MCMC & Inference

While the reference CRML runtime focuses on Monte Carlo simulation, it also provides a lightweight Metropolis–Hastings (MH) engine for 1D parameters.

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1. Metropolis–Hastings in one dimension

We want to sample from a posterior distribution:

$$p(\theta \mid \mathcal{D}) \propto p(\mathcal{D} \mid \theta) p(\theta)$$

MH constructs a Markov chain:

1. Start at $\theta^{(0)}$.
2. At step $s$, propose $\theta^* \sim q(\theta^* \mid \theta^{(s-1)})$.
3. Compute the acceptance probability:

$$\alpha = \min\left(1, \frac{p(\mathcal{D}\mid\theta^*) p(\theta^*) q(\theta^{(s-1)} \mid \theta^*)} {p(\mathcal{D}\mid\theta^{(s-1)}) p(\theta^{(s-1)}) q(\theta^* \mid \theta^{(s-1)})} \right)$$

1. Accept with probability $\alpha$.

In the reference runtime, the proposal is symmetric (normal), so $q$ cancels.

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2. Reference implementation

from crml.mcmc import metropolis_hastings_1d
import numpy as np

def log_posterior(theta):
    # Example: Normal prior N(0, 10^2) and Normal likelihood around 5
    lp_prior = -0.5 * (theta / 10.0) ** 2
    lp_like = -0.5 * ((theta - 5.0) / 2.0) ** 2
    return lp_prior + lp_like

samples = metropolis_hastings_1d(
    log_posterior,
    initial=0.0,
    steps=5000,
    proposal_std=0.5,
    burn_in=1000,
)

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3. How MCMC fits into QBER-style CRML

In a full QBER implementation:

• $\theta$ could be:
• a frequency hyperparameter (e.g., $\alpha$, $\beta$)
• a mixture weight
• a tail index
• The likelihood $p(\mathcal{D}\mid\theta)$ is computed from observed loss data or incident counts.

CRML does not encode the full Bayesian graph; it encodes:

• the structure of frequency and severity models
• hyperparameters that a Bayesian runtime could treat as priors

A more advanced runtime can:

1. Read CRML.
2. Construct a probabilistic model in PyMC, NumPyro, or Stan.
3. Use HMC/NUTS or advanced MCMC rather than MH.

The current MH implementation is intentionally simple to keep the reference runtime light but conceptually aligned to QBER.