🚀 Master Guide to Bayes Theorem The Probabilistic Foundation Of Modern Decision Making That Will Boost Your!
Hey there! Ready to dive into Bayes Theorem The Probabilistic Foundation Of Modern Decision Making? This friendly guide will walk you through everything step-by-step with easy-to-follow examples. Perfect for beginners and pros alike!
🚀
💡 Pro tip: This is one of those techniques that will make you look like a data science wizard! Bayesian Fundamentals - Made Simple!
Bayes’ theorem provides a mathematical framework for updating probabilities based on new evidence. This example shows you the core calculation of posterior probabilities using Python, establishing the foundation for more complex Bayesian applications.
Ready for some cool stuff? Here’s how we can tackle this:
def bayes_theorem(prior, likelihood, marginal):
"""
Calculate posterior probability using Bayes' Theorem
P(A|B) = P(B|A) * P(A) / P(B)
"""
posterior = (likelihood * prior) / marginal
return posterior
# Example: Disease testing
prior_disease = 0.001 # Prior probability of disease
sensitivity = 0.99 # P(positive|disease)
false_positive = 0.01 # P(positive|no disease)
# Calculate marginal probability
marginal = (sensitivity * prior_disease +
false_positive * (1 - prior_disease))
# Calculate posterior probability
posterior = bayes_theorem(prior_disease, sensitivity, marginal)
print(f"Posterior probability: {posterior:.4f}")
# Output:
# Posterior probability: 0.0901
🚀
🎉 You’re doing great! This concept might seem tricky at first, but you’ve got this! Bayesian Parameter Estimation - Made Simple!
Parameter estimation is a crucial application of Bayesian methods. This example shows how to estimate the probability of success in a binomial distribution using conjugate prior distributions and updating beliefs with observed data.
Let’s break this down together! Here’s how we can tackle this:
import numpy as np
from scipy import stats
def beta_binomial_update(prior_alpha, prior_beta, successes, trials):
"""
Update Beta distribution parameters using binomial observations
"""
post_alpha = prior_alpha + successes
post_beta = prior_beta + (trials - successes)
return post_alpha, post_beta
# Initial beliefs (Beta prior)
alpha_prior = 2
beta_prior = 2
# Observed data
successes = 7
trials = 10
# Update parameters
alpha_post, beta_post = beta_binomial_update(
alpha_prior, beta_prior, successes, trials
)
# Calculate MAP estimate
map_estimate = (alpha_post - 1) / (alpha_post + beta_post - 2)
print(f"MAP estimate: {map_estimate:.4f}")
# Generate posterior distribution samples
x = np.linspace(0, 1, 100)
posterior = stats.beta.pdf(x, alpha_post, beta_post)
🚀
✨ Cool fact: Many professional data scientists use this exact approach in their daily work! Naive Bayes Classifier Implementation - Made Simple!
The Naive Bayes classifier applies Bayes’ theorem with strong independence assumptions between features. This example creates a classifier from scratch, demonstrating the mathematical principles behind this popular machine learning algorithm.
Let’s break this down together! Here’s how we can tackle this:
import numpy as np
from collections import defaultdict
class NaiveBayesClassifier:
def __init__(self):
self.class_probs = {}
self.feature_probs = defaultdict(lambda: defaultdict(dict))
def fit(self, X, y):
n_samples = len(y)
classes = np.unique(y)
# Calculate class probabilities
for c in classes:
self.class_probs[c] = np.sum(y == c) / n_samples
# Calculate feature probabilities for each class
for c in classes:
class_samples = X[y == c]
for feature in range(X.shape[1]):
unique_vals, counts = np.unique(
class_samples[:, feature],
return_counts=True
)
probs = counts / len(class_samples)
for val, prob in zip(unique_vals, probs):
self.feature_probs[c][feature][val] = prob
def predict(self, X):
predictions = []
for sample in X:
class_scores = {}
for c in self.class_probs:
score = np.log(self.class_probs[c])
for feature, value in enumerate(sample):
if value in self.feature_probs[c][feature]:
score += np.log(
self.feature_probs[c][feature][value]
)
class_scores[c] = score
predictions.append(
max(class_scores.items(), key=lambda x: x[1])[0]
)
return np.array(predictions)
🚀
🔥 Level up: Once you master this, you’ll be solving problems like a pro! Real-world Example: Credit Risk Assessment - Made Simple!
This example shows you a practical application of Bayesian inference in credit risk assessment, incorporating multiple risk factors and updating probability estimates based on new customer data.
