Understanding Probabilistic Regression Model Python Tutorial: Predicting with Confidence | Techniculus

Understanding Probabilistic Regression Model Python Tutorial

what is probabilistic regression model?

Probabilistic regression model is a statistical technique that models the relationship between a dependent variable and one or more independent variables, while also considering the uncertainty in the predictions. Unlike traditional regression models, which provide a single predicted value for each set of input variables, probabilistic regression models provide a distribution of possible outcomes, along with an associated probability for each outcome.

One of the most common forms of probabilistic regression is Bayesian regression. Bayesian regression uses Bayes' theorem to update the probability distribution of the model parameters based on new data. The model initially assumes a prior distribution for the parameters, which is then updated with the likelihood of the data given the parameters. The resulting distribution of the parameters is then used to make predictions about new data.

In probabilistic regression, the output is not a single value but rather a distribution. This distribution can be used to estimate the probability of any particular outcome, or to construct confidence intervals around the predicted value. Confidence intervals provide a range of values within which the predicted value is likely to fall, with a given degree of confidence.

Probabilistic regression can be particularly useful in cases where the data is noisy or the relationship between the dependent and independent variables is complex. By taking into account the uncertainty in the predictions, probabilistic regression models can provide a more accurate and nuanced understanding of the data.

One example of the application of probabilistic regression is in the field of finance. In finance, predicting future stock prices is a notoriously difficult task, with many factors affecting the outcome. A probabilistic regression model could be used to predict the future price of a stock, along with a probability distribution for the predicted price. This could be used by investors to make informed decisions about whether to buy or sell a particular stock.

Thus the probabilistic regression is a powerful statistical technique that provides a more nuanced understanding of the relationship between variables, while also considering the uncertainty in the predictions. By providing a distribution of possible outcomes and associated probabilities, probabilistic regression models can be used to make more informed decisions in a wide range of fields, from finance to healthcare to engineering.

Types of probabilistic regression model:

Here are several types of probabilistic regression models that can be used depending on the specific needs and characteristics of the data. Some common types of probabilistic regression models are:

1) Bayesian Linear Regression: As mentioned earlier, this is one of the most common types of probabilistic regression models. It uses Bayes' theorem to update the probability distribution of the model parameters based on new data, allowing for uncertainty in the predictions.

2) Logistic Regression: This type of probabilistic regression model is used when the dependent variable is binary, such as yes or no. It models the probability of the binary outcome as a function of the independent variables, and provides a probability distribution for the predicted outcome.

3) Poisson Regression: This type of probabilistic regression model is used when the dependent variable is a count, such as the number of occurrences of an event. It models the distribution of the counts as a function of the independent variables, and provides a probability distribution for the predicted counts.

4) Gaussian Process Regression: This type of probabilistic regression model is a non-parametric approach that does not assume any particular functional form for the relationship between the dependent and independent variables. It uses a kernel function to model the covariance between observations, and provides a distribution of possible functions that could describe the data.

5) Time-Series Regression: This type of probabilistic regression model is used when the data is collected over time, and the dependent variable is a function of both time and the independent variables. It models the relationship between the variables over time, and provides a probability distribution for future values of the dependent variable.

These are just a few examples of the types of probabilistic regression models that exist. The choice of model will depend on the specific characteristics of the data and the goals of the analysis.

We'll discuss about Bayesian Linear Regression and Logistic Regression in detail, the rest will be covered in the upcoming blogs.

Bayesian Linear Regression:

Bayesian linear regression is a statistical technique that models the relationship between a dependent variable and one or more independent variables, while taking into account the uncertainty in the predictions. It uses Bayes' theorem to update the probability distribution of the model parameters based on new data, allowing for uncertainty in the predictions.

In Bayesian linear regression, the model initially assumes a prior distribution for the parameters, which reflects the researcher's beliefs about the likely values of the parameters before seeing the data. The prior distribution can be chosen based on previous knowledge or data, or it can be chosen to be non-informative, allowing the data to have a stronger influence on the model.

The prior distribution is then updated with the likelihood of the data given the parameters. The likelihood function represents the probability of observing the data given the model parameters. It is calculated using the normal distribution, which assumes that the errors between the predicted values and the actual values are normally distributed.

