From fc2e11b20dca5b7bbef74c03838f8afd193b30ba Mon Sep 17 00:00:00 2001 From: antoniofrancaib Date: Fri, 22 Nov 2024 15:31:59 +0000 Subject: [PATCH] nw --- ...-ranking.md => 3-probabilistic-ranking.md} | 290 ---------- 4F13/4-factor-graphs.md | 531 ++++++++++++++++++ ...{4-text-modeling.md => 5-text-modeling.md} | 0 .../Pasted image 20241121174428.png | Bin .../Pasted image 20241121214704.png | Bin .../Pasted image 20241121214716.png | Bin .../Pasted image 20241121215227.png | Bin assets/Pasted image 20241122102423.png | Bin 0 -> 77109 bytes assets/Pasted image 20241122102907.png | Bin 0 -> 280165 bytes assets/Pasted image 20241122122750.png | Bin 0 -> 89522 bytes assets/Pasted image 20241122124517.png | Bin 0 -> 70217 bytes index.md | 5 +- 12 files changed, 534 insertions(+), 292 deletions(-) rename 4F13/{3-ranking.md => 3-probabilistic-ranking.md} (56%) create mode 100644 4F13/4-factor-graphs.md rename 4F13/{4-text-modeling.md => 5-text-modeling.md} (100%) rename Pasted image 20241121174428.png => assets/Pasted image 20241121174428.png (100%) rename Pasted image 20241121214704.png => assets/Pasted image 20241121214704.png (100%) rename Pasted image 20241121214716.png => assets/Pasted image 20241121214716.png (100%) rename Pasted image 20241121215227.png => assets/Pasted image 20241121215227.png (100%) create mode 100644 assets/Pasted image 20241122102423.png create mode 100644 assets/Pasted image 20241122102907.png create mode 100644 assets/Pasted image 20241122122750.png create mode 100644 assets/Pasted image 20241122124517.png diff --git a/4F13/3-ranking.md b/4F13/3-probabilistic-ranking.md similarity index 56% rename from 4F13/3-ranking.md rename to 4F13/3-probabilistic-ranking.md index 652d0f3..0f6e3a6 100644 --- a/4F13/3-ranking.md +++ b/4F13/3-probabilistic-ranking.md @@ -2,8 +2,6 @@ - [15-Introduction-to-Probabilistic-Ranking](#15-Introduction-to-Probabilistic-Ranking) - [16-Gibbs-Sampling-for-Inference](#16-Gibbs-Sampling-for-Inference) - [17-Gibbs-Sampling-in-TrueSkill](#17-Gibbs-Sampling-in-TrueSkill) -- [18-Factor-Graphs](#18-Factor-Graphs) -- [19-Applying-Message-Passing-to-TrueSkill™](#19-Applying-Message-Passing-to-TrueSkill™) --- @@ -373,291 +371,3 @@ $$p(y_{ij} = +1) \approx \frac{1}{N} \sum_{s=1}^N \Phi\left(\frac{w_i^{(s)} - w_ where $w_i^{(s)}$ and $w_j^{(s)}$ are samples from the posterior. ---- - -# 18-Factor-Graphs - -Probabilistic graphical models provide a powerful framework for representing complex distributions and performing efficient inference. Among these models, factor graphs are particularly useful for representing the factorization of probability distributions and facilitating efficient computation of marginal and conditional probabilities through message passing algorithms. - -### What are Factor Graphs? -A factor graph is a bipartite graphical model that represents the factorization of a function, typically a joint probability distribution. It consists of two types of nodes: -1. **Variable Nodes**: Represent the variables in the function. -2. **Factor Nodes**: Represent local functions (factors) that depend on a subset of variables. - -Edges connect factor nodes to variable nodes if the factor depends on that variable. -#### Purpose of Factor Graphs -1. **Visualization**: Provides a clear graphical representation of the dependencies between variables and factors. -2. **Computation**: Facilitates efficient computation of marginal and conditional distributions through message passing algorithms. -3. **Generalization**: Factor graphs generalize other graphical models like Bayesian networks (directed graphs) and Markov networks (undirected graphs). -#### Example of a Factor Graph -Consider a joint probability distribution that factors as: - -$$p(v,w,x,y,z)=f_1(v,w)\cdot f_2(w,x)\cdot f_3(x,y)\cdot f_4(x,z)$$ - -The corresponding factor graph has: -- **Variable Nodes**: $v, w, x, y, z$ -- **Factor Nodes**: $f_1, f_2, f_3, f_4$ -- **Edges**: Connect factors to the variables they depend on. - -![[Pasted image 20241121214716.png]] -#### Questions We Can Answer Using Factor Graphs -1. **Marginal Distributions**: What is $p(w)$? -2. **Conditional Distributions**: What is $p(w\mid y)$? -3. **Efficient Computation**: How can we compute these distributions efficiently using the structure of the factor graph? - -### Efficient Computation with Factor Graphs - -#### Challenges with Naïve Computation -Computing marginals directly can be computationally expensive due to the high dimensionality and combinatorial explosion of possible variable configurations. - -For example, computing $p(w)$ naively involves: - -$$p(w)=\sum_v \sum_x \sum_y \sum_z f_1(v,w)f_2(w,x)f_3(x,y)f_4(x,z)$$ - -If each variable can take $K$ values, the computational complexity is $O(K^5)$, which becomes infeasible as $K$ and the number of variables grow. - -#### Exploiting the Factor Graph Structure -The key to efficient computation lies in exploiting the distributive property of multiplication over addition and the separability of the factor graph: -1. **Distributive Property**: Allows us to rearrange sums and products to reduce computations. -2. **Tree Structure**: In tree-structured graphs (graphs without loops), each node separates the graph into disjoint subgraphs, enabling recursive computations. - -### Step-by-Step Computation -1. **Group Terms**: Start by grouping terms associated with each variable or factor. -2. **Local Computations**: Compute messages locally at each node, passing summarized information to neighboring nodes. -3. **Recursive Summations**: Use the distributive