Credible Interval

Understanding credible intervals

Credible intervals emerge from Bayesian updating of probabilities, combining prior distributions with observed data through the likelihood function to produce a posterior distribution. The interval represents a range containing the parameter of interest with a specific probability (e.g., 95%).

The mathematical foundation can be expressed as:

$P(\theta_L \leq \theta \leq \theta_U | \text{data}) = 1 - \alpha$

Where:

• $\theta$ is the parameter of interest
• $\theta_L$ and $\theta_U$ are the lower and upper bounds
• $\alpha$ is the complement of the desired probability level
• $|$ denotes conditional probability

Types of credible intervals

Equal-tailed credible intervals

The most common type, where probability mass is equally distributed in both tails:

$P(\theta < \theta_L | \text{data}) = P(\theta > \theta_U | \text{data}) = \frac{\alpha}{2}$

Highest posterior density (HPD) intervals

The shortest possible interval containing the specified probability mass, particularly useful for asymmetric distributions:

$\text{HPD} = \{\theta: p(\theta|\text{data}) \geq k\}$ where $k$ is chosen to give the desired probability content.

Applications in financial markets

Risk assessment

• Portfolio Value-at-Risk (VaR) estimation
• Option pricing parameter uncertainty
• Credit risk modeling

Trading strategy evaluation

• Parameter uncertainty in signal generation
• Robust portfolio optimization
• Performance attribution analysis

Comparison with confidence intervals

Implementation considerations

1. Prior selection

   • Informative vs non-informative priors
   • Sensitivity analysis to prior assumptions

2. Computational methods

   • Markov Chain Monte Carlo (MCMC)
   • Variational inference
   • Numerical integration

3. Model validation

   • Posterior predictive checks
   • Prior-posterior analysis
   • Robustness assessment

Best practices for financial applications

1. Documentation: Clearly state prior assumptions and methodology

2. Validation: Regular backtesting and model performance assessment

3. Communication: Present results in context with clear interpretation guidelines

4. Monitoring: Continuous evaluation of model assumptions and performance

Practical example

Consider estimating the expected return of a trading strategy:

# Posterior distribution parameters
mu_posterior = 0.05  # mean
sigma_posterior = 0.02  # standard deviation

# 95% credible interval
alpha = 0.05
z_score = stats.norm.ppf(1 - alpha/2)
lower_bound = mu_posterior - z_score * sigma_posterior
upper_bound = mu_posterior + z_score * sigma_posterior

This example demonstrates a simple normal approximation to the posterior distribution, though more complex models often require MCMC methods.

Conclusion

Credible intervals provide a natural and intuitive framework for quantifying parameter uncertainty in financial applications. Their direct probabilistic interpretation makes them particularly valuable for risk management and decision-making processes where clear communication of uncertainty is crucial.