The resulting distribution of the parameters is called the posterior distribution, which represents the updated belief about the parameters given the observed data. The posterior distribution can be used to make predictions about new data, by sampling from the distribution to generate a range of possible outcomes.

One advantage of Bayesian linear regression is that it provides a natural way to incorporate regularization into the model. Regularization is a technique used to prevent overfitting, which occurs when the model fits the noise in the data instead of the underlying relationship between the variables. In Bayesian linear regression, regularization is achieved by adding a penalty term to the likelihood function, which biases the model towards simpler solutions.

Another advantage of Bayesian linear regression is that it allows for uncertainty in the predictions. Instead of providing a single predicted value, Bayesian linear regression provides a distribution of possible outcomes, along with an associated probability for each outcome. This can be particularly useful when making decisions based on the predictions, as it allows for a more nuanced understanding of the possible outcomes.

So we can say that Bayesian linear regression is a powerful statistical technique that provides a more nuanced understanding of the relationship between variables, while also considering the uncertainty in the predictions. By providing a distribution of possible outcomes and associated probabilities, Bayesian linear regression models can be used to make more informed decisions in a wide range of fields, from finance to healthcare to engineering.

Bayesian linear regression python sklearn code tutorial:

Here is an example code that demonstrates how to perform Bayesian linear regression using the scikit-learn (sklearn) package:

from sklearn.linear_model import BayesianRidge import numpy as np import matplotlib.pyplot as plt # Generate some toy data np.random.seed(123) x = np.linspace(0, 1, 50) y = 0.5 + 2 * x + np.random.normal(0, 0.2, size=50) # Reshape the data to fit the model X = x[:, np.newaxis] # Fit the model model = BayesianRidge() model.fit(X, y) # Predict the output for new data x_new = np.linspace(0, 1, 100) X_new = x_new[:, np.newaxis] y_new, y_std = model.predict(X_new, return_std=True) # Plot the results plt.scatter(x, y, color='black') plt.plot(x_new, y_new, color='blue', linewidth=2) plt.fill_between(x_new, y_new - y_std, y_new + y_std, color='gray') plt.title('Bayesian Linear Regression') plt.show()

Let's go through each step of the code in detail:

1) We import the BayesianRidge class from the linear_model module of the sklearn package. This class provides an implementation of Bayesian linear regression using a ridge prior.

2) We generate some toy data to use as an example. In this case, we generate 50 data points that follow a linear relationship with some random noise added to it.

3) We reshape the data to fit the input format required by the BayesianRidge class. Specifically, we reshape the x array to be a column vector, and store it as the X variable.

4) We fit the model using the BayesianRidge class. This fits the Bayesian linear regression model to the training data, using the default hyperparameters.

5) We predict the output for new data using the predict() method of the BayesianRidge object. We pass the X_new variable, which contains a range of input values we want to predict the output for. We also set the return_std parameter to True, which returns the standard deviation of the predictions.

We plot the results using Matplotlib. We plot the original data points as a scatter plot, and the predicted output as a line. We also fill in the area between the upper and lower bounds of the standard deviation to illustrate the uncertainty in the predictions.

This code demonstrates how to perform Bayesian linear regression in Python using scikit-learn. Unlike the PyMC3 approach, the sklearn approach provides a more familiar and simple way to perform Bayesian linear regression using a familiar package. However, it may not offer as much flexibility as PyMC3 for more complex models.

Logistic Regression:

Logistic regression is a popular machine learning algorithm that is used for binary classification tasks, where the goal is to predict a binary outcome, such as yes or no, true or false, 0 or 1. The algorithm is named logistic regression because it uses a logistic function to model the relationship between the input variables and the output.

Logistic regression is a supervised learning algorithm, meaning that it requires a labeled training set to learn from. In binary classification problems, the labeled training set consists of a set of input-output pairs, where each input corresponds to a set of features, and the output corresponds to a binary label indicating the class of the input.