property to perform sums over variables in a recursive manner, reducing the overall computational complexity. - -#### Example -Compute $p(w)$ by rearranging terms: - -$$p(w)=\left(\sum_v f_1(v,w)\right)\cdot\left(\sum_x f_2(w,x)\cdot\left(\sum_y f_3(x,y)\cdot\sum_z f_4(x,z)\right)\right)$$ - -The complexity reduces from $O(K^5)$ to $O(K^4)$ or even $O(K^2)$ with further optimizations. - -![[Pasted image 20241121215227.png]] - -### The Sum-Product Algorithm - -The sum-product algorithm is a message passing algorithm used to compute marginal distributions in factor graphs efficiently. It is also known as belief propagation or factor-graph propagation. - -#### Key Steps in the Sum-Product Algorithm -1. **Initialization**: Set initial messages, typically starting with uniform distributions or prior information. -2. **Message Passing**: Iteratively compute messages from factors to variables and from variables to factors until convergence. -3. **Marginal Computation**: Once messages have stabilized, compute the marginal distributions by combining incoming messages at each variable node. - -$$p(t) = \prod_{f \in F_t} m_{f \to t}(t)$$ - -#### Message Computation Rules -##### Messages from Factors to Variables -For a factor $f$ connected to variables $x_1, x_2, \ldots, x_n$, the message from factor $f$ to variable $x_i$ is: - -$$m_{f \to t_1}(t_1) = \sum_{t_2} \sum_{t_3} \cdots \sum_{t_n} f(t_1, t_2, \ldots, t_n) \prod_{i \neq 1} m_{t_i \to f}(t_i)$$ - -**Interpretation**: Sum over all variables except $x_i$, multiplying the factor $f$ with messages from neighboring variables. - -#### Messages from Variables to Factors -For a variable $x$ connected to factors $f_1, f_2, \ldots, f_k$, the message from variable $x$ to factor $f$ is: - -$$m_{t \to f}(t) = \prod_{f_j \in F_t \setminus \{f\}} m_{f_j \to t}(t) = \frac{p(t)}{m_{f \to t}(t)}$$ - -**Interpretation**: Multiply all incoming messages from neighboring factors except the recipient factor $f$. - -### Computing Marginals -The marginal distribution for a variable $x$ is: - -$$p(x)=\prod_{f\in ne(x)} m_{f\to x}(x)$$ - -**Interpretation**: Multiply all incoming messages from factors connected to $x$. - ---- - -## 19-Applying-Message-Passing-to-TrueSkill™ -[-](#index) -### The TrueSkill™ Model -TrueSkill™ is a Bayesian rating system that models player skills and predicts match outcomes. It consists of: -1. **Player Skills ($w_i$)**: Random variables representing the skill levels of players. -2. **Performance Differences ($t_g$)**: Observed performance differences in games. -3. **Game Outcomes ($y_g$)**: Observed outcomes of games, where $y_g=+1$ if Player $I_g$ wins and $y_g=-1$ if Player $J_g$ wins. - -### TrueSkill™ Factor Graph -The factor graph for TrueSkill™ includes: -1. **Prior Factors**: Representing prior beliefs about player skills. - - $$f_i(w_i)=N(w_i;\mu_0,\sigma_0^2)$$ - -2. **Game Factors**: Modeling the relationship between skills and performance differences. - - $$h_g(w_{I_g},w_{J_g},t_g)=N(t_g;w_{I_g}-w_{J_g},\sigma_n^2)$$ - -3. **Outcome Factors**: Incorporating observed game outcomes. - - $$k_g(t_g,y_g)=\delta(y_g-\text{sign}(t_g))$$ - -### Goals -1. **Compute Marginals**: Determine the marginal distributions of player skills $p(w_i)$. -2. **Update Beliefs**: Incorporate observed game outcomes to update beliefs about player skills. - ---- - -## Part 5: Addressing Challenges in TrueSkill™ - -### Handling Loops in the Factor Graph -1. **Approximate Inference**: When the factor graph is not a tree, we can still apply message passing algorithms approximately. -2. **Iterative Message Passing**: Messages are passed iteratively until convergence, similar to belief propagation in loopy graphs. - -### Approximation of Non-Standard Messages -1. **Expectation Propagation (EP)**: An approximation method that replaces complex messages with approximations (e.g., Gaussian distributions) by matching moments. -2. **Moment Matching**: Adjusting the parameters of the approximating distribution so that its first and second moments match those of the true distribution. - -# Part 6: Expectation Propagation (EP) in TrueSkill™ - -## Overview of Expectation Propagation -EP is an iterative algorithm used to approximate complex probability distributions by simpler ones (e.g., Gaussians). It involves: - -- **Approximate Factors**: Replace intractable factors with approximate ones that are tractable (e.g., Gaussian approximations). -- **Moment Matching**: Ensure that the approximate distribution matches certain moments (mean and variance) of the true distribution. -- **Iterative Updates**: Repeat the process until convergence. - -## Steps in EP for TrueSkill™ -1. **Initialize Messages**: Start with initial messages, typically set to uniform or prior distributions. -2. **Update Skill Marginals**: Compute the marginal distributions for skills using incoming messages. -3. **Compute Messages from Skills to Games**: Use the current skill marginals to send messages to game factors. -4. **Compute Messages from Games to Performances**: Combine messages to compute the distribution of performance differences. -5. **Approximate Performance Marginals**: Use moment matching to approximate the true distribution of performance differences with a Gaussian. -6. **Compute Messages from Performances to Games**: Update messages based on the approximate performance marginals. -7. **Compute Messages from Games to Skills**: Update skill messages based on incoming messages from performances. -8. **Iterate**: Repeat the process until messages and marginals