This next part is really neat! Here’s how we can tackle this:
import numpy as np
from scipy.stats import beta
class CreditRiskAssessor:
def __init__(self, base_default_rate=0.05):
self.base_rate = base_default_rate
self.alpha_prior = 2
self.beta_prior = 38 # Gives prior mean of 0.05
def update_risk(self, risk_factors):
"""
Update default probability based on risk factors
"""
# Risk factors: payment history, debt ratio, income level
risk_weights = {
'payment_history': 0.4,
'debt_ratio': 0.35,
'income_level': 0.25
}
total_risk_score = sum(
score * risk_weights[factor]
for factor, score in risk_factors.items()
)
# Update Beta distribution parameters
likelihood = total_risk_score
n_observations = 1
alpha_post = self.alpha_prior + (likelihood * n_observations)
beta_post = self.beta_prior + (
(1 - likelihood) * n_observations
)
# Calculate updated default probability
updated_prob = alpha_post / (alpha_post + beta_post)
return updated_prob
# Example usage
risk_assessor = CreditRiskAssessor()
customer_risk = {
'payment_history': 0.8, # Good history
'debt_ratio': 0.6, # Moderate debt
'income_level': 0.7 # Above average income
}
updated_risk = risk_assessor.update_risk(customer_risk)
print(f"Updated default probability: {updated_risk:.4f}")
# Output:
# Updated default probability: 0.0623
🚀 Bayesian Time Series Analysis - Made Simple!
Implementing Bayesian methods for time series analysis allows for dynamic probability updates as new data arrives. This example showcases a basic Bayesian approach to forecasting with uncertainty quantification.
Here’s a handy trick you’ll love! Here’s how we can tackle this:
import numpy as np
from scipy.stats import norm
class BayesianTimeSeriesAnalyzer:
def __init__(self, prior_mean=0, prior_var=1):
self.mean = prior_mean
self.variance = prior_var
self.data_history = []
def update(self, observation, observation_var):
"""
Update beliefs using Kalman filter equations
"""
# Calculate Kalman gain
k_gain = self.variance / (self.variance + observation_var)
# Update mean and variance
self.mean = self.mean + k_gain * (observation - self.mean)
self.variance = (1 - k_gain) * self.variance
self.data_history.append(observation)
def predict(self, steps=1, confidence=0.95):
"""
Make predictions with confidence intervals
"""
z_score = norm.ppf((1 + confidence) / 2)
forecast = np.array([self.mean] * steps)
std = np.sqrt(self.variance)
ci_lower = forecast - z_score * std
ci_upper = forecast + z_score * std
return forecast, ci_lower, ci_upper
# Example usage
analyzer = BayesianTimeSeriesAnalyzer()
data = [1.2, 1.4, 1.1, 1.3, 1.5]
observation_var = 0.1
for obs in data:
analyzer.update(obs, observation_var)
forecast, lower, upper = analyzer.predict(steps=3)
print(f"Forecast: {forecast[0]:.3f}")
print(f"95% CI: [{lower[0]:.3f}, {upper[0]:.3f}]")
🚀 Bayesian A/B Testing Framework - Made Simple!
This example shows you how to perform Bayesian A/B testing, providing a more nuanced approach to experimental analysis than traditional frequentist methods by incorporating prior beliefs and updating them with evidence.
Here’s where it gets exciting! Here’s how we can tackle this:
import numpy as np
from scipy import stats
class BayesianABTest:
def __init__(self, prior_alpha_a=1, prior_beta_a=1,
prior_alpha_b=1, prior_beta_b=1):
self.alpha_a = prior_alpha_a
self.beta_a = prior_beta_a
self.alpha_b = prior_alpha_b
self.beta_b = prior_beta_b
def update(self, successes_a, trials_a, successes_b, trials_b):
"""
Update Beta distributions for both variants
"""
self.alpha_a += successes_a
self.beta_a += trials_a - successes_a
self.alpha_b += successes_b
self.beta_b += trials_b - successes_b
def probability_b_better_than_a(self, samples=10000):
"""
Monte Carlo estimation of P(B > A)
"""
samples_a = np.random.beta(self.alpha_a, self.beta_a, samples)
samples_b = np.random.beta(self.alpha_b, self.beta_b, samples)
return np.mean(samples_b > samples_a)
def expected_loss(self, samples=10000):
"""
Calculate expected loss of choosing wrong variant
"""
samples_a = np.random.beta(self.alpha_a, self.beta_a, samples)
samples_b = np.random.beta(self.alpha_b, self.beta_b, samples)
return np.mean(np.maximum(samples_a - samples_b, 0))
# Example usage
ab_test = BayesianABTest()
# Update with results: A(45/100) vs B(55/100)
ab_test.update(45, 100, 55, 100)
prob_b_better = ab_test.probability_b_better_than_a()
exp_loss = ab_test.expected_loss()
print(f"P(B > A): {prob_b_better:.3f}")
print(f"Expected Loss: {exp_loss:.3f}")
# Calculate credible intervals
ci_a = stats.beta.interval(0.95, ab_test.alpha_a, ab_test.beta_a)
ci_b = stats.beta.interval(0.95, ab_test.alpha_b, ab_test.beta_b)
🚀 Bayesian Linear Regression - Made Simple!