The goal of logistic regression is to learn a model that can predict the probability of the positive class (i.e., the output being 1) given a set of input features. The logistic function, also known as the sigmoid function, is used to model this probability. The sigmoid function is defined as:

sigmoid(x) = 1 / (1 + exp(-x))

The sigmoid function has an S-shaped curve that ranges from 0 to 1, which makes it useful for modeling probabilities. The input to the sigmoid function is a linear combination of the input features, weighted by a set of coefficients (also known as weights) and added to a bias term. The coefficients and bias term are learned from the training set using a method called gradient descent.

The logistic regression model can be represented mathematically as:

p(y=1 | x) = sigmoid(wTx + b)

where p(y=1 | x) is the probability of the positive class given the input features x, w is the weight vector, b is the bias term, and T is the transpose operator.

The logistic regression model can be trained using a maximum likelihood estimation approach. The goal is to find the values of the weight vector w and bias term b that maximize the likelihood of the observed data. This can be done using gradient descent, a common optimization algorithm that iteratively updates the weights to minimize the loss function.

 

Logistic regression python sklearn code tutorial:

Here is an example code that demonstrates how to perform logistic regression in Python using the scikit-learn (sklearn) package:

from sklearn.linear_model import LogisticRegression from sklearn.model_selection import train_test_split from sklearn.datasets import load_breast_cancer # Load the breast cancer dataset data = load_breast_cancer() # Split the data into training and testing sets X_train, X_test, y_train, y_test = train_test_split(data.data, data.target, test_size=0.2, random_state=123) # Fit the logistic regression model model = LogisticRegression() model.fit(X_train, y_train) # Evaluate the model on the testing set score = model.score(X_test, y_test) print("Accuracy:", score)

Let's go through each step of the code in detail:

1) We import the LogisticRegression class from the linear_model module, which provides an implementation of logistic regression in scikit-learn. We also import the train_test_split function from the model_selection module to split the data into training and testing sets. Finally, we import the load_breast_cancer function from the datasets module to load a built-in dataset of breast cancer diagnosis.

2) We load the breast cancer dataset using the load_breast_cancer() function, which returns a dictionary-like object containing the features and labels.

3) We split the data into training and testing sets using the train_test_split() function. We pass the data.data array as the input features, and data.target as the binary labels. We also set the test_size parameter to 0.2, which means we will use 20% of the data for testing and 80% for training. We also set the random_state parameter to 123 to ensure reproducibility.

4) We fit the logistic regression model using the LogisticRegression() class. This fits the logistic regression model to the training data.

5) We evaluate the model on the testing set using the score() method of the LogisticRegression object. This returns the accuracy of the model on the testing data.

6) We print the accuracy of the model on the testing set.

This code demonstrates how to perform logistic regression in Python using scikit-learn. The LogisticRegression class provides a simple and efficient way to perform binary classification tasks, and the train_test_split() function provides a convenient way to split the data into training and testing sets. The code can be easily adapted to work with other datasets and classification problems.

Difference between deterministic and probabilistic regression model: 

The main difference between deterministic and probabilistic regression models lies in the way they model the relationship between the input variables and the output variable.

In a deterministic regression model, the output variable is a deterministic function of the input variables. This means that given the same input variables, the output variable will always have the same value. The function used to model the relationship between the input and output variables is usually a linear function, such as y = mx + b in simple linear regression, where m and b are constants.

On the other hand, in a probabilistic regression model, the output variable is a probabilistic function of the input variables. This means that given the same input variables, the output variable will have a different value with some probability. The function used to model the relationship between the input and output variables is usually a non-linear function, such as a sigmoid function in logistic regression or a Gaussian distribution in Bayesian linear regression.

The goal of a deterministic regression model is to find the values of the parameters (such as m and b in linear regression) that minimize the difference between the predicted output and the actual output in the training data. This is done by minimizing a cost function, such as the mean squared error.

In a probabilistic regression model, the goal is to learn the underlying probability distribution of the output variable given the input variables. This is done by estimating the parameters of the probability distribution using maximum likelihood estimation or Bayesian inference. The probabilistic nature of the model allows us to quantify the uncertainty in the predictions, which can be useful in many applications.

In summary, the main difference between deterministic and probabilistic regression models is that the former models the relationship between the input and output variables as a deterministic function, while the latter models it as a probabilistic function. Deterministic models aim to minimize the difference between the predicted and actual outputs, while probabilistic models aim to estimate the underlying probability distribution of the output given the input.