converge. - -## Detailed Equations - -### Step 1: Initialize Messages -Set initial messages from game factors to skills to be uniform or prior distributions. - -### Step 2: Update Skill Marginals -Compute the marginal for each skill $w_i$: - -$$q(w_i) = f_i(w_i) \prod_{g \in \text{games involving } w_i} m_{h_g \to w_i}(w_i)$$ - -### Step 3: Compute Messages from Skills to Games -For each game $g$: - -$$m_{w_{I_g} \to h_g}(w_{I_g}) = \frac{q(w_{I_g})}{m_{h_g \to w_{I_g}}(w_{I_g})}$$ - -$$m_{w_{J_g} \to h_g}(w_{J_g}) = \frac{q(w_{J_g})}{m_{h_g \to w_{J_g}}(w_{J_g})}$$ - -### Step 4: Compute Messages from Games to Performances -Compute the message from game factor $h_g$ to performance difference $t_g$: - -$$m_{h_g \to t_g}(t_g) = \int \int h_g(w_{I_g}, w_{J_g}, t_g) m_{w_{I_g} \to h_g}(w_{I_g}) m_{w_{J_g} \to h_g}(w_{J_g}) \, dw_{I_g} \, dw_{J_g}$$ - -This results in a Gaussian distribution because $h_g$ and the messages are Gaussian. - -### Step 5: Approximate Performance Marginals -Compute the marginal for $t_g$: - -$$q(t_g) = m_{h_g \to t_g}(t_g) \cdot k_g(t_g, y_g)$$ - -Since $k_g(t_g, y_g)$ involves a step function, $q(t_g)$ is not Gaussian. - -**Moment Matching**: Approximate $q(t_g)$ with a Gaussian by matching the first and second moments. - -### Step 6: Compute Messages from Performances to Games -Update the message from performance difference $t_g$ back to the game factor $h_g$: - -$$m_{t_g \to h_g}(t_g) = \frac{q(t_g)}{m_{h_g \to t_g}(t_g)}$$ - -### Step 7: Compute Messages from Games to Skills -Update messages from game factors back to skills: - -$$m_{h_g \to w_{I_g}}(w_{I_g}) = \int h_g(w_{I_g}, w_{J_g}, t_g) m_{t_g \to h_g}(t_g) m_{w_{J_g} \to h_g}(w_{J_g}) \, dt_g \, dw_{J_g}$$ - -This results in a Gaussian message after integration. - -### Step 8: Iterate Until Convergence -Repeat steps 2-7 until the messages and marginals stabilize. - ---- - -# Part 7: Moment Matching Approximation - -## Why Moment Matching? -1. **Intractable Distributions**: The true distributions may involve step functions or other non-Gaussian components that are difficult to handle analytically. -2. **Gaussian Approximation**: By approximating these distributions with Gaussians, we can leverage the tractability and closed-form solutions available for Gaussian distributions. - -## Calculating Moments for Truncated Gaussians - -### Truncated Gaussian Distribution -Consider a truncated Gaussian distribution: - -$$p(t) = \frac{1}{Z} N(t; \mu, \sigma^2) \cdot I(y \cdot t > 0)$$ - -where $I(\cdot)$ is the indicator function, and $y \in \{-1, +1\}$. - -### Normalization Constant -The normalization constant $Z$ is: - -$$Z = \Phi\left(\frac{y\mu}{\sigma}\right)$$ - -where $\Phi(\cdot)$ is the standard normal cumulative distribution function (CDF). - -### First Moment (Mean) -The mean $E[t]$ of the truncated Gaussian is: - -$$E[t] = \mu + \sigma \frac{N\left(\frac{y\mu}{\sigma}\right)}{\Phi\left(\frac{y\mu}{\sigma}\right)} = \mu + \sigma \lambda\left(\frac{y\mu}{\sigma}\right)$$ - -where: - -$$\lambda(z) = \frac{N(z)}{\Phi(z)}$$ - -and $N(z)$ is the standard normal PDF. - -### Second Moment (Variance) -The variance $V[t]$ is: - -$$V[t] = \sigma^2 \left(1 - \lambda\left(\frac{y\mu}{\sigma}\right) \left(\lambda\left(\frac{y\mu}{\sigma}\right) + \frac{y\mu}{\sigma}\right)\right)$$ - ---- - -# Part 8: Detailed Example of Message Passing in TrueSkill™ - -## Notations -- **Means and Variances**: - - $\mu_i, \sigma_i^2$: Parameters of skill marginals. - - $\mu_{t_g}, \sigma_{t_g}^2$: Parameters of performance difference marginals. -- **Precisions**: - - $\tau_i = \sigma_i^{-2}$ - - $\tau_{t_g} = \sigma_{t_g}^{-2}$ - -## Steps - -### Step 1: Initialize Messages -Messages from game factors to skills are initialized as: - -$$m_{h_g \to w_{I_g}}(w_{I_g}) = 1, \, m_{h_g \to w_{J_g}}(w_{J_g}) = 1$$ - -### Step 2: Update Skill Marginals -For each player $i$: - -$$q(w_i) = f_i(w_i) \prod_{g \in \text{games involving } i} m_{h_g \to w_i}(w_i)$$ - -Since all factors are Gaussian, $q(w_i)$ remains Gaussian with updated mean and variance. - ---- - -# Part 9: Summary and Key Takeaways -1. **Factor Graphs**: Provide a framework for representing factorized functions and enable efficient computation through message passing. -2. **Sum-Product Algorithm**: An algorithm for exact inference in tree-structured factor graphs using message passing. -3. **TrueSkill™ Application**: Message passing can be applied to the TrueSkill™ model to update player skill estimates based on game outcomes. - -**Challenges in TrueSkill™**: -- Non-Tree Structure: Requires approximate inference methods due to loops in the graph. -- Non-Gaussian Messages: Handled by approximation methods like Expectation Propagation. - -**Expectation Propagation**: -- An iterative method that approximates complex distributions with tractable ones by matching moments. - -**Moment Matching**: -- A key technique in EP for approximating intractable distributions (e.g., truncated Gaussians) with Gaussians. - -## Conclusion -Understanding factor graphs and message passing algorithms is crucial for performing efficient inference in complex probabilistic models. By delving into the details of the sum-product algorithm and approximation methods like Expectation Propagation, we gain the tools to tackle challenges in real-world applications like TrueSkill™. - diff --git a/4F13/4-factor-graphs.md b/4F13/4-factor-graphs.md new file mode 100644 index 0000000..37281a8 --- /dev/null +++ b/4F13/4-factor-graphs.md @@ -0,0 +1,531 @@ +# Index +- [18-Factor-Graphs](#18-Factor-Graphs) +- [19-Applying-Message-Passing-to-TrueSkill™](#19-Applying-Message-Passing-to-TrueSkill™) +- [20-Moment-Matching-Approximation](#20-Moment-Matching-Approximation) + +# 18-Factor-Graphs + +Probabilistic graphical models provide a powerful framework for representing complex distributions and performing efficient inference. Among these models, factor graphs are particularly useful for representing the factorization of probability distributions and facilitating efficient computation of marginal and conditional probabilities through message passing algorithms. + +### What are Factor Graphs? +A factor graph is a bipartite graphical model that represents the factorization of a function, typically a joint probability distribution. It consists of two types of nodes: +1. **Variable Nodes**: Represent the variables in the function. +2. **Factor Nodes**: Represent local functions (factors) that depend on a subset of variables. + +Edges connect factor nodes to variable nodes if the factor depends on that variable. +#### Purpose of Factor Graphs +1. **Visualization**: Provides a clear graphical representation of the dependencies between variables and factors. +2. **Computation**: Facilitates efficient computation of marginal and conditional distributions through message passing algorithms. +3. **Generalization**: Factor graphs generalize other graphical models like Bayesian networks (directed graphs) and Markov networks (undirected graphs). +#### Example of a Factor Graph +Consider a joint probability distribution that factors as: + +$$p(v,w,x,y,z)=f_1(v,w)\cdot f_2(w,x)\cdot f_3(x,y)\cdot f_4(x,z)$$ + +The corresponding factor graph has: +- **Variable Nodes**: $v, w, x, y, z$ +- **Factor Nodes**: $f_1, f_2, f_3, f_4$ +- **Edges**: Connect factors to the variables they depend on. + +![[Pasted image 20241121214716.png]] +#### Questions We Can Answer Using Factor Graphs +1. **Marginal Distributions**: What is $p(w)$? +2. **Conditional Distributions**: What is $p(w\mid y)$? +3. **Efficient Computation**: How can we compute these distributions efficiently using the structure of the factor graph? + +### Efficient Computation with Factor Graphs + +#### Challenges with Naïve Computation +Computing marginals directly can be computationally expensive due to the high dimensionality and combinatorial explosion of possible variable configurations. + +For example, computing $p(w)$ naively involves: + +$$p(w)=\sum_v \sum_x \sum_y \sum_z f_1(v,w)f_2(w,x)f_3(x,y)f_4(x,z)$$ + +If each variable can take $K$ values, the computational complexity is $O(K^5)$, which becomes infeasible as $K$ and the number of variables grow. + +#### Exploiting the Factor Graph Structure +The key to efficient computation lies in exploiting the distributive property of multiplication over addition and the separability of the factor graph: +1. **Distributive Property**: Allows us to rearrange sums and products to reduce computations. +2. **Tree Structure**: In tree-structured graphs (graphs without loops), each node separates the graph into disjoint subgraphs, enabling recursive computations. + +### Step-by-Step Computation +1. **Group Terms**: Start by grouping terms associated with each variable or factor. +2. **Local Computations**: Compute messages locally at each node, passing summarized information to neighboring nodes. +3. **Recursive Summations**: Use the distributive property to perform sums over variables in a recursive manner, reducing the overall computational complexity. + +#### Example +Compute $p(w)$ by rearranging terms: + +$$p(w)=\left(\sum_v f_1(v,w)\right)\cdot\left(\sum_x f_2(w,x)\cdot\left(\sum_y f_3(x,y)\cdot\sum_z f_4(x,z)\right)\right)$$ + +The complexity reduces from $O(K^5)$ to $O(K^4)$ or even $O(K^2)$ with further optimizations. + +![[Pasted image 20241121215227.png]] + +### The Sum-Product Algorithm + +The sum-product algorithm is a message passing algorithm used to compute marginal distributions in factor graphs efficiently. It is also known as belief propagation or factor-graph propagation. + +#### Key Steps in the Sum-Product Algorithm +1. **Initialization**: Set initial messages, typically starting with uniform distributions or prior information. +2. **Message Passing**: Iteratively compute messages from factors to variables and from variables to factors until convergence. +3. **Marginal Computation**: Once messages have stabilized, compute the marginal distributions by combining incoming messages at each variable node. + +$$p(t) = \prod_{f \in F_t} m_{f \to t}(t)$$ + +#### Message Computation Rules +##### Messages from Factors to Variables +For a factor $f$ connected to variables $x_1, x_2, \ldots, x_n$, the message from factor $f$ to variable $x_i$ is: + +$$m_{f \to t_1}(t_1) = \sum_{t_2} \sum_{t_3} \cdots \sum_{t_n} f(t_1, t_2, \ldots, t_n) \prod_{i \neq 1} m_{t_i \to f}(t_i)$$ + +**Interpretation**: Sum over all variables except $x_i$, multiplying the factor $f$ with messages from neighboring variables. + +#### Messages from Variables to Factors +For a variable $x$ connected to factors $f_1, f_2, \ldots, f_k$, the message from variable $x$ to factor $f$ is: + +$$m_{t \to f}(t) = \prod_{f_j \in F_t \setminus \{f\}} m_{f_j \to t}(t) = \frac{p(t)}{m_{f \to t}(t)}$$ + +**Interpretation**: Multiply all incoming messages from neighboring factors except the recipient factor $f$. + +### Computing Marginals +The marginal distribution for a variable $x$ is: + +$$p(x)=\prod_{f\in ne(x)} m_{f\to x}(x)$$ + +**Interpretation**: Multiply all incoming messages from factors connected to $x$. + +--- + +## 