Julia Assignment 2 version 2:

Stochastic Variational Inference in the TrueSkill Model

STA414/STA2014 and CSC412/CSC2506 Winter 2020

David Duvenaud and Jesse Bettencourt Student Number

March  15, 2020

 

 

The goal of this assignment is to get you familiar with the basics of Bayesian inference in large models with continuous latent variables, and the basics of stochastic variational inference.

 

Background We’ll implement a variant of the TrueSkill model, a player ranking system for competitive games originally developed for Halo 2. It is a generalization of the Elo rating system in Chess. For the curious, the original 2007 NIPS paper introducing the trueskill paper can be found here: http://papers. nips.cc/paper/3079-trueskilltm-a-bayesian-skill-rating-system.pdf

This assignment is based on one developed by Carl Rasmussen at Cambridge for his course on probabilistic machine learning: http://mlg.eng.cam.ac.uk/teaching/4f13/1920/

 

0.1 Model definition

We’ll consider a slightly simplified version of the original trueskill model. We assume that each player has a true, but unknown skill z_i ∈ R. We use N to denote the number of players.

The prior. The prior over each player’s skill is a standard normal distribution, and all player’s skills are

a prior independent.

 

The likelihood. For each observed game, the probability that player i beats player j, given the player’s skills z_A and z_B, is:

 

 

where

p(A beat B|z_A, z_B) = σ(z_i − z_j)

1

σ(y) =

1 + exp(−y)

 

There can be more than one game played between a pair of players, and in this case the outcome of each game is independent given the players’ skills. We use M to denote the number of games.

 

The data. The data will be an array of game outcomes. Each row contains a pair of player indices. The first index in each pair is the winner of the game, the second index is the loser. If there were M games played, then the array has shape M × 2.

 

• Implementing the model [10points]
  • [2 points] Implement a function log prior that computes the log of the prior over all player’s skills. Specifically, given a K × N  array where each row is a setting of the skills for all N  players, it returns          aK × 1 array, where each row contains a scalar giving the log-prior for that set of
  • [3 points] Implement a function logp a beats b that, given a pair of skills z_aand z_b evaluates the log-likelihood that player with skill z_a beat player with skill z_b under the model detailed above. To ensure numerical stability, use the function log1pexp that computes log(1 + exp(x)) in a numerically stable way.  This function is provided by  jl and imported already, and also by Python’s numpy.
  • [3points] Assuming all game outcomes are i.d. conditioned on all players’ skills, implement a function all games log likelihood that takes a batch of player skills zs and a collection of observed games games and gives a batch of log-likelihoods for those observations. Specifically, given a K N array where each row is a setting of the skills for all N players, and an M 2 array of game outcomes, it returns a K 1 array, where each row contains a scalar giving the log-likelihood of all games for that set of skills. Hint: You should be able to write this function without using for loops, although you might want to start that way to make sure what you’ve written is correct. If A is an array of integers, you can index the corresponding entries of another matrix B for every entry in A by writing B[A].
  • [2 points] Implement a function joint log density which combines the log-prior and log-likelihood ofthe observations to give p(z₁, z₂, . . . , z_N , all game outcomes)

 

2 Examining the posterior for only two players and toy data [10 points]

To get a feel for this model, we’ll first consider the case where we only have 2 players, A and B. We’ll examine how the prior and likelihood interact when conditioning on different sets of games.

Provided in the starter code is a function skillcontour! which evaluates a provided function on a grid of z_A and z_B’s and plots the isocontours of that function.  As well there is a function plot line equal skill!. We have included an example for how you can use these functions.

We also provided a function two player toy games which produces toy data for two players. I.e.

two player toy games(5,3) produces a dataset where player A wins 5 games and player B wins 3 games.

 

• [2  points]  For  two  players A  and B, plot the isocontours of the joint prior over  their skills.  Also plot  theline of equal skill, z_A = z_B. Hint: you’ve already implemented the log of the likelihood
• [2 points] Plot the isocontours of the likelihood function. Also plot the line of equal skill, z_A= z_B.
• [2 points] Plot isocountours of the joint posterior over  z_Aand z_B  given that player A beat player B  in one match.  Since the contours don’t  depend on the normalization constant, you can simply plot    the isocontours of the log of joint distribution of p(z_A, z_B, A beat B) Also plot the line of equal skill,      z_A = z_B.
• [2points] Plot isocountours of the joint posterior over z_A and z_B given that 10 matches were played, and player A beat player B all 10 times. Also plot the line of equal skill, z_A = z_B.
• [2  points] Plot isocountours of the joint posterior over  z_Aand z_B  given that 20 matches were played,  and each player beat the other 10  Also plot the line of equal skill, z_A = z_B.

For all plots, label both axes.

 

3 Stochastic Variational Inference on Two Players and Toy Data [18 points]

One nice thing about a Bayesian approach is that it separates the model specification from the approxi- mate inference strategy. The original Trueskill paper from 2007 used message passing. Carl Rasmussen’s assignment uses Gibbs sampling, a form of Markov Chain Monte Carlo. We’ll use gradient-based stochastic variational inference, which wasn’t invented until around 2014.

