# MCMC Sampling

**Markov Chain Monte Carlo (MCMC)** methods are a class of algorithms for sampling from 
probability distributions by constructing a Markov chain whose stationary distribution 
equals the target distribution. MCMC is fundamental to Bayesian inference, computational 
statistics, and quantitative finance.

This module provides a high-performance **Metropolis-Hastings sampler** with Rust 
acceleration, designed for Bayesian parameter estimation and posterior exploration.

---

## Mathematical Foundations

### The Monte Carlo Goal

Sample from a target distribution $\pi(\theta)$ where:

- Direct sampling is difficult or impossible
- We can evaluate $\pi(\theta)$ **up to a normalization constant**

Given samples $\theta^{(1)}, \ldots, \theta^{(N)} \sim \pi(\theta)$, we approximate:

**Expectations:**

$$
\mathbb{E}_\pi[f(\theta)] \approx \frac{1}{N}\sum_{i=1}^N f(\theta^{(i)})
$$

**Probabilities:**

$$
P(\theta \in A) \approx \frac{1}{N}\sum_{i=1}^N \mathbb{1}[\theta^{(i)} \in A]
$$

**Quantiles**, **posterior intervals**, and other distributional properties.

---

### Markov Chains

A sequence $\theta^{(0)}, \theta^{(1)}, \theta^{(2)}, \ldots$ is a **Markov chain** if:

$$
P(\theta^{(t+1)} \mid \theta^{(0)}, \ldots, \theta^{(t)}) = P(\theta^{(t+1)} \mid \theta^{(t)})
$$

The next state depends only on the current state.

### Transition Kernel

$$
K(\theta' \mid \theta) = P(\theta^{(t+1)} = \theta' \mid \theta^{(t)} = \theta)
$$

### Stationary Distribution

A distribution $\pi(\theta)$ is **stationary** if:

$$
\pi(\theta') = \int K(\theta' \mid \theta) \, \pi(\theta) \, d\theta
$$

If we start with $\theta^{(0)} \sim \pi$, then $\theta^{(t)} \sim \pi$ for all $t$.

### Ergodicity

A Markov chain is **ergodic** if:

1. **Irreducible:** Can reach any state from any state
2. **Aperiodic:** No cyclic behavior

For ergodic chains with stationary distribution $\pi$:

$$
\lim_{t \to \infty} P(\theta^{(t)} \in A) = \pi(A)
$$

regardless of initial state $\theta^{(0)}$.

### Detailed Balance

A sufficient condition for $\pi$ to be stationary:

$$
\pi(\theta) \, K(\theta' \mid \theta) = \pi(\theta') \, K(\theta \mid \theta')
$$

**Reversibility:** The probability of going $\theta \to \theta'$ equals that of $\theta' \to \theta$.

---

## Metropolis-Hastings Algorithm

The MH algorithm constructs a Markov chain whose stationary distribution is the target $\pi(\theta)$.

### Algorithm

**Input:** Target distribution $\pi(\theta)$, proposal distribution $q(\theta' \mid \theta)$

```
Algorithm: Metropolis-Hastings
──────────────────────────────
1. Initialize θ⁽⁰⁾

2. For t = 0, 1, 2, ..., N-1:
   
   a. Propose: Draw θ* ~ q(θ* | θ⁽ᵗ⁾)
   
   b. Compute acceptance probability:
      α = min(1, [π(θ*) · q(θ⁽ᵗ⁾|θ*)] / [π(θ⁽ᵗ⁾) · q(θ*|θ⁽ᵗ⁾)])
   
   c. Accept or reject:
      u ~ Uniform(0, 1)
      if u < α:
          θ⁽ᵗ⁺¹⁾ = θ*      # accept
      else:
          θ⁽ᵗ⁺¹⁾ = θ⁽ᵗ⁾    # reject

3. Return samples {θ⁽¹⁾, θ⁽²⁾, ..., θ⁽ᴺ⁾}
```

### Acceptance Probability

$$
\alpha = \min\left(1, \frac{\pi(\theta^*) \, q(\theta^{(t)} \mid \theta^*)}{\pi(\theta^{(t)}) \, q(\theta^* \mid \theta^{(t)})}\right)
$$

The ratio $\pi(\theta^*)/\pi(\theta^{(t)})$ compares likelihoods. The ratio 
$q(\theta^{(t)} \mid \theta^*)/q(\theta^* \mid \theta^{(t)})$ corrects for asymmetric proposals.

### Why It Works

**Theorem:** The MH algorithm produces a Markov chain with stationary distribution $\pi(\theta)$.

The acceptance rule ensures **detailed balance** holds, guaranteeing convergence to $\pi$.

---

## Special Cases

### Metropolis Algorithm (Symmetric Proposal)

When the proposal is **symmetric:** $q(\theta' \mid \theta) = q(\theta \mid \theta')$

Acceptance probability simplifies to:

$$
\alpha = \min\left(1, \frac{\pi(\theta^*)}{\pi(\theta^{(t)})}\right)
$$

Always accept moves to higher probability; sometimes accept moves to lower probability.

### Random Walk Metropolis

Use a Gaussian proposal centered at the current state:

$$
q(\theta' \mid \theta) = \mathcal{N}(\theta' \mid \theta, \sigma^2 \mathbf{I})
$$

This is symmetric, so Metropolis acceptance applies.