19-Applying-Message-Passing-to-TrueSkill™ +[-](#index) +### The TrueSkill™ Model +TrueSkill™ is a Bayesian rating system that models player skills and predicts match outcomes. It consists of: +1. **Player Skills ($w_i$)**: Random variables representing the skill levels of players. +2. **Performance Differences ($t_g$)**: Observed performance differences in games. +3. **Game Outcomes ($y_g$)**: Observed outcomes of games, where $y_g=+1$ if Player $I_g$ wins and $y_g=-1$ if Player $J_g$ wins. + +### TrueSkill™ Factor Graph +The factor graph for TrueSkill™ includes: +1. **Prior Factors**: Representing prior beliefs about player skills. + $$f_i(w_i)=N(w_i;\mu_0,\sigma_0^2)$$ + +2. **Game Factors**: Modeling the relationship between skills and performance differences. + $$h_g(w_{I_g},w_{J_g},t_g)=N(t_g;w_{I_g}-w_{J_g},\sigma_n^2)$$ + +3. **Outcome Factors**: Incorporating observed game outcomes. + $$k_g(t_g,y_g)=\delta(y_g-\text{sign}(t_g))$$ + +![[Pasted image 20241122102423.png]] +### Goals +1. **Compute Marginals**: Determine the marginal distributions of player skills $p(w_i)$. +2. **Update Beliefs**: Incorporate observed game outcomes to update beliefs about player skills. + + +### Addressing Challenges in TrueSkill™ + +#### Handling Loops in the Factor Graph +1. **Approximate Inference**: When the factor graph is not a tree, we can still apply message passing algorithms approximately. +2. **Iterative Message Passing**: Messages are passed iteratively until convergence, similar to belief propagation in loopy graphs. + +#### Approximation of Non-Standard Messages +1. **Expectation Propagation (EP)**: An approximation method that replaces complex messages with approximations (e.g., Gaussian distributions) by matching moments. +2. **Moment Matching**: Adjusting the parameters of the approximating distribution so that its first and second moments match those of the true distribution. + +### Expectation Propagation (EP) in TrueSkill™ + +EP is an iterative algorithm used to approximate complex probability distributions by simpler ones (e.g., Gaussians). It involves: + +- **Approximate Factors**: Replace intractable factors with approximate ones that are tractable (e.g., Gaussian approximations). +- **Moment Matching**: Ensure that the approximate distribution matches certain moments (mean and variance) of the true distribution. +- **Iterative Updates**: Repeat the process until convergence. + +#### Steps in EP for TrueSkill™ +1. **Initialize Messages**: Start with initial messages, typically set to uniform or prior distributions. +2. **Update Skill Marginals**: Compute the marginal distributions for skills using incoming messages. +3. **Compute Messages from Skills to Games**: Use the current skill marginals to send messages to game factors. +4. **Compute Messages from Games to Performances**: Combine messages to compute the distribution of performance differences. +5. **Approximate Performance Marginals**: Use moment matching to approximate the true distribution of performance differences with a Gaussian. +6. **Compute Messages from Performances to Games**: Update messages based on the approximate performance marginals. +7. **Compute Messages from Games to Skills**: Update skill messages based on incoming messages from performances. +8. **Iterate**: Repeat the process until messages and marginals converge. + +![[Pasted image 20241122102907.png]] +### Detailed Steps: + +##### Step 0: Initialization +**Objective**: Initialize the messages from game factors to skill variables. + +###### Equations: +For each game $g$: +$$m_{h_g \to w_{I_g}}^{\tau=0}(w_{I_g}) = 1$$ +$$m_{h_g \to w_{J_g}}^{\tau=0}(w_{J_g}) = 1$$ + +This means that initially, the messages from the game factors to the player skills are uniform distributions. We have no information from the games yet. + +##### Step 1: Compute Marginal Skills +**Objective**: Compute the marginal distribution for each player's skill based on the prior and incoming messages. + +###### Equation: +For player $i$: +$$q^{\tau}(w_i) = f_i(w_i) \prod_g m_{h_g \to w_i}^{\tau}(w_i)$$ + +- $f_i(w_i)$: Prior belief about player $i$'s skill. +- $m_{h_g \to w_i}^{\tau}(w_i)$: Message from game $g$ to player $i$ at iteration $\tau$. + +Since both the prior and the messages are Gaussians (or initialized as uniform, which is a special case of a Gaussian with infinite variance), the marginal $q^{\tau}(w_i)$ is also Gaussian. + +**Interpretation**: +We update our belief about each player's skill by combining the prior belief with the information received from all games involving that player. + +##### Step 2: Compute Skill-to-Game Messages +**Objective**: Calculate the messages from player skills to game factors, reflecting the influence of the player's current skill estimate on the game. + +###### Equations: +For each game $g$ involving player $i$: +$$m_{w_i \to h_g}^{\tau}(w_i) = \frac{q^{\tau}(w_i)}{m_{h_g \to w_i}^{\tau}(w_i)}$$ + +- $q^{\tau}(w_i)$: The marginal skill distribution computed in Step 1. +- $m_{h_g \to w_i}^{\tau}(w_i)$: The incoming message from game $g$ to player $i$. + +**Interpretation**: +This calculation effectively removes the influence of game $g$ from the marginal skill to isolate the impact of player $i$'s skill on that game. + +##### Step 3: Compute Game-to-Performance Messages +**Objective**: Compute the message from the game factor to the performance difference variable $t_g$. + +###### Equation: +$$m_{h_g \to t_g}^{\tau}(t_g) = \int \int h_g(w_{I_g}, w_{J_g}, t_g) m_{w_{I_g} \to h_g}^{\tau}(w_{I_g}) m_{w_{J_g} \to h_g}^{\tau}(w_{J_g}) \, dw_{I_g} \, dw_{J_g}$$ + +**Explanation**: +- We integrate over the skills $w_{I_g}$ and $w_{J_g}$ of the two players in game $g$. +- $h_g(w_{I_g}, w_{J_g}, t_g)$ is the game factor relating the skills to the performance difference. +- $m_{w_{I_g} \to h_g}^{\tau}(w_{I_g})$ and $m_{w_{J_g} \to h_g}^{\tau}(w_{J_g})$ are the messages from the player skills to the game factor. + +**Result**: +The resulting message $m_{h_g \to t_g}^{\tau}(t_g)$ is a Gaussian distribution over the performance difference $t_g$. + +**Interpretation**: +This message represents our belief about the performance difference $t_g$ based on the current estimates of the players' skills. + +##### Step 4: Compute Marginal Performances (Approximation) +**Objective**: Update the marginal distribution of the performance difference $t_g$ by incorporating the observed game outcome. + +###### Equation: +$$q^{\tau+1}(t_g) = \text{Approx}\big(m_{h_g \to t_g}^{\tau}(t_g) \cdot k_g(t_g, y_g)\big)$$ + +**Explanation**: +- $m_{h_g \to t_g}^{\tau}(t_g)$: Message from the game factor to $t_g$. +- $k_g(t_g, y_g)$: The outcome factor, enforcing that the sign of $t_g$ matches the observed outcome $y_g$. + +The product $m_{h_g \to t_g}^{\tau}(t_g) \cdot k_g(t_g, y_g)$ is not Gaussian because $k_g$ introduces a nonlinearity (it's a step function). + +**Approximation**: +We approximate this product with a Gaussian distribution by matching the first two moments (mean and variance). This is where EP comes into play. + +**Interpretation**: +We're updating our belief about the performance difference $t_g$ considering the actual game result, while keeping the distribution Gaussian for tractability. + +##### Step 5: Compute Performance-to-Game Messages +**Objective**: Update the message from the performance difference $t_g$ back to the game factor $h_g$. + +###### Equation: +$$m_{t_g \to h_g}^{\tau+1}(t_g) = \frac{q^{\tau+1}(t_g)}{m_{h_g \to t_g}^{\tau}(t_g)}$$ + +**Explanation**: +- $q^{\tau+1}(t_g)$: The approximated marginal from Step 4. +- $m_{h_g \to t_g}^{\tau}(t_g)$: The message computed in Step 3. + +By dividing the updated marginal by the incoming message, we obtain the outgoing message from $t_g$ to $h_g$. + +**Interpretation**: +This message captures the updated information about the performance difference that will be sent back to the game factor. + +##### Step 6: Compute Game-to-Skill Messages +**Objective**: Update the messages from the game factor $h_g$ back to the player skills $w_{I_g}$ and $w_{J_g}$. + +###### Equations: +For Player $I_g$: +$$m_{h_g \to w_{I_g}}^{\tau+1}(w_{I_g}) = \int h_g(w_{I_g}, w_{J_g}, t_g) m_{t_g \to h_g}^{\tau+1}(t_g) m_{w_{J_g} \to h_g}^{\tau}(w_{J_g}) \, dt_g \, dw_{J_g}$$ + +For Player $J_g$: +$$m_{h_g \to w_{J_g}}^{\tau+1}(w_{J_g}) = \int h_g(w_{I_g}, w_{J_g}, t_g) m_{t_g \to h_g}^{\tau+1}(t_g) m_{w_{I_g} \to h_g}^{\tau}(w_{I_g}) \, dt_g \, dw_{I_g}$$ + +**Explanation**: +We integrate over the performance difference $t_g$ and the opponent's skill. + +**Result**: +These integrals result in Gaussian messages back to the player skills. + +**Interpretation**: +These messages update our beliefs about each player's skill based on the updated performance difference and the opponent's skill estimate. + +##### Step 7: Iterate Until Convergence +We repeat Steps 1 to 6, incrementing $\tau$ each time, until the messages and marginals stabilize. This iterative process refines our estimates of the player skills with each pass. + + +### Detailed Computations Using Natural Parameters + +To make the computations more tractable, we can represent Gaussians using their natural parameters: + +- **Precision ($r$)**: The inverse of the variance ($r = \nu^{-1}$). +- **Natural Mean ($\lambda$)**: The product of the precision and the mean ($\lambda = r \mu$). + +This representation simplifies the multiplication and division of Gaussian messages. + +### Computations in Terms of Natural Parameters + +#### Initialization: +The incoming messages $m_{h_g \to w_i}^{\tau=0}$ are initialized with zero precision and zero natural mean: +$$r_{h_g \to w_i}^{\tau=0} = 0, \; \lambda_{h_g \to w_i}^{\tau=0} = 0$$ + + +#### Step 1: Compute Marginal Skills +For player $i$: +$$r_i^{\tau} = r_0 + \sum_g r_{h_g \to w_i}^{\tau}$$ +$$\lambda_i^{\tau} = \lambda_0 + \sum_g \lambda_{h_g \to w_i}^{\tau}$$ + +- $r_0$ and $\lambda_0$: Precision and natural mean from the prior $f_i(w_i)$. +- The sum is over all games involving player $i$. + +#### Step 2: Compute Skill-to-Game Messages +For game $g$ involving player $i$: +$$r_{w_i \to h_g}^{\tau} = r_i^{\tau} - r_{h_g \to w_i}^{\tau}$$ +$$\lambda_{w_i \to h_g}^{\tau} = \lambda_i^{\tau} - \lambda_{h_g \to w_i}^{\tau}$$ + + +#### Step 3: Compute Game-to-Performance Messages +Compute the variance ($\nu$) and mean ($\mu$): +$$\nu_{h_g \to t_g}^{\tau} = \sigma_n^2 + \nu_{w_{I_g} \to h_g}^{\tau} + \nu_{w_{J_g} \to h_g}^{\tau}$$ +$$\mu_{h_g \to t_g}^{\tau} = \mu_{w_{I_g} \to h_g}^{\tau} - \mu_{w_{J_g} \to h_g}^{\tau}$$ + +#### Step 4: Approximate Marginal Performances +The exact product $m_{h_g \to t_g}^{\tau}(t_g) \cdot k_g(t_g, y_g)$ is not Gaussian due to the step function in $k_g$. + +##### Moment Matching: +We compute the mean ($\tilde{\mu}_g^{\tau+1}$) and variance ($\tilde{\nu}_g^{\tau+1}$) of the true distribution and set the approximating Gaussian to have these moments. + +##### Formulas: +Let $\Phi(x)$ be the cumulative distribution function (CDF) of the standard normal distribution, and $\phi(x)$ be the probability density function (PDF). + +Define: +- **Standardized Mean**: + $$\hat{\mu}_g = \frac{\mu_{h_g \to t_g}^{\tau}}{\nu_{h_g \to t_g}^{\tau}}$$ +- **Correction Factors**: + $$\Psi(\hat{\mu}_g) = \frac{\phi(\hat{\mu}_g)}{\Phi(\hat{\mu}_g)}$$ + $$\Lambda(\hat{\mu}_g) = \Psi(\hat{\mu}_g)(\Psi(\hat{\mu}_g) + \hat{\mu}_g)$$ + +Then: +- **Updated Variance**: + $$\tilde{\nu}_g^{\tau+1} = \nu_{h_g \to t_g}^{\tau}(1 - \Lambda(\hat{\mu}_g))$$ +- **Updated Mean**: + $$\tilde{\mu}_g^{\tau+1} = \mu_{h_g \to t_g}^{\tau} + \nu_{h_g \to t_g}^{\tau} \Psi(\hat{\mu}_g)$$ + +**Interpretation**: +These updates adjust the mean and variance of the performance difference to account for the observed game outcome. + +#### Step 5: Compute Performance-to-Game Messages +Compute the natural parameters of the message: +$$r_{t_g \to h_g}^{\tau+1} = \tilde{r}_g^{\tau+1} - r_{h_g \to t_g}^{\tau}$$ +$$\lambda_{t_g \to h_g}^{\tau+1} = \tilde{\lambda}_g^{\tau+1} - \lambda_{h_g \to t_g}^{\tau}$$ + +Where: +$$\tilde{r}_g^{\tau+1} = (\tilde{\nu}_g^{\tau+1})^{-1}$$ +$$\tilde{\lambda}_g^{\tau+1} = \tilde{r}_g^{\tau+1} \tilde{\mu}_g^{\tau+1}$$ + +**Interpretation**: +We adjust the message from $t_g$ to $h_g$ based on the updated marginal of $t_g$. + +#### Step 6: Compute Game-to-Skill Messages +For Player $I_g$: +$$\nu_{h_g \to w_{I_g}}^{\tau+1} = \sigma_n^2 + \nu_{t_g \to h_g}^{\tau+1} + \nu_{w_{J_g} \to h_g}^{\tau}$$ +$$\mu_{h_g \to w_{I_g}}^{\tau+1} = \mu_{w_{J_g} \to h_g}^{\tau} + \mu_{t_g \to h_g}^{\tau+1}$$ + +For Player $J_g$: +$$\nu_{h_g \to w_{J_g}}^{\tau+1} = \sigma_n^2 + \nu_{t_g \to h_g}^{\tau+1} + \nu_{w_{I_g} \to h_g}^{\tau}$$ +$$\mu_{h_g \to w_{J_g}}^{\tau+1} = \mu_{w_{I_g} \to h_g}^{\tau} - \mu_{t_g \to h_g}^{\tau+1}$$ + +**Interpretation**: +These messages incorporate the updated performance information and the opponent's skill to refine the player's skill estimate. + + +## Iterative Process: +We repeat the steps, updating messages and marginals in each iteration. The algorithm continues until the changes in the skill estimates are below a certain threshold, indicating convergence. + +--- + +## 20-Moment-Matching-Approximation + +### Why Moment Matching? +1. **Intractable Distributions**: The true distributions may involve step functions or other non-Gaussian components that are difficult to handle analytically. +2. **Gaussian Approximation**: By approximating these distributions with Gaussians, we can leverage the tractability and closed-form solutions available for Gaussian distributions. + +### Handling Binary Outcomes with Gaussian Approximations + +#### The Challenge + +In TrueSkill™, game outcomes ($y_g$) are binary ($+1$ or $-1$), indicating which player won. However, the Bayesian framework and message passing rely on Gaussian distributions for computational efficiency. The challenge arises when attempting to incorporate binary information into a Gaussian framework. + +#### Key Idea: Moment Matching + +To reconcile binary outcomes with Gaussian approximations, moment matching is employed. This technique approximates the influence of the binary variable by adjusting the moments (mean and variance) of the Gaussian distribution to reflect the observed outcome. + +#### Moment Matching for Truncated Gaussian Densities + +##### Truncated Gaussian Density + +When incorporating a binary outcome into a Gaussian framework, we effectively truncate the Gaussian distribution based on the observed outcome. For instance, if Player $I_g$ wins ($y_g = +1$), the performance difference $t_g$ must be positive, leading to a truncated Gaussian over $t_g$. + +##### Defining the Truncated Gaussian + +Consider the truncated Gaussian density function: + +$$p(t) = \frac{1}{Z_t} \delta(y - \text{sign}(t)) N(t; \mu, \sigma^2)$$ + +where: + +- $N(t; \mu, \sigma^2)$ is the normal distribution with mean $\mu$ and variance $\sigma^2$. +- $\delta(x)$ is the Dirac delta function (aka indicator function), ensuring $y = \text{sign}(t)$. +- $Z_t$ is the normalization constant. +- $y \in \{-1, +1\}$ represents the observed game outcome. + +![[Pasted image 20241122124517.png]] +##### Objective: Approximate $p(t)$ with a Gaussian $q(t)$ + +We aim to approximate the truncated Gaussian $p(t)$ with a Gaussian $q(t)$ that has the same first and second moments (mean and variance): + +$$q(t) = N(t; \tilde{\mu}, \tilde{\sigma}^2)$$ + +###### First Moment (Mean) + +$$E[t] = \langle t \rangle_{p(t)}$$ + +###### Second Central Moment (Variance) + +$$V[t] = \langle t^2 \rangle_{p(t)} - \langle t \rangle_{p(t)}^2$$ + +##### Calculating the Moments + +To determine $\tilde{\mu}$ and $\tilde{\sigma}^2$, we compute the first and second moments of $p(t)$ and set them equal to those of $q(t)$. + +##### Detailed Derivation of Moments + +###### Normalization Constant $Z_t$ + +$$Z_t = \Phi\left(\frac{y\mu}{\sigma}\right)$$ + +where $\Phi$ is the cumulative distribution function (CDF) of the standard normal distribution. + +###### Derivation: + +**i. Case 1: $y = +1$** + +When $y = +1$, $p(t)$ is non-zero only for $t > 0$. Therefore: + +$$Z_t = \int_{-\infty}^{+\infty} \delta(+1 - \text{sign}(t)) \mathcal{N}(t; \mu, \sigma^2) \, dt = \int_{0}^{+\infty} \mathcal{N}(t; \mu, \sigma^2) \, dt = \int_{-\mu/\sigma}^{+\infty} \mathcal{N}(z; 0, 1) \, dz = \Phi\left(\frac{\mu}{\sigma}\right).