Building an Automated PyMC3 Model for Chess Rankings and Championships

Automatically inferring relationships to predict future chess outcomes

Key Takeaways

Comprehensive Modeling: Leveraging PyMC3 to create a Bayesian model that captures dependencies between player rankings and championship outcomes.
Latent Variables Integration: Utilizing latent variables to represent underlying factors influencing chess rankings and championship probabilities.
Automated Inference: Employing Markov Chain Monte Carlo (MCMC) sampling to infer posterior distributions and understand relationships between variables.

Introduction

Predicting future outcomes in competitive environments, such as chess rankings and championships, involves understanding the intricate relationships between various factors. By leveraging Bayesian modeling with PyMC3, we can create a robust framework that automatically infers these relationships, providing insightful predictions based on given probabilities.

Understanding the Problem

The objective is to build a PyMC3 model that captures the relationships between specific chess-related probabilities:

Will Magnus Carlsen, the world chess #1 since 2011, be ranked #1 in 2029? 24%
Will Arjun Erigaisi be the next world number 1 in chess? 28%
Will Fabiano Caruana be the next world number 1 in chess? 26%
Will Magnus Carlsen become the FIDE World Chess Champion again by 2031? 21%

Approach to Modeling

Defining the Variables and Probabilities

Each of the questions poses a specific event with an associated probability. These probabilities serve as observed data points that the model will use to infer underlying factors influencing these outcomes. The primary variables to consider include:

Carlsen Ranked #1 in 2029 – Probability: 24%
Arjun Erigaisi Next World #1 – Probability: 28%
Fabiano Caruana Next World #1 – Probability: 26%
Carlsen Becoming FIDE Champion by 2031 – Probability: 21%

Establishing Relationships Between Variables

Understanding how these variables interact is crucial. For instance, Magnus Carlsen's continued dominance at the top rank might influence the probabilities of others like Erigaisi and Caruana ascending to the #1 spot. Similarly, Carlsen's performance can impact his chances of reclaiming the championship title.

Incorporating Latent Variables

Latent variables represent unobserved factors that affect the observed probabilities. In this context, latent variables might include players' current performance ratings, historical performance trends, psychological factors, and other relevant metrics. By modeling these latent factors, we can capture the nuanced dependencies between the different outcomes.

Step-by-Step Implementation

1. Installing Necessary Libraries

Before building the model, ensure that PyMC3 and its dependencies are installed. This can be done using pip:

!pip install pymc3 theano matplotlib

2. Defining the Observed Probabilities

Start by defining the observed probabilities for each of the questions. These will serve as the foundation for the Bayesian model.

question_probs = {
    "carlsen_rank1_2029": 0.24,
    "erigaisi_next_n1": 0.28,
    "caruana_next_n1": 0.26,
    "carlsen_champion_2031": 0.21
}

3. Converting Probabilities to Log-Odds

PyMC3 often works more naturally with log-odds rather than raw probabilities. This conversion facilitates the modeling process.

def prob_to_logit(p):
    return np.log(p / (1 - p))

logit_probs = {k: prob_to_logit(v) for k, v in question_probs.items()}

4. Building the PyMC3 Model

Create a Bayesian model that incorporates both the observed probabilities and latent variables representing underlying factors.

import pymc3 as pm
import numpy as np
import theano.tensor as tt
import matplotlib.pyplot as plt

with pm.Model() as model:
    # Latent variables representing underlying performance factors
    performance_carlsen = pm.Normal('performance_carlsen', mu=0, sigma=1)
    performance_erigaisi = pm.Normal('performance_erigaisi', mu=0, sigma=1)
    performance_caruana = pm.Normal('performance_caruana', mu=0, sigma=1)
    championship_carlsen = pm.Normal('championship_carlsen', mu=0, sigma=1)
    
    # Define relationships between latent variables and observed probabilities
    logits = {
        "carlsen_rank1_2029": performance_carlsen,
        "erigaisi_next_n1": performance_erigaisi,
        "caruana_next_n1": performance_caruana,
        "carlsen_champion_2031": performance_carlsen  # Championship linked to Carlsen's performance
    }
    
    # Observed data modeled as Bernoulli distributions
    observed_carlsen_rank1 = pm.Bernoulli(
        'obs_carlsen_rank1_2029',
        logit_p=logits["carlsen_rank1_2029"],
        observed=(question_probs["carlsen_rank1_2029"] > 0.5).astype(int)
    )
    
    observed_erigaisi_n1 = pm.Bernoulli(
        'obs_erigaisi_next_n1',
        logit_p=logits["erigaisi_next_n1"],
        observed=(question_probs["erigaisi_next_n1"] > 0.5).astype(int)
    )
    
    observed_caruana_n1 = pm.Bernoulli(
        'obs_caruana_next_n1',
        logit_p=logits["caruana_next_n1"],
        observed=(question_probs["caruana_next_n1"] > 0.5).astype(int)
    )
    
    observed_carlsen_champion = pm.Bernoulli(
        'obs_carlsen_champion_2031',
        logit_p=logits["carlsen_champion_2031"],
        observed=(question_probs["carlsen_champion_2031"] > 0.5).astype(int)
    )
    
    # Define dependencies: Carlsen's performance influences others
    performance_erigaisi = pm.Normal('performance_erigaisi', mu=performance_carlsen, sigma=1)
    performance_caruana = pm.Normal('performance_caruana', mu=performance_carlsen, sigma=1)
    championship_carlsen = pm.Normal('championship_carlsen', mu=performance_carlsen, sigma=1)

5. Sampling from the Posterior

Use Markov Chain Monte Carlo (MCMC) sampling to draw samples from the posterior distributions of the latent variables.

