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Introduction to Approximate Inference
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Today, weβre going to discuss approximate inference in graphical models. Can anyone tell me why we sometimes can't rely on exact inference methods?
Maybe because the system is too complex?
What if it has cycles? Those might make it harder, right?
Exactly! High dimensions and cycles complicate the computations. That's where approximate inference comes into play. We use sampling or variational methods to navigate through these challenges.
What are some of the methods used in approximation?
Good question! Weβll cover techniques like Gibbs Sampling and Metropolis-Hastings under sampling methods and discuss variational inference.
To summarize, approximate inference is essential when exact methods are impractical due to complexity and dimensionality.
Sampling Methods: Monte Carlo
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Letβs dive into the first category of approximate inference, which is sampling methods. Who knows how Gibbs Sampling works?
Isnβt it something about taking turns to sample each variable based on others?
Exactly! Gibbs Sampling takes turns sampling each variable from its conditional distribution based on the values of other variables. This method helps us draw samples until we get a representative distribution.
And what about Metropolis-Hastings? Iβve heard of that too.
Great! Metropolis-Hastings constructs a proposal distribution to generate samples. After proposing a new sample, it accepts or rejects it based on a probability criterion, thus ensuring we sample from our desired distribution.
In summary, both Gibbs Sampling and Metropolis-Hastings provide ways to approximate complex distributions through iterative sampling techniques.
Variational Inference
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Now letβs shift gears to variational inference. Can anyone explain what this approach focuses on?
I think it tries to approximate the true distribution by using simpler distributions, right?
Spot on! Variational inference optimizes a simpler distribution such that it's as close as possible to the true distribution. The optimization process often involves maximizing the Evidence Lower Bound, or ELBO.
How does this method help in practice?
It allows us to handle complex models more easily by reducing computational costs while still being accurate.
To summarize, variational inference is critical in approximating complex distributions, especially when dealing with high dimensions.
Introduction & Overview
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Quick Overview
Standard
The section discusses approximate inference techniques necessary for dealing with graphical models in scenarios where exact inference methods are impractical. Key techniques include sampling methods, specifically Monte Carlo methods like Gibbs Sampling and Metropolis-Hastings, as well as variational inference approaches that approximate complex distributions with simpler ones.
Detailed
Approximate Inference
In scenarios where exact inference in graphical models becomes intractable, approximate inference offers viable solutions, primarily when dealing with complex systems that have cycles or high-dimensional datasets. This section introduces two principal methods:
- Sampling Methods: These include Monte Carlo techniques, where two significant methods are highlighted:
- Gibbs Sampling: A method that allows the sampling of the conditional distribution of each variable, iteratively updating them based on sampled values from other variables.
- Metropolis-Hastings: This algorithm generates samples by constructing a proposal distribution and accepting or rejecting samples based on a predefined probability criterion, ultimately leading to a distribution that approximates the desired target distribution.
- Variational Inference: This approach attempts to approximate the true distribution of the model with a simpler distribution. It does this by optimizing the Evidence Lower Bound (ELBO), ensuring that the approximated distribution remains close to the actual one.
Both of these methods play a vital role in inference tasks where conventional algorithms fail or are computationally prohibitive, thereby enabling the practical use of graphical models in various complex applications.
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Introduction to Approximate Inference
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Used when exact inference is intractable due to cycles or high dimensions.
Detailed Explanation
Approximate inference is a method used in probabilistic modeling when exact calculations are not feasible. This situation often arises in complex systems where the number of potential outcomes is extremely large or when the graphical model contains cycles. In these cases, we cannot reliably compute exact probabilities due to computational constraints or the complexity of the relationships between variables.
Examples & Analogies
Consider trying to predict the weather for a whole month based on complex meteorological data. If we attempted to evaluate every single possible weather condition exactly, it would take an impractical amount of time and resources. Instead, we might use approximate methods, like weather models, which give us a reasonable estimate based on a subset of data rather than trying to compute absolute certainty.