Bayesian linear regression extends traditional linear regression by treating model parameters as probability distributions rather than point estimates, providing uncertainty quantification in predictions.
This next part is really neat! Here’s how we can tackle this:
import numpy as np
from scipy import stats
class BayesianLinearRegression:
def __init__(self, alpha=1.0, beta=1.0):
self.alpha = alpha # Precision of prior
self.beta = beta # Precision of likelihood
self.mean = None
self.precision = None
def fit(self, X, y):
n_samples, n_features = X.shape
# Calculate posterior precision matrix
self.precision = (
self.alpha * np.eye(n_features) +
self.beta * X.T @ X
)
# Calculate posterior mean
self.mean = self.beta * np.linalg.solve(
self.precision,
X.T @ y
)
def predict(self, X_new, return_std=True):
y_mean = X_new @ self.mean
if return_std:
# Calculate predictive variance
precision_inv = np.linalg.inv(self.precision)
var = 1/self.beta + np.sum(
X_new @ precision_inv * X_new,
axis=1
)
std = np.sqrt(var)
return y_mean, std
return y_mean
# Example usage
np.random.seed(42)
X = np.random.randn(100, 2)
true_weights = np.array([2.0, -1.0])
y = X @ true_weights + np.random.randn(100) * 0.1
# Create and fit model
model = BayesianLinearRegression(alpha=0.1, beta=10.0)
model.fit(X, y)
# Make predictions with uncertainty
X_test = np.random.randn(10, 2)
y_pred, y_std = model.predict(X_test)
print("Predictions and uncertainties:")
for pred, std in zip(y_pred[:3], y_std[:3]):
print(f"Prediction: {pred:.3f} ± {2*std:.3f}")
🚀 Bayesian Model Selection - Made Simple!
Implementing Bayesian model selection using Bayes factors provides a principled way to compare competing models while naturally penalizing complexity and avoiding overfitting.
Here’s a handy trick you’ll love! Here’s how we can tackle this:
import numpy as np
from scipy.special import gammaln
class BayesianModelSelector:
def __init__(self):
self.models = {}
self.prior_probs = {}
def add_model(self, name, model, prior_prob):
self.models[name] = model
self.prior_probs[name] = prior_prob
def compute_log_evidence(self, X, y, model):
"""
Compute log marginal likelihood (model evidence)
using Laplace approximation
"""
n_samples = len(y)
n_params = len(model.mean)
# Compute log likelihood at MAP
residuals = y - X @ model.mean
log_likelihood = (-0.5 * n_samples * np.log(2 * np.pi) +
0.5 * n_samples * np.log(model.beta) -
0.5 * model.beta * np.sum(residuals**2))
# Add log prior
log_prior = (-0.5 * n_params * np.log(2 * np.pi) +
0.5 * n_params * np.log(model.alpha) -
0.5 * model.alpha * np.sum(model.mean**2))
# Add log determinant term from Laplace approximation
sign, logdet = np.linalg.slogdet(model.precision)
log_laplace = -0.5 * logdet
return log_likelihood + log_prior + log_laplace
def select_model(self, X, y):
log_evidences = {}
posterior_probs = {}
# Compute log evidence for each model
for name, model in self.models.items():
model.fit(X, y)
log_evidences[name] = (
self.compute_log_evidence(X, y, model) +
np.log(self.prior_probs[name])
)
# Compute posterior probabilities
log_total = np.logaddexp.reduce(list(log_evidences.values()))
for name in self.models:
posterior_probs[name] = np.exp(
log_evidences[name] - log_total
)
return posterior_probs
# Example usage
# Create two competing models with different complexities
model1 = BayesianLinearRegression(alpha=0.1, beta=10.0)
model2 = BayesianLinearRegression(alpha=1.0, beta=10.0)
selector = BayesianModelSelector()
selector.add_model("Simple", model1, prior_prob=0.5)
selector.add_model("Complex", model2, prior_prob=0.5)
# Generate data and select model
X = np.random.randn(100, 2)
y = X @ np.array([2.0, -1.0]) + np.random.randn(100) * 0.1
probs = selector.select_model(X, y)
for model, prob in probs.items():
print(f"{model} model probability: {prob:.3f}")
🚀 Markov Chain Monte Carlo (MCMC) Implementation - Made Simple!