In this question we will optimize an approximate posterior distribution with stochastic variational infer- ence to approximate the true posterior.

 

• [5 points] Implement a function elbo which computes an unbiased estimate of the evidence lower As discussed in class, the ELBO is equal to the KL divergence between the true posterior p(z|data), and an approximate posterior, q_φ(z|data), plus an unknown constant. Use a fully-factorized Gaussiandistribution for q_φ(z|data). This estimator takes the following arguments:
  • params, the parameters φ of the approximate posteriorq_φ(z|data).

A function logp, which is equal to the true posterior plus a constant.  This function must take a  batch of samples of z. If we have N  players, we can consider B-many samples from the joint over all players’ skills. This batch of samples zs will be an array with dimensions (N, B).

• num samples, the number of samples to

This function should return a single scalar. Hint: You will need to use the reparamterization trick when sampling zs.

• [2  points]  Write  a loss function called neg toy elbo that takes variational distribution parameters   andan array of game outcomes, and returns the negative elbo estimate with 100
• [5 points] Write an optimization function called fit toy variational dist which takes initial vari- ational parameters, and the evidence. Inside it will perform a number of iterations of gradient descent where for each iteration:

• Computethe gradient of the loss with respect to the parameters using automatic
• Updatethe parameters by taking an lr-scaled step in the direction of the descending
• Reportthe loss with the new parameters (using @info or print statements)
• On the same set of axes plot the target distribution in red and the variational approximation in blue.

Return the parameters resulting from training.

• [2 points]  Initialize a variational distribution parameters and optimize them to approximate the joint  where we observe player A winning 1 game. Report the final loss. Also plot the optimized variational approximationcontours (in blue) aand the target distribution (in red) on the same
• [2 points]  Initialize a variational distribution parameters and optimize them to approximate the joint  where we observe player A winning 10 games. Report the final loss. Also plot the optimized variational approximationcontours (in blue) aand the target distribution (in red) on the same
• [2 points]  Initialize a variational distribution parameters and optimize them to approximate the joint  where we  observe player A winning 10 games and player B winning 10 games.  Report the final loss.       Also plot the optimized  variational  approximation  contours  (in  blue)  aand  the  target  distribution  (in  red) on the same

For all plots, label both axes.

 

4 Approximate inference conditioned on real data [24 points]

Load the dataset from tennis data.mat containing two matrices:

• Wis a 107 by 1 matrix, whose i’th entry is the name of player i.

G is a 1801 by 2 matrix of game outcomes (actually tennis matches), one row per game. The first column contains the indices of the players who won. The second column contains the indices of the player who lost.

Compute the following using your code from the earlier questions in the assignment, but conditioning on the tennis match outcomes:

• [1point] For any two players i and j, p(z_i, z_j|all games) is always proportional to p(z_i, z_j, all games).  In general, are the isocontours of p(z_i, z_j all games) the same as those of p(z_i, z_j games between i and j)? That is, do the games between other players besides i and j provide information about the skill of players i and j? A simple yes or no

Hint: One way  to answer this is to draw the graphical model for three players,  i,  j, and k, and the  results of games between all three pairs, and then examine conditional independencies. If you do this, there’s no need to include the graphical models in your assignment.

• [5 points] Write a new optimization function fit variational dist like the one from the previous question except it does not plot anything. Initialize a variational distribution and fit it to the joint distribution with all the observed tennis games from the dataset. Report the final negative ELBO estimate after
• [2points] Plot the approximate mean and variance of all players, sorted by  For example, in Julia, you can use: perm = sortperm(means); plot(means[perm], yerror=exp.(logstd[perm])) There’s no need to include the names of the players.
• [2points] List the names of the 10 players with the highest mean skill under the variational
• [3 points] Plot the joint approximate posterior over  the skills of Roger Federerand Rafael Nadal.  Use  the approximate posterior that you fit in question 4 part
• [5 points] Derive the exact probability under a factorized Guassian over two players’ skills that one has higher skill than the other, as a function of the two means and variances over their skills. Express  your answer in terms of the cumulative distribution function of a one-dimensional Gaussian random variable.

Hint 1:  Use a linear change of variables y_A, y_B = z_A z_B, z_B. What does the line of equal skill look like after this transformation?

• Hint 2:If X ∼ N (µ, Σ), then AX ∼ N (Aµ, AΣA^T ) where A is a linear
• Hint 3: Marginalization in Gaussians is easy: if X ∼ N (µ, Σ), then the ith element of X has a marginaldistribution X_i ∼ N (µ_i, Σ_ii)

• [2 points] Using the formula from part c, compute the exact probability under  your  approximate  posterior that Roger Federer has higher skill than Rafael Nadal. Then,  estimate  it using  simple  Monte  Carlowith 10000 examples, again using your approximate posterior.
• [2 points] Using the formula from part c, compute the probability that Roger Federer is better than theplayer with the lowest mean  Compute this quantity exactly, and then estimate it using simple Monte Carlo with 10000 examples, again using your approximate posterior.
• [2 points] Imagine that we knew ahead of time that we were examining the skills of top tennis players, and so changed our prior on all players to Normal(10, 1). Which answers in this section would this change? No need to show your work, just list the letters of the questions whose answers would be different in

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