Beta Distributions: A Cornerstone of Bayesian Calibration
Hi there!
Distributions may not seem like a complex concept at first glance, but they are incredibly powerful and fundamental in the world of data analysis and statistics. Think about it this way: if you were to gather 50 shirts in various sizes and colors, you would have created a color distribution, a size distribution, and perhaps even a "how much does this shirt annoy you" distribution (jokingly, of course). The point is that as long as you have a category to measure, there's a distribution waiting to be explored.
So, what exactly is a distribution? It's essentially a way to show how a category spreads across a scale of probabilities or likelihoods. You can figure this out either from the data you have or from what you know about a particular topic. You've probably heard of terms like the normal distribution, skewed distribution, long-tailed distribution, and so on – each of these describes how data points are shaped.
Today I wanted to touch on the Beta Distribution and specifically its application in Bayesian Calibration. Bayesian Calibration is an approach that updates Bayesian inference with new data to find the best-fitting values for a model's parameters. It considers both the prior information available about these parameters and the likelihood of the observed data given those parameters.
Before we dive into Bayesian Calibration with the Beta Distribution, let's cover some technical details. Once we have those basics down, we'll explore the Bayesian Calibration with Beta Distributions with an intriguing scenario.
Beta Distribution
The beta distribution, denoted as Beta(α, β), is a probability distribution characterized by two parameters. Its probability density function (pdf) is expressed as follows:
In this equation, both α and β represent the hyperparameters, and it's important to note that they must always be greater than 0. Additionally, for the purpose of this article, we will focus on integer values exclusively.
Before we begin, let's add a visual aid to see a colorful assortment of beta distribution PDFs with α and β ranging from 0.5 all the way to 50.
Now that we have a good idea of what a beta distribution looks like, let's jump into a scenario.
Our Scenario
Our fictional company MM Manufacturing, is renowned for producing precision weights. Their system is top-notch, ensuring near-perfect calibration for the majority of their products. However, in recent times, an unusual issue has cropped up – a surge in customer complaints about weights that fall short of perfection. In response, MM Manufacturing introduced an additional layer of human verification to guarantee that every weight dispatched to customers is flawless.
But in order to analyze the trend in actual production weights, they tasked their Data Science team to analyze the likelihood of encountering these irregular weights and, more importantly, to monitor the distribution of such occurrences, in order to gain insight into the path to improved performance. Fortunately, all weights get their exact values recorded on a conveyor belt.
The Data Science team's approach is rooted in Bayesian calibration. Every month, they update a beta distribution probability density function (pdf) to assess the anomaly in weights and how it has fared over time. To do this, they use data from the conveyor belt, serving as observations to ultimately determine the posterior. They also need to establish a prior, which can be based on historical data, domain knowledge, or a non-specific, uninformative distribution.
For the alpha (α) and beta (β) values in the beta distribution for the observable data or likelihood estimation, they opt for the following strategy:
α = Number of correctly calibrated weights + 1 (ensuring α > 0)
β = Number of incorrectly calibrated weights + 1 (ensuring β > 0)
As for their choice of prior, they initially select a uninformative one, represented by Beta(1,1) (uniform distribution as shown below), which minimizes influence of the prior on the posterior and place the primary reliance on observational data.
It maybe worthwhile to deviate a little bit to note the role of prior in this context
The Role of the Prior
In the realm of Bayesian inference, the prior is where you can incorporate your perspective – well, not quite your opinion, but rather your informed perspective alongside previous observations. It comes in various forms, ranging from highly informative to completely uninformative, and it plays a crucial role in shaping the posterior distribution.
In our Bayesian calibration process, the posterior distribution is proportionally influenced by both the likelihood and the prior.
Posterior Distribution ∝ Likelihood × Prior
Furthermore, the Beta distribution serves as a conjugate prior in Bayesian inference for many distribution. This means that if you're dealing with distributions like Bernoulli, binomial, negative binomial and geometric, for the likelihood function then the resulting posterior will also belong to the Beta distribution family. In our case, the situation with anomalous weights the likelihood distribution is based on a success failure scenario much like a binomial distribution.