Building a Metropolis-Hastings MCMC sampler provides a powerful tool for sampling from complex posterior distributions when analytical solutions are intractable. This example shows you the core concepts of MCMC sampling.
Here’s a handy trick you’ll love! Here’s how we can tackle this:
import numpy as np
from scipy import stats
class MetropolisHastings:
def __init__(self, target_log_prob, proposal_width=0.1):
self.target_log_prob = target_log_prob
self.proposal_width = proposal_width
self.samples = []
def run(self, n_samples, initial_state):
current_state = initial_state
current_log_prob = self.target_log_prob(current_state)
for _ in range(n_samples):
# Propose new state
proposal = current_state + np.random.normal(
0, self.proposal_width, size=len(current_state)
)
# Calculate acceptance ratio
proposal_log_prob = self.target_log_prob(proposal)
log_ratio = proposal_log_prob - current_log_prob
# Accept or reject
if np.log(np.random.random()) < log_ratio:
current_state = proposal
current_log_prob = proposal_log_prob
self.samples.append(current_state.copy())
def get_samples(self, burnin=0.2):
samples = np.array(self.samples)
n_burnin = int(len(samples) * burnin)
return samples[n_burnin:]
def diagnostics(self):
samples = self.get_samples()
acceptance_rate = len(np.unique(samples, axis=0)) / len(samples)
return {
'acceptance_rate': acceptance_rate,
'mean': np.mean(samples, axis=0),
'std': np.std(samples, axis=0)
}
# Example: Sampling from a mixture of Gaussians
def target_distribution(x):
"""Log probability of mixture of two Gaussians"""
return np.log(
0.3 * stats.norm.pdf(x[0], -2, 1) +
0.7 * stats.norm.pdf(x[0], 2, 1.5)
)
# Run sampler
mcmc = MetropolisHastings(target_distribution)
mcmc.run(10000, initial_state=np.array([0.0]))
samples = mcmc.get_samples()
diagnostics = mcmc.diagnostics()
print("MCMC Diagnostics:")
for key, value in diagnostics.items():
print(f"{key}: {value}")
🚀 Online Bayesian Learning - Made Simple!
This example showcases online learning using Bayesian methods, allowing for real-time updates of model parameters as new data arrives sequentially.
Here’s a handy trick you’ll love! Here’s how we can tackle this:
class OnlineBayesianLearner:
def __init__(self, prior_mean=0, prior_var=1):
self.mean = prior_mean
self.variance = prior_var
self.n_updates = 0
self.learning_rate_schedule = lambda n: 1 / (n + 1)
def update(self, observation, observation_var):
"""
Perform online Bayesian update
"""
self.n_updates += 1
lr = self.learning_rate_schedule(self.n_updates)
# Precision-weighted update
precision = 1 / self.variance
obs_precision = 1 / observation_var
# Update precision
new_precision = precision + lr * obs_precision
# Update mean
innovation = observation - self.mean
self.mean += (lr * obs_precision / new_precision) * innovation
# Update variance
self.variance = 1 / new_precision
return self.mean, self.variance
def predict(self, confidence=0.95):
"""
Make prediction with confidence interval
"""
z_score = stats.norm.ppf((1 + confidence) / 2)
ci_lower = self.mean - z_score * np.sqrt(self.variance)
ci_upper = self.mean + z_score * np.sqrt(self.variance)
return {
'prediction': self.mean,
'variance': self.variance,
'confidence_interval': (ci_lower, ci_upper)
}
# Example usage
learner = OnlineBayesianLearner()
data_stream = np.random.normal(1.5, 0.5, size=1000)
# Online learning
results = []
for obs in data_stream[:10]: # Show first 10 updates
mean, var = learner.update(obs, observation_var=0.5)
pred = learner.predict()
results.append(pred)
print("\nOnline Learning Results:")
for i, res in enumerate(results):
print(f"Step {i+1}:")
print(f"Prediction: {res['prediction']:.3f}")
print(f"95% CI: [{res['confidence_interval'][0]:.3f}, "
f"{res['confidence_interval'][1]:.3f}]")
🚀 Bayesian Change Point Detection - Made Simple!