$$ + +**ii. Case 2: $y = -1$** + +When $y = -1$, $p(t)$ is non-zero only for $t < 0$. Therefore: + +$$ +Z_t = \int_{-\infty}^{+\infty} \delta(-1 - \text{sign}(t)) \mathcal{N}(t; \mu, \sigma^2) \, dt = \int_{-\infty}^{0} \mathcal{N}(t; \mu, \sigma^2) \, dt = \int_{-\infty}^{-\mu/\sigma} \mathcal{N}(z; 0, 1) \, dz = \Phi\left(-\frac{\mu}{\sigma}\right) . +$$ + +We can generalize this using $y \in \{-1, +1\}$: + +$$ +Z_t = \Phi\left(\frac{y\mu}{\sigma}\right). +$$ + +--- +###### First Moment (Mean) + +To determine the first moment (mean) of the truncated Gaussian distribution, we differentiate the normalization constant $Z_t$ with respect to the mean parameter $\mu$: + +$$ +\frac{\partial Z_t}{\partial \mu} = \frac{\partial}{\partial \mu} \int_{0}^{+\infty} \mathcal{N}(t; y\mu, \sigma^2) \, dt = \int_{0}^{+\infty} \frac{\partial}{\partial \mu} \mathcal{N}(t; y\mu, \sigma^2) \, dt. +$$ + +Evaluating the derivative inside the integral, we obtain: + +$$ += \int_{0}^{+\infty} y\sigma^{-2}(t - y\mu)\mathcal{N}(t; y\mu, \sigma^2) \, dt. +$$ + +This expression can be rewritten by factoring out the constants $y$ and $\sigma^{-2}$: + +$$ += y\sigma^{-2} \int_{0}^{+\infty} (t - y\mu)\mathcal{N}(t; y\mu, \sigma^2) \, dt. +$$ + +Recognizing that $p(t) = \frac{\mathcal{N}(t; y\mu, \sigma^2)}{Z_t}$ is the normalized truncated Gaussian density, the integral simplifies to: + +$$ += y\sigma^{-2} Z_t \left( \langle t \rangle_{p(t)} - y\mu \right) = yZ_t\sigma^{-2} \langle t \rangle_{p(t)} - \mu Z_t\sigma^{-2}., +$$ + +where $\langle t \rangle_{p(t)}$ denotes the expectation of $t$ under the distribution $p(t)$. + +**Crucial Observation:** Since $y \in \{-1, +1\}$, it follows that $y^2 = 1$, effectively neutralizing the $y^2$ term in the equation. + +Alternatively, the derivative of the normalization constant can also be expressed using the properties of the cumulative distribution function (CDF): + +$$ +\frac{\partial Z_t}{\partial \mu} = \frac{\partial}{\partial \mu} \Phi\left(\frac{y\mu}{\sigma}\right) = \frac{y}{\sigma} \mathcal{N}\left(\frac{y\mu}{\sigma}; 0, 1\right), +$$ + +where $\Phi$ is the CDF of the standard normal distribution, and $\mathcal{N}$ represents the probability density function (PDF) of the standard normal distribution. + +By equating both expressions for $\frac{\partial Z_t}{\partial \mu}$, we solve for the expectation $\langle t \rangle_{p(t)}$: + +$$\langle t \rangle_{p(t)} = y\mu + \sigma \frac{\mathcal{N}\left(\frac{y\mu}{\sigma}; 0, 1 \right)}{\Phi\left(\frac{y\mu}{\sigma}\right)} = y\mu + \sigma \Psi\left(\frac{y\mu}{\sigma}\right).$$ + +where we introduced the **tilting function** $\Psi(z) = \frac{\mathcal{N}(z)}{\Phi(z)}$. + +##### Second Moment (Variance) + +To compute the second moment (variance) of the truncated Gaussian distribution, we differentiate the normalization constant $Z_t$ twice with respect to $\mu$: + +$$ +\frac{\partial^2 Z_t}{\partial \mu^2} = \frac{\partial}{\partial \mu} \int_{0}^{+\infty} y\sigma^{-2}(t - y\mu)\mathcal{N}(t; y\mu, \sigma^2) \, dt. +$$ + +Expanding the derivative inside the integral: + +$$ += \int_{0}^{+\infty} y\sigma^{-2} \frac{\partial}{\partial \mu} \left[ (t - y\mu)\mathcal{N}(t; y\mu, \sigma^2) \right] \, dt. +$$ + +Carrying out the differentiation: + +$$ += \int_{0}^{+\infty} y\sigma^{-2} \left[ -\sigma^{-2} + \sigma^{-4}(t - y\mu)^2 \right] \mathcal{N}(t; y\mu, \sigma^2) \, dt. +$$ + +Simplifying, we express this in terms of expectations under $p(t)$: + +$$ += \Phi\left(\frac{y\mu}{\sigma}\right) \left\langle -\sigma^{-2} + \sigma^{-4}(t - y\mu)^2 \right\rangle_{p(t)}. +$$ + +Alternatively, differentiating the earlier expression for the first derivative provides another perspective: + +$$ +\frac{\partial^2 Z_t}{\partial \mu^2} = \frac{\partial}{\partial \mu} \left[ y \mathcal{N}\left(\frac{y\mu}{\sigma}; 0, 1\right) \right] = -\sigma^{-2} y\mu \mathcal{N}\left(\frac{y\mu}{\sigma}; 0, 1\right). +$$ + +By equating both expressions for $\frac{\partial^2 Z_t}{\partial \mu^2}$, we solve for the variance $V[t]$: + +$$ +V[t] = \sigma^2 \left(1 - \Lambda\left(\frac{y\mu}{\sigma}\right)\right), +$$ + +where we define: + +$$ +\Lambda(z) = \Psi(z) \left( \Psi(z) + z \right). +$$ + +**Clarification:** The term involving $y^2$ emerges during the derivation but simplifies to 1 due to $y \in \{-1, +1\}$, thereby making it redundant in the final variance expression. diff --git a/4F13/4-text-modeling.md b/4F13/5-text-modeling.md similarity index 100% rename from 4F13/4-text-modeling.md rename to 4F13/5-text-modeling.md diff --git a/Pasted image 20241121174428.png b/assets/Pasted image 20241121174428.png similarity index 100% rename from Pasted image 20241121174428.png rename to assets/Pasted image 20241121174428.png diff --git a/Pasted image 20241121214704.png b/assets/Pasted image 20241121214704.png similarity index 100% rename from Pasted image 20241121214704.png rename to assets/Pasted image 20241121214704.png diff --git a/Pasted image 20241121214716.png b/assets/Pasted image 20241121214716.png similarity index 100% rename from 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z^>8}C0m|E42!+|o!BMO4rd2YyY+lsX)oFlNrnw)&TmJ866P3Wt4z}|yaM7we!3C>w z19|WdU_OZZd)GF94+1nvLNIH4+#r<~jk z`rnWM{@0IcFt5b>cRXe_m>@yof5veD7sF#gr@#Lu`hR~R?gyDgqE)SNum8_D=8*W| z@a}(S%zux#kqiROsrRzSum3X+(kF1yykHvk|2|?K9LN&K_bUr@|Iavca8KkES_i?l T+NOt4;E#fgs&u8KS@8b@WdAE; literal 0 HcmV?d00001 diff --git a/index.md b/index.md index e23e307..cdcac91 100644 --- a/index.md +++ b/index.md @@ -12,8 +12,9 @@ Reference Book: - [Parametric Models](4F13/1-parametric-models.md) - [Gaussian Processes](4F13/2-gaussian-processes.md) -- [Probabilistic Ranking](4F13/3-ranking.md) -- [Text Modeling](4F13/4-text-modeling.md) +- [Probabilistic Ranking](4F13/3-probabilistic-ranking.md) +- [Factor Graphs](4F13/4-factor-graphs.md) +- [Text Modeling](4F13/5-text-modeling.md) Reference Book: Docs: https://mlg.eng.cam.ac.uk/teaching/4f13/2425/