Now, let's explore the options for the prior, considering the nature of the distribution:
Uninformative Prior
An uninformative prior has the least impact on the posterior, making it suitable when you possess minimal information about how the distribution should appear. In the Beta Distribution, examples of uninformative priors can include:
- Beta(0.5, 0.5) or Jeffreys prior
- Beta(1, 1) or the uniform prior.
This choice is ideal when you want the likelihood to be the dominant factor without substantial influence from the prior.
Mildly Informative Prior
Mildly informative priors convey some information to the posterior. In the Beta Distribution, options for mildly informative priors can be Beta(2, 2) and Beta(1, 2).
These priors provide a gentle nudge to the posterior based on partial knowledge.
Informative Prior
When you possess substantial information about the distribution and wish to make slight adjustments based on new observations, informative priors come into play. In the Beta Distribution context, informative priors could take the form of Beta(10, 10) and Beta(20, 2) and many values on the larger end. These priors carry more weight in shaping the posterior.
With a better understanding of the different types of priors and their roles, let's return to our specific scenario of mapping the anomalous weights by MM Manufacturing into an observable posterior distribution
Python Implementation
So let's do a little bit of Anomaly Detection using the Beta Distribution prior and bayesian calibration just to make the concept clearer.
First of all, to simulate the weights produced by the conveyor belt, we'll generated synthetic data with 500 data points twice for the two scenarios below.
Scenario 1: Bayesian Calibration the first time
For the first time calibration, we use an uninformative prior denoted as Beta(1,1). We define the likelihood Beta(α , β) where α, β are:
α = correctly calibrated weights + 1 (since alpha should be > 0)
β = incorrectly calibrated weights + 1 (again for no events beta > 0)
We also generate our synthetic data where weight is considered correctly calibrated if the value is between 4.85 and 5.15, inclusive for a 5 pound weight and it's incorrectly calibrated if weight lies outside these values.
We initially generate data with 10% anomalous values.
import random
import matplotlib.pyplot as plt
from scipy.stats import beta
# Simulated data: 500 observations with 90% normal weight and 10% anomalous weights
np.random.seed(42)
normal_instances = [random.uniform(4.85, 5.15) for i in range(450)]
anomalous_instances_1 = [random.uniform(3, 4.85) for i in range(25)]
anomalous_instances_2 = [random.uniform(5.15, 6) for i in range(25)]
data = np.concatenate((normal_instances, anomalous_instances_1, anomalous_instances_2))
# Initial prior belief using a Beta distribution (uninformative uniform prior)
prior_alpha = 1
prior_beta = 1
# Beta Distribution as inferred by Observed data
likelihood_alpha = len(data[(data >= 4.85) & (data <= 5.15)]) + 1
likelihood_beta = len(data) - likelihood_alpha + 1
# Calculate posterior parameters based on observed data and prior
posterior_alpha = prior_alpha + likelihood_alpha
posterior_beta = prior_beta + likelihood_beta
# Plot the prior, likelihood and posterior Beta distributions
x = np.linspace(0, 1, 1000)
prior_distribution = beta.pdf(x, prior_alpha, prior_beta)
likelihood_distribution = beta.pdf(x, likelihood_alpha, likelihood_beta)
posterior_distribution = beta.pdf(x, posterior_alpha, posterior_beta)
plt.plot(x, prior_distribution, label='Prior Distribution')
plt.plot(x, likelihood_distribution, label='Likelihood Distribution')
plt.plot(x, posterior_distribution, label='Posterior Distribution')
plt.title('Bayesian Calibration for Anomalous Weight Detection')
plt.xlabel('Anomaly Probability')
plt.ylabel('Probability Density')
plt.legend()
plt.show()As intended, our posterior is almost exactly like the likelihood so this wasn't much of calibration. This also shows the impact of the uniform prior on the posterior.
