Data-driven decision-making has become essential across various fields, from finance and economics to medicine and engineering. Understanding probability and statistics is crucial for making informed choices today. Bayesian inference, a powerful tool in probabilistic reasoning, allows us to update our beliefs about an event based on new evidence.
Bayes’s theorem, a fundamental concept in probability theory, forms the foundation of Bayesian inference. This theorem provides a way to calculate the probability of an event occurring given prior knowledge and observed data. Combining our initial beliefs with the likelihood of the evidence, we can arrive at a more accurate posterior probability.
Bayesian inference is a statistical method based on Bayes’s theorem, which updates the probability of an event as new data becomes available. It is widely used in various fields, such as finance, medicine, and engineering, to make predictions and decisions based on prior knowledge and observed data. In Python, Bayesian inference can be implemented using libraries like NumPy and Matplotlib to generate and visualize posterior distributions.
This article will explore Bayesian inference and its implementation using Python, a popular programming language for data analysis and scientific computing. We will start by understanding the fundamentals of Bayes’s theorem and formula, then move on to a step-by-step guide on implementing Bayesian inference in Python. Along the way, we will discuss a real-world example of predicting website conversion rates to illustrate the practical application of this powerful technique.
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What is Bayesian Inference?
Bayesian inference is based on Bayes’s theorem, which is based on the prior probability of an event. As events happen, the probability of the event keeps updating. Let us look at the formula of Baye’s theorem.
Implementing Bayesian Inference in Python
Let us try to implement the same in Python with the code below.
import numpy as npimport matplotlib.pyplot as plt# Generate some synthetic datanp.random.seed(42)true_mu = 5true_sigma = 2data = np.random.normal(true_mu, true_sigma, size=100)# Define the prior hyperparametersprior_mu_mean = 0prior_mu_precision = 1 # Variance = 1 / precisionprior_sigma_alpha = 2prior_sigma_beta = 2 # Beta = alpha / beta# Update the prior hyperparameters with the dataposterior_mu_precision = prior_mu_precision + len(data) / true_sigma**2posterior_mu_mean = (prior_mu_precision * prior_mu_mean + np.sum(data)) / posterior_mu_precisionposterior_sigma_alpha = prior_sigma_alpha + len(data) / 2posterior_sigma_beta = prior_sigma_beta + np.sum((data - np.mean(data))**2) / 2# Calculate the posterior parametersposterior_mu = np.random.normal(posterior_mu_mean, 1 / np.sqrt(posterior_mu_precision), size=10000)posterior_sigma = np.random.gamma(posterior_sigma_alpha, 1 / posterior_sigma_beta, size=10000)# Plot the posterior distributionsplt.figure(figsize=(10, 4))plt.subplot(1, 2, 1)plt.hist(posterior_mu, bins=30, density=True, color='skyblue', edgecolor='black')plt.title('Posterior distribution of $\mu$')plt.xlabel('$\mu$')plt.ylabel('Density')plt.subplot(1, 2, 2)plt.hist(posterior_sigma, bins=30, density=True, color='lightgreen', edgecolor='black')plt.title('Posterior distribution of $\sigma$')plt.xlabel('$\sigma$')plt.ylabel('Density')plt.tight_layout()plt.show()# Calculate summary statisticsmean_mu = np.mean(posterior_mu)std_mu = np.std(posterior_mu)print("Mean of mu:", mean_mu)print("Standard deviation of mu:", std_mu)mean_sigma = np.mean(posterior_sigma)std_sigma = np.std(posterior_sigma)print("Mean of sigma:", mean_sigma)print("Standard deviation of sigma:", std_sigma)Let us look at the output of the same.
Real-World Example: Predicting Website Conversion Rates with Bayesian Inference
Let us now look at the case of a website. We are trying to predict how many people will buy the product to the ratio of the number of visitors. We have also created some parameters.
import numpy as npimport matplotlib.pyplot as plt# Observed datanum_visitors = 1000 # Total number of visitors to the websitenum_conversions = 50 # Number of conversions (desired actions)# Prior hyperparameters for the Beta distributionprior_alpha = 1 # Shape parameterprior_beta = 1 # Shape parameter# Update the prior with the observed data to get the posterior parametersposterior_alpha = prior_alpha + num_conversionsposterior_beta = prior_beta + (num_visitors - num_conversions)# Generate samples from the posterior Beta distributionposterior_samples = np.random.beta(posterior_alpha, posterior_beta, size=10000)# Plot the posterior distributionplt.figure(figsize=(8, 6))plt.hist(posterior_samples, bins=30, density=True, color='skyblue', edgecolor='black', alpha=0.7)plt.title('Posterior Distribution of Conversion Rate')plt.xlabel('Conversion Rate')plt.ylabel('Density')plt.xlim(0, 0.1) # Limiting x-axis to focus on conversion rates close to zeroplt.show()# Calculate summary statisticsmean_conversion_rate = posterior_alpha / (posterior_alpha + posterior_beta)mode_conversion_rate = (posterior_alpha - 1) / (posterior_alpha + posterior_beta - 2) # Mode of the Beta distributionprint("Mean conversion rate:", mean_conversion_rate)print("Mode conversion rate:", mode_conversion_rate)Let us look at the output of the above code and try to deduce some information from it.
According to our predictions, the probability of conversion is around 5%.
Conclusion
Here you go! Now, you know a lot more about Bayesian inference. In this article, we learned what Bayesian inference is and also touched upon how to implement it in the Python programming language. We also learned about a very simple case study where we calculated the probability of customers’ conversions if they visited a particular website. Bayesian inference, as mentioned, is also used heavily in the fields of finance, economics, and engineering.
Hope you enjoyed reading it!!
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