Sampling Methods
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Monte Carlo methods:
- Gibbs Sampling
- Metropolis-Hastings
Detailed Explanation
Sampling methods are a main approach to approximate inference. They involve generating random samples from a probability distribution to estimate characteristics of that distribution. Monte Carlo methods are a category of algorithms that provide numerical results utilizing random sampling. Two common Monte Carlo methods are Gibbs Sampling and Metropolis-Hastings.
- Gibbs Sampling: This technique involves iteratively sampling from the conditional distributions of each variable, given the others. It's useful when the joint distribution is complex, but the conditional distributions are easier to work with.
- Metropolis-Hastings: This method proposes new samples based on previous ones and accepts or rejects them based on a criteria that ensures the correct distribution is approached over time.
Examples & Analogies
Imagine a scavenger hunt where you want to find treasure hidden at various places in a large park. Instead of checking every single spot (which could take forever), you decide to randomly check a few spots based on a rule: if one spot looks promising based on what others found, youβll keep checking similar areas. Over time, this strategy helps you hone in on the treasure's likely location much faster than exhaustive searching.
Variational Inference
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Approximate true distribution with a simpler one.
β’ Optimizes a lower bound on the log-likelihood (ELBO).
Detailed Explanation
Variational inference is another method of approximate inference that aims to simplify the problem of estimating a probability distribution. Instead of sampling, it approximates the true distribution with a simpler one that is easier to work with. This is done by finding parameters for the simpler distribution that make it as close as possible to the true distribution. The process involves optimizing a quality measure known as the Evidence Lower Bound (ELBO), which helps gauge how well the chosen distribution approximates the actual distribution.
Examples & Analogies
Think of a musician trying to play a complex symphony on a simplified instrument. While the original symphony captures every note of a rich orchestra, the musician might adapt it to fit into a solo performance on a guitar. They find the best way to express the symphony's main themes within the limitations of their instrument, creating a simpler version that is still enjoyable and representative of the original work.
Definitions & Key Concepts
Learn essential terms and foundational ideas that form the basis of the topic.
Key Concepts
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Approximate Inference: Techniques for estimating probability distributions when exact methods are impractical.
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Gibbs Sampling: A method for sampling from the conditional distribution of variables iteratively.
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Metropolis-Hastings: An algorithm that accepts or rejects samples to approximate a target distribution.
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Variational Inference: A strategy for approximating complex distributions using simpler functions.
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Evidence Lower Bound (ELBO): A metric used in variational inference to determine how well an approximate distribution performs.
Examples & Real-Life Applications
See how the concepts apply in real-world scenarios to understand their practical implications.
Examples
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In a Bayesian network for spam detection, Gibbs sampling could be used to iteratively update probabilities of features based on observed data.
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Metropolis-Hastings can be applied to social network analysis to infer the likelihood of a person being friends with another based on existing connections.
Memory Aids
Use mnemonics, acronyms, or visual cues to help remember key information more easily.
π΅ Rhymes Time
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Sampling schemes for graphs can be grand, Gibbs and Hastings lend a hand.
π Fascinating Stories
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Imagine a baker creating a perfect cake with ingredients that vary. Just like baking, you need a method to blend the right proportions; Gibbs and Hastings help gather those ingredients into the best mix.
π§ Other Memory Gems
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GHV - Gibbs, Hastings, Variational; remember this trio for approximation in graphical models!
π― Super Acronyms
SOME - Sampling, Optimization, Monte Carlo, Evidence. This helps remember key techniques used in approximate inference.
Flash Cards
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Glossary of Terms
Review the Definitions for terms.
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Term: Approximate Inference
Definition:
Techniques used in graphical models to approximate probabilities when exact inference is computationally infeasible.
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Term: Gibbs Sampling
Definition:
A Monte Carlo method for obtaining a sequence of observations from a multivariate probability distribution.
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Term: MetropolisHastings
Definition:
A sampling method that generates samples from a probability distribution by constructing a proposal distribution.
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Term: Variational Inference
Definition:
An approach that approximates the true distribution with a simpler one by optimizing a lower bound on the log-likelihood.
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Term: Evidence Lower Bound (ELBO)
Definition:
A lower bound on the log likelihood used in variational inference methods.