The next month we have more data and now our prior is the posterior for previous month, similarly we could have had some information of the internal system and adjusted the prior accordingly.
Scenario 2: Bayesian Calibration update
Assuming the MM Manufacturing paid attention and made some changes to the system, now only 6% of the weights are anomalous. Now we have a more informative prior given the posterior from our previous data.
# Simulated data: 500 observations with 94% normal weight and 6% anomalous weights
np.random.seed(42)
normal_instances = [random.uniform(4.85, 5.15) for i in range(470)]
anomalous_instances_1 = [random.uniform(3, 4.85) for i in range(15)]
anomalous_instances_2 = [random.uniform(5.15, 6) for i in range(15)]
data = np.concatenate((normal_instances, anomalous_instances_1, anomalous_instances_2))
# Initial prior belief about normal behavior using a Beta distribution
prior_alpha = posterior_alpha
prior_beta = posterior_beta
# Beta Distribution as inferred by Observed data
likelihood_alpha = len(data[(data >= 4.85) & (data <= 5.15)]) + 1
likelihood_beta = len(data) - likelihood_alpha + 1
# Calculate posterior parameters based on observed data and prior
posterior_alpha = prior_alpha + likelihood_alpha
posterior_beta = prior_beta + likelihood_beta
# Plot the prior, likelihood and posterior Beta distributions
x = np.linspace(0, 1, 1000)
prior_distribution = beta.pdf(x, prior_alpha, prior_beta)
likelihood_distribution = beta.pdf(x, likelihood_alpha, likelihood_beta)
posterior_distribution = beta.pdf(x, posterior_alpha, posterior_beta)
plt.plot(x, prior_distribution, label='Prior Distribution')
plt.plot(x, likelihood_distribution, label='Likelihood Distribution')
plt.plot(x, posterior_distribution, label='Posterior Distribution')
plt.title('Bayesian Calibration for Anomalous Weight Detection')
plt.xlabel('Anomaly Probability')
plt.ylabel('Probability Density')
plt.legend()
plt.show()This time we see the impact of prior on the posterior and how much more defined the distribution is. The relation between prior, posterior and likelihood is much more clearly visible here.
Considering the two scenarios described above, it becomes evident that a variety of these outcomes can be leveraged to acquire insights into system performance, make comparisons to observe enhancements, and improve data calibration across a broad temporal spectrum.
The Beta Distribution's appeal lies in its adaptability and versatility in defining various distributions, while Bayesian Calibration's strength lies in its ability to flexibly embrace and integrate intricate model parameters.
Let's talk about some other applications.
Other Applications
No discussion about the Beta Distribution would be complete without recognizing its wide-ranging uses. It's not just used in the realm of Bayesian Inference and calibration, like we saw in the success-failure scenario earlier. The Beta Distribution also plays a crucial role in A/B testing, where it helps model the conversion rates of different web page or web app versions – a scenario similar to success and failure, just in a different context.
Furthermore the Beta Distribution can also come into play in risk analysis, where a probabilistic approach is highly informative for estimating the probability of success of a project.
Wrapping Up
In conclusion, the Beta Distribution, especially when applied in the context of Bayesian calibration, is an exceptionally valuable and elegant concept. It excels at handling the intricacies of a model while offering an intuitive approach to decision-making. Moreover, its relevance extends to a wide array of applications across various industries, where it plays a pivotal role in gaining valuable insights into the performance of the systems undergoing calibration.
The Beta Distribution is not just a theoretical concept; it's a practical and valuable asset in the data scientist's toolkit. Understanding its applications and adaptability opens doors to insights that can enhance decision-making and improve system performance. As you delve deeper into data science, remember the Beta Distribution's significance in handling complex models and gaining valuable insights across various industries.
A cool way to visualize beta distribution
Don't forget to read some of my other intriguing articles!
P-Values: Understanding Statistical Significance in Plain Language
Exploring Counterfactual Insights: From Correlation to Causation in Data Analysis
Feel free to share your thoughts in the comments.