    # Sample from the posterior
    trace = pm.sample(1000, tune=1000, cores=2, random_seed=42)

6. Analyzing the Results

After sampling, visualize and summarize the posterior distributions to understand the inferred relationships and probabilities.

    # Plot the posterior distributions
    pm.traceplot(trace)
    plt.show()
    
    # Summary of the posterior
    print(pm.summary(trace))

Comprehensive PyMC3 Model Code

The following complete code encapsulates all the steps discussed above, providing a functional PyMC3 model to infer relationships between the specified chess rankings and championship probabilities.

import pymc3 as pm
import numpy as np
import matplotlib.pyplot as plt

# Define observed probabilities
question_probs = {
    "carlsen_rank1_2029": 0.24,
    "erigaisi_next_n1": 0.28,
    "caruana_next_n1": 0.26,
    "carlsen_champion_2031": 0.21
}

# Function to convert probability to logit
def prob_to_logit(p):
    return np.log(p / (1 - p))

# Convert probabilities to log-odds
logit_probs = {k: prob_to_logit(v) for k, v in question_probs.items()}

with pm.Model() as model:
    # Latent variables representing underlying performance
    performance_carlsen = pm.Normal('performance_carlsen', mu=0, sigma=1)
    performance_erigaisi = pm.Normal('performance_erigaisi', mu=0, sigma=1)
    performance_caruana = pm.Normal('performance_caruana', mu=0, sigma=1)
    championship_carlsen = pm.Normal('championship_carlsen', mu=0, sigma=1)
    
    # Define logits based on performance
    logits = {
        "carlsen_rank1_2029": performance_carlsen,
        "erigaisi_next_n1": performance_erigaisi,
        "caruana_next_n1": performance_caruana,
        "carlsen_champion_2031": performance_carlsen  # Championship linked to Carlsen's performance
    }
    
    # Observed data as Bernoulli distributions
    observed_carlsen_rank1 = pm.Bernoulli(
        'obs_carlsen_rank1_2029',
        logit_p=logits["carlsen_rank1_2029"],
        observed=(question_probs["carlsen_rank1_2029"] > 0.5).astype(int)
    )
    
    observed_erigaisi_n1 = pm.Bernoulli(
        'obs_erigaisi_next_n1',
        logit_p=logits["erigaisi_next_n1"],
        observed=(question_probs["erigaisi_next_n1"] > 0.5).astype(int)
    )
    
    observed_caruana_n1 = pm.Bernoulli(
        'obs_caruana_next_n1',
        logit_p=logits["caruana_next_n1"],
        observed=(question_probs["caruana_next_n1"] > 0.5).astype(int)
    )
    
    observed_carlsen_champion = pm.Bernoulli(
        'obs_carlsen_champion_2031',
        logit_p=logits["carlsen_champion_2031"],
        observed=(question_probs["carlsen_champion_2031"] > 0.5).astype(int)
    )
    
    # Define dependencies: Carlsen's performance influences others
    performance_erigaisi = pm.Normal('performance_erigaisi', mu=performance_carlsen, sigma=1)
    performance_caruana = pm.Normal('performance_caruana', mu=performance_carlsen, sigma=1)
    championship_carlsen = pm.Normal('championship_carlsen', mu=performance_carlsen, sigma=1)
    
    # Sample from the posterior
    trace = pm.sample(1000, tune=1000, cores=2, random_seed=42)
    
    # Plot the posterior distributions
    pm.traceplot(trace)
    plt.show()
    
    # Summary of the posterior
    print(pm.summary(trace))

Model Interpretation and Insights

Once the model is run, the posterior distributions offer insights into the likelihood of each event given the observed probabilities and the modeled dependencies. Key interpretations include:

Carlsen's Dominance: The posterior distribution for Carlsen maintaining the #1 rank in 2029 reflects the initial probability adjusted by the influence on other players' rankings.
Emergence of New Leaders: The probabilities of Erigaisi and Caruana ascending to the #1 spot are influenced by Carlsen's performance, showcasing the interconnected nature of their rankings.
Championship Prospects: Carlsen's chances of reclaiming the FIDE World Chess Championship are directly tied to his overall performance, as modeled by the latent variable.

By analyzing the summary statistics and trace plots, one can assess the credibility intervals, mean estimates, and potential correlations between the variables, facilitating informed predictions about future chess standings.

Conclusion

Robust Bayesian Framework: Utilizing PyMC3 allows for a comprehensive Bayesian approach to model complex dependencies between chess rankings and championships.
Latent Variables Highlight Key Factors: Incorporating latent variables effectively captures the underlying performance metrics influencing the observed probabilities.
Automated Inference Enhances Predictions: MCMC sampling provides a powerful mechanism to infer posterior distributions, offering deeper insights into the relationships between variables.
Flexible Model Structure: The model can be further refined by introducing more nuanced relationships or additional data, enhancing its predictive capabilities.
Actionable Insights for Stakeholders: The resulting model aids players, coaches, and analysts in understanding potential future outcomes, facilitating strategic decision-making.

Last updated January 9, 2025