_y=True,
                n_restarts_optimizer=5
            )
            model.fit(np.array(self.X).reshape(-1, 1), self.y)
            
            # Find next point to evaluate
            result = minimize(
                lambda x: self.acquisition_function(x, model),
                x0=np.random.uniform(self.bounds[0], self.bounds[1]),
                bounds=[self.bounds],
                method='L-BFGS-B'
            )
            
            next_x = result.x[0]
            next_y = objective_func(next_x)
            
            self.X.append(next_x)
            self.y.append(next_y)
            
        best_idx = np.argmax(self.y)
        return self.X[best_idx], self.y[best_idx]

# Example usage
def objective(x):
    return -(x - 2)**2 + 10

optimizer = BayesianOptimizer(bounds=[-5, 5])
best_x, best_y = optimizer.optimize(objective)

print(f"Best found value: {best_y:.3f} at x = {best_x:.3f}")

Bayesian Change Point Detection

Implementation of a Bayesian approach to detecting changes in time series data using conjugate priors and MCMC sampling.

import numpy as np
import pymc3 as pm

class BayesianChangePointDetector:
    def __init__(self, data):
        self.data = np.array(data)
        self.n = len(data)
        self.trace = None
        
    def fit(self, max_changepoints=2, samples=2000):
        with pm.Model() as model:
            # Prior on changepoint locations
            changepoint_locs = pm.DiscreteUniform(
                'changepoint_locs',
                lower=1,
                upper=self.n-2,
                shape=max_changepoints
            )
            
            # Prior on segment parameters
            sigma = pm.HalfNormal('sigma', sd=1)
            means = pm.Normal('means', 
                            mu=np.mean(self.data), 
                            sd=np.std(self.data),
                            shape=max_changepoints+1)
            
            # Generate segments
            idx = tt.arange(self.n)
            segment_means = means[0] * tt.ones_like(idx)
            
            for i in range(max_changepoints):
                segment_means = tt.switch(
                    idx >= changepoint_locs[i],
                    means[i+1],
                    segment_means
                )
            
            # Likelihood
            y = pm.Normal('y',
                         mu=segment_means,
                         sd=sigma,
                         observed=self.data)
            
            # Inference
            self.trace = pm.sample(samples,
                                 tune=1000,
                                 return_inferencedata=False)
            
        return self.trace
    
    def get_changepoints(self):
        cp_samples = self.trace['changepoint_locs']
        mean_cps = np.mean(cp_samples, axis=0)
        std_cps = np.std(cp_samples, axis=0)
        return mean_cps, std_cps

# Example usage
np.random.seed(42)
n_points = 200
data = np.concatenate([
    np.random.normal(0, 1, 80),
    np.random.normal(3, 1, 70),
    np.random.normal(-1, 1, 50)
])

detector = BayesianChangePointDetector(data)
trace = detector.fit()
mean_cps, std_cps = detector.get_changepoints()

print("Detected changepoints:")
for i, (mean, std) in enumerate(zip(mean_cps, std_cps)):
    print(f"Changepoint {i+1}: {mean:.1f} ± {std:.1f}")

Frequentist vs Bayesian Clustering

A comparative implementation of clustering methods using both frequentist (K-means) and Bayesian (Dirichlet Process Mixture Model) approaches, showing key differences in uncertainty handling.

import numpy as np
from sklearn.cluster import KMeans
import pymc3 as pm

class ClusteringComparison:
    def __init__(self, data):
        self.data = data
        self.n_samples = len(data)
        
    def frequentist_clustering(self, n_clusters=3):
        kmeans = KMeans(n_clusters=n_clusters, random_state=42)
        labels = kmeans.fit_predict(self.data)
        centers = kmeans.cluster_centers_
        
        return {
            'labels': labels,
            'centers': centers,
            'inertia': kmeans.inertia_
        }
    
    def bayesian_clustering(self, max_clusters=10, samples=2000):
        with pm.Model() as model:
            # Concentration parameter
            alpha = pm.Gamma('alpha', 2, 2)
            
            # Stick-breaking weights
            w = pm.Dirichlet('w', 
                            a=tt.ones(max_clusters)*alpha/max_clusters,
                            shape=max_clusters)
            
            # Cluster parameters
            mu = pm.Normal('mu', 
                          mu=0,
                          sd=10,
                          shape=(max_clusters, self.data.shape[1]))
            sigma = pm.HalfNormal('sigma', 
                                sd=1,
                                shape=(max_clusters, self.data.shape[1]))
            
            # Mixture likelihood
            y = pm.Mixture('y',
                          w=w,
                          comp_dists=[pm.Normal.dist(mu=mu[i],
                                                   sd=sigma[i])
                                    for i in range(max_clusters)],
                          observed=self.data)
            
            # Inference
            self.trace = pm.sample(samples,
                                 tune=1000,
                                 return_inferencedata=False)
        
        return self.trace

# Example usage
np.random.seed(42)
n_points = 300
data = np.vstack([
    np.random.normal(0, 1, (100, 2)),
    np.random.normal(5, 1, (100, 2)),
    np.random.normal([2, 5], 1, (100, 2))
])

clustering = ClusteringComparison(data)
freq_results = clustering.frequentist_clustering()
bayes_results = clustering.bayesian_clustering()

print("Frequentist Results:")
print(f"Inertia: {freq_results['inertia']:.2f}")
print("Cluster centers:")
print(freq_results['centers'])

print("\nBayesian Results:")
print(f"Estimated number of active clusters: "
      f"{(bayes_results['w'].mean(axis=0) > 0.05).sum()}")

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  1. Central Limit Theorem for Confidence Intervals and Hypothesis Testing in Python11 min

    The Central Limit Theorem (CLT) is a fundamental concept in probability theory and statistics. It states that the distribution of sample means approximates a...

  2. Bayesian Online Natural Gradient (BONG) in Python6 min

    The Bayesian Online Natural Gradient (BONG) is a principled approach to online learning of overparametrized models, combining Bayesian inference with natural...

  3. Choosing the Right Statistical Test for Your Data10 min

    Understanding variable types is crucial in programming and data analysis. It affects how we store, manipulate, and interpret data. Let's explore this concept...

  4. Bayesian Learning in Nonlinear Multiscale State-Space Models15 min

    Bayesian learning in nonlinear multiscale state-space models combines probabilistic inference with complex dynamical systems. This approach allows us to...

  5. Common Hypothesis Tests in Python12 min

    Hypothesis testing is a fundamental tool in statistical analysis, allowing researchers to make inferences about populations based on sample data. This...

  6. Bayesian Inference Concepts in Python11 min

    Bayesian inference is a statistical method that uses Bayes' theorem to update the probability of a hypothesis as more evidence becomes available. It forms...