Probabilistic Models
Probabilistic models treat learning as the management of uncertainty. Instead of giving absolute answers, they assign probabilities to outcomes (for example, an 85% chance that an email is spam), reflecting our incomplete knowledge.
Example Scenario 1: Weather forecasting estimates a range based on historical data. Example Scenario 2: A medical AI might give a diagnosis with a 73% confidence. They output probability distributions that make them robust to noisy or incomplete data.
Frequentist models interpret probability as the long-term frequency of events in repeated trials. They estimate parameters directly from observed data without incorporating prior beliefs.
Everyday example: Flipping a coin 100 times and observing 55 heads leads to a 55% estimate for heads. Such methods rely on hypothesis testing and objective data analysis, as used in many scientific fields.
Logistic regression is a statistical model for binary classification. Despite its name, it is used for classification tasks rather than regression.
Example: Similar to how a doctor evaluates multiple symptoms to assess disease probability rather than giving a simple yes/no answer. It is widely used in credit scoring, spam detection, and medical diagnosis.
Bayesian models use probability theory to represent uncertainty and update beliefs as new evidence is obtained. They are based on Bayes' theorem, which combines prior knowledge with observed data.
Example: A doctor using both historical data and current symptoms to estimate the probability of a disease. Applications include spam filters, recommendation systems, and weather forecasting.
Naive Bayes is a simple probabilistic classifier based on Bayes' theorem with a strong (naive) independence assumption between features. Despite this assumption, it performs remarkably well for many tasks.
Key characteristics include computational efficiency and suitability for high-dimensional data. An everyday example is a spam filter that uses word frequencies to classify emails.
Bayesian networks extend Naive Bayes by representing complex probabilistic relationships between variables using directed acyclic graphs (DAGs). They explicitly model conditional dependencies.
Key characteristics include capturing causal relationships and handling missing data. Imagine a doctor mapping how smoking increases lung cancer risk and leads to shortness of breath.
Gaussian processes are non-parametric Bayesian models that define distributions over functions rather than parameters. They excel at modeling continuous data and quantifying uncertainty.
Key characteristics include principled uncertainty estimates and automatic adaptation of complexity. Imagine predicting temperature throughout a day with confidence intervals.
They are particularly useful in regression tasks where uncertainty estimation is crucial.
Markov models describe systems in which the future state depends only on the current state, not on past states. This memoryless property makes them tractable for various sequential tasks.
Example: A board game where only the current position matters for the next move. They are used for forecasting, stock market analysis, and other time-dependent phenomena.
Hidden Markov Models (HMMs) model sequential data with a series of hidden states that produce observable outputs. They solve evaluation, decoding, and learning problems for sequences.
Key concept: HMMs have a hidden state process (which follows the Markov property) and an observation process dependent on the current state. Example: Inferring the weather in a windowless room by observing people’s clothing.
Markov chains model sequences where only the current state determines the next state. Over time, they often settle into a stationary distribution that reflects long-term probabilities.
Everyday example: Weather patterns where if today is sunny, there is an 80% chance tomorrow will also be sunny. Applications include stock market predictions and website navigation analysis.
Monte Carlo methods use random sampling to approximate solutions for problems that are difficult to solve analytically. They rely on the law of large numbers to estimate values through repeated simulation.
Example: Estimating the area of an irregular lake by randomly throwing darts at a map. They are used in financial risk assessment, weather forecasting, computer graphics, drug discovery, and reinforcement learning.
Probabilistic methods extend beyond basic models, offering sophisticated tools for uncertainty quantification across diverse applications. These approaches provide robust frameworks for reasoning under uncertainty, enabling more nuanced and reliable predictions in complex domains.
- Kalman Filters: Recursive estimators that optimally track dynamic systems in the presence of noise. They maintain a probability distribution over the system state and update it with each new measurement, making them essential for navigation systems, financial forecasting, and sensor fusion.
- Particle Filters: Non-parametric implementations of Bayes filters that approximate posterior distributions using random samples (particles). They excel at tracking non-linear, non-Gaussian systems where traditional methods fail, with applications in robotics, computer vision, and target tracking.
- Markov Chain Monte Carlo (MCMC): A family of algorithms that sample from probability distributions by constructing Markov chains with the desired distribution as equilibrium. MCMC methods like Metropolis-Hastings and Gibbs sampling tackle problems too complex for analytical solutions, revolutionizing fields from physics to genomics.