Implementing a Bayesian approach to detect significant changes in time series data by modeling the probability of change points and updating beliefs as new data arrives.
Here’s where it gets exciting! Here’s how we can tackle this:
import numpy as np
from scipy.stats import norm
class BayesianChangePointDetector:
def __init__(self, hazard_rate=0.01):
self.hazard_rate = hazard_rate
self.run_length = 0
self.means = [0]
self.variances = [1]
self.probs = [1.0]
def update(self, observation):
# Calculate predictive probabilities
pred_probs = np.zeros(len(self.means))
for i in range(len(self.means)):
pred_probs[i] = norm.pdf(
observation,
self.means[i],
np.sqrt(self.variances[i])
)
# Calculate growth probabilities
growth_probs = (1 - self.hazard_rate) * np.array(self.probs)
# Calculate changepoint probability
cp_prob = self.hazard_rate * sum(self.probs)
# Update run length distribution
self.probs = np.append(cp_prob, growth_probs)
self.probs /= sum(self.probs)
# Update sufficient statistics
self.run_length += 1
n = self.run_length
# Update parameters for existing runs
for i in range(len(self.means)-1, -1, -1):
delta = observation - self.means[i]
self.means[i] += delta / (n - i)
if n - i > 1:
self.variances[i] *= (n-i-2)/(n-i-1)
self.variances[i] += (delta**2)/(n-i)
# Add new run
self.means.append(observation)
self.variances.append(1.0)
return self.get_change_point_probability()
def get_change_point_probability(self):
"""Return probability of most recent change point"""
return self.probs[0]
# Example usage
np.random.seed(42)
# Generate synthetic data with change point
n_points = 200
data = np.concatenate([
np.random.normal(0, 1, 100),
np.random.normal(3, 1, 100)
])
# Detect change points
detector = BayesianChangePointDetector()
change_probs = []
for obs in data:
prob = detector.update(obs)
change_probs.append(prob)
# Print results for key points
print("\nChange Point Detection Results:")
for i in [98, 99, 100, 101, 102]:
print(f"Time {i}: Change probability = {change_probs[i]:.3f}")
🚀 Bayesian Neural Network - Made Simple!
This example shows you a simple Bayesian Neural Network using variational inference, providing uncertainty estimates in neural network predictions.
Here’s a handy trick you’ll love! Here’s how we can tackle this:
import numpy as np
from scipy.special import expit
class BayesianNeuralNetwork:
def __init__(self, input_dim, hidden_dim, output_dim):
self.input_dim = input_dim
self.hidden_dim = hidden_dim
self.output_dim = output_dim
# Initialize variational parameters
self.w1_mu = np.random.randn(input_dim, hidden_dim) * 0.1
self.w1_rho = np.random.randn(input_dim, hidden_dim) * 0.1
self.w2_mu = np.random.randn(hidden_dim, output_dim) * 0.1
self.w2_rho = np.random.randn(hidden_dim, output_dim) * 0.1
def sample_weights(self):
"""Sample weights using reparameterization trick"""
w1_sigma = np.log(1 + np.exp(self.w1_rho))
w2_sigma = np.log(1 + np.exp(self.w2_rho))
epsilon1 = np.random.randn(*self.w1_mu.shape)
epsilon2 = np.random.randn(*self.w2_mu.shape)
w1 = self.w1_mu + w1_sigma * epsilon1
w2 = self.w2_mu + w2_sigma * epsilon2
return w1, w2
def forward(self, X, n_samples=10):
"""Forward pass with multiple samples"""
predictions = []
for _ in range(n_samples):
w1, w2 = self.sample_weights()
# Forward pass
h = np.tanh(X @ w1)
y = h @ w2
predictions.append(y)
# Stack predictions
predictions = np.stack(predictions)
# Calculate mean and variance
mean = np.mean(predictions, axis=0)
var = np.var(predictions, axis=0)
return mean, var
def kl_divergence(self):
"""Calculate KL divergence for variational inference"""
w1_sigma = np.log(1 + np.exp(self.w1_rho))
w2_sigma = np.log(1 + np.exp(self.w2_rho))
kl = np.sum(np.log(w1_sigma)) + np.sum(np.log(w2_sigma))
kl += np.sum(self.w1_mu**2 + w1_sigma**2)
kl += np.sum(self.w2_mu**2 + w2_sigma**2)
kl -= len(self.w1_mu.flatten()) + len(self.w2_mu.flatten())
return 0.5 * kl
# Example usage
X = np.random.randn(100, 2)
y = np.sin(X[:, 0]) + np.cos(X[:, 1])
# Create and use model
model = BayesianNeuralNetwork(2, 10, 1)
mean, var = model.forward(X)
print("\nPrediction Results:")
for i in range(3):
print(f"Sample {i+1}:")
print(f"Prediction: {mean[i,0]:.3f} ± {2*np.sqrt(var[i,0]):.3f}")
🚀 Bayesian Optimization - Made Simple!
Implementing Bayesian optimization for hyperparameter tuning using Gaussian Processes as the surrogate model and Expected Improvement as the acquisition function.
Let’s break this down together! Here’s how we can tackle this:
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 _kernel(self, X1, X2, l=1.0, sigma_f=1.0):
"""RBF kernel"""
sqdist = np.sum(X1**2, 1).reshape(-1, 1) + \
np.sum(X2**2, 1) - \
2 * np.dot(X1, X2.T)
return sigma_f**2 * np.exp(-0.5 / l**2 * sqdist)
def _expected_improvement(self, X, xi=0.01):
"""Calculate expected improvement"""
mu, sigma = self._predict(X)
mu = mu.reshape(-1)
sigma = sigma.reshape(-1)
mu_sample_opt = np.max(self.y)
with np.errstate(divide='warn'):
imp = mu - mu_sample_opt - xi
Z = imp / sigma
ei = imp * norm.cdf(Z) + sigma * norm.pdf(Z)
ei[sigma == 0.0] = 0.0
return -ei
def _predict(self, X_new):
"""GP prediction"""
if len(self.X) == 0:
return np.zeros(len(X_new)), np.ones(len(X_new))
K = self._kernel(np.array(self.X), np.array(self.X))
K_s = self._kernel(np.array(self.X), X_new)
K_ss = self._kernel(X_new, X_new)
K_inv = np.linalg.inv(K + 1e-8 * np.eye(len(K)))
mu = K_s.T.dot(K_inv).dot(self.y)
sigma = np.diag(K_ss - K_s.T.dot(K_inv).dot(K_s))
return mu, sigma
def propose_location(self):
"""Propose next sampling location"""
if len(self.X) < self.n_init:
return np.random.uniform(
self.bounds[:, 0],
self.bounds[:, 1]
)
dim = self.bounds.shape[0]
def min_obj(X):
return self._expected_improvement(X.reshape(-1, dim))
X_tries = np.random.uniform(
self.bounds[:, 0],
self.bounds[:, 1],
size=(100, dim)
)
ei_tries = [-min_obj(x) for x in X_tries]
X_max = X_tries[np.argmax(ei_tries)]
res = minimize(
min_obj,
X_max,
bounds=self.bounds,
method='L-BFGS-B'
)
return res.x
def update(self, X, y):
"""Add new observation"""
self.X.append(X)
self.y.append(y)
# Example usage
def objective(x):
"""Example objective function"""
return -(x[0]**2 + x[1]**2)
# Initialize optimizer
bounds = [(-5, 5), (-5, 5)]
optimizer = BayesianOptimizer(bounds)
# Run optimization
n_iters = 10
for i in range(n_iters):
next_point = optimizer.propose_location()
f_value = objective(next_point)
optimizer.update(next_point, f_value)
print(f"\nIteration {i+1}:")
print(f"Next point: {next_point}")
print(f"Value: {f_value:.3f}")
🚀 Additional Resources - Made Simple!
- “Probabilistic Programming and Bayesian Methods for Hackers” https://arxiv.org/abs/1507.08050
- “A Tutorial on Bayesian Optimization” https://arxiv.org/abs/1807.02811
- “Practical Bayesian Optimization of Machine Learning Algorithms” https://arxiv.org/abs/1206.2944
- “An Introduction to Probabilistic Programming” https://arxiv.org/abs/1809.10756
- “Bayesian Methods for Hackers: Probabilistic Programming and Bayesian Inference” https://arxiv.org/abs/1503.02531
🎊 Awesome Work!
You’ve just learned some really powerful techniques! Don’t worry if everything doesn’t click immediately - that’s totally normal. The best way to master these concepts is to practice with your own data.
What’s next? Try implementing these examples with your own datasets. Start small, experiment, and most importantly, have fun with it! Remember, every data science expert started exactly where you are right now.
Keep coding, keep learning, and keep being awesome! 🚀