**This is what OptimizR implements.**

---

## Bayesian Inference with MCMC

### Bayes' Theorem

$$
p(\theta \mid D) = \frac{p(D \mid \theta) \, p(\theta)}{p(D)}
$$

| Term | Name | Description |
|------|------|-------------|
| $p(\theta \mid D)$ | Posterior | What we want |
| $p(D \mid \theta)$ | Likelihood | How well parameters explain data |
| $p(\theta)$ | Prior | Beliefs before seeing data |
| $p(D)$ | Evidence | Normalizing constant (often intractable) |

### MCMC for Posterior Sampling

The evidence $p(D)$ is often intractable, but we can evaluate:

$$
\pi(\theta) \propto p(D \mid \theta) \cdot p(\theta)
$$

MCMC only needs $\pi$ **up to a constant**, so we can sample from the posterior!

### Log-Posterior

In practice, work with log-probabilities to avoid underflow:

$$
\log \pi(\theta) = \log p(D \mid \theta) + \log p(\theta) + \text{const}
$$

---

## Python API

### Basic Usage

```python
import numpy as np
from optimizr import mcmc_sample

# Define log-likelihood for a Gaussian model
def log_likelihood(params, data):
    mu, sigma = params
    if sigma <= 0:
        return -np.inf  # invalid parameter
    residuals = (data - mu) / sigma
    return -0.5 * np.sum(residuals**2) - len(data) * np.log(sigma)

# Generate synthetic data: N(1.2, 1.0)
np.random.seed(42)
observations = np.random.randn(1000) + 1.2

# Run MCMC sampling
samples = mcmc_sample(
    log_likelihood_fn=log_likelihood,
    data=observations,
    initial_params=np.array([0.0, 1.0]),
    param_bounds=[(-5, 5), (0.1, 5.0)],
    n_samples=8000,
    burn_in=500,
    proposal_std=0.2,
)

print("Posterior mean:", samples.mean(axis=0))
print("Posterior std:", samples.std(axis=0))
```

**Expected output:**

```
Posterior mean: [1.198 0.987]
Posterior std: [0.032 0.022]
```

The true values (1.2, 1.0) are recovered within posterior uncertainty.

### Configuration Options

```python
samples = mcmc_sample(
    log_likelihood_fn=log_likelihood,
    data=observations,
    initial_params=np.array([0.0, 1.0]),
    param_bounds=[(-5, 5), (0.1, 5.0)],
    n_samples=10000,      # total samples to generate
    burn_in=1000,         # discard initial samples
    proposal_std=0.15,    # step size for random walk
    thin=2,               # keep every 2nd sample
    seed=42,              # for reproducibility
)
```

### Posterior Analysis

```python
import matplotlib.pyplot as plt

# Trace plots
fig, axes = plt.subplots(2, 2, figsize=(12, 8))

# Mu trace
axes[0, 0].plot(samples[:, 0], alpha=0.7)
axes[0, 0].set_ylabel('μ')
axes[0, 0].set_title('Trace: μ')
axes[0, 0].axhline(1.2, color='r', linestyle='--', label='True')

# Sigma trace
axes[0, 1].plot(samples[:, 1], alpha=0.7)
axes[0, 1].set_ylabel('σ')
axes[0, 1].set_title('Trace: σ')
axes[0, 1].axhline(1.0, color='r', linestyle='--', label='True')

# Mu histogram
axes[1, 0].hist(samples[:, 0], bins=50, density=True, alpha=0.7)
axes[1, 0].axvline(1.2, color='r', linestyle='--', label='True')
axes[1, 0].set_xlabel('μ')
axes[1, 0].set_title('Posterior: μ')

# Sigma histogram
axes[1, 1].hist(samples[:, 1], bins=50, density=True, alpha=0.7)
axes[1, 1].axvline(1.0, color='r', linestyle='--', label='True')
axes[1, 1].set_xlabel('σ')
axes[1, 1].set_title('Posterior: σ')

plt.tight_layout()
plt.savefig('mcmc_posterior.png', dpi=150)
```

---

## Convergence Diagnostics

### Burn-in Period

Discard initial samples before the chain has converged to the stationary distribution.

**How to choose:**

- Plot trace plots and look for stabilization
- Typically 1000–10000 iterations
- Conservative: discard first 50% of samples

### Effective Sample Size (ESS)

Due to autocorrelation, MCMC samples are not independent:

$$
\text{ESS} = \frac{N}{1 + 2\sum_{k=1}^\infty \rho_k}
$$

where $\rho_k$ is the autocorrelation at lag $k$.

**Interpretation:** ESS ≈ number of independent samples.

**Goal:** ESS > 400 for reliable posterior estimates.

### Autocorrelation

$$
\rho_k = \frac{\text{Cov}(\theta^{(t)}, \theta^{(t+k)})}{\text{Var}(\theta^{(t)})}
$$

| Autocorrelation | Interpretation |
|-----------------|----------------|
| Low (< 0.1) | Fast mixing, efficient sampling |
| High (> 0.5) | Slow mixing, need more samples or better tuning |

### Gelman-Rubin Diagnostic ($\hat{R}$)

Run multiple chains with different starting points:

$$
\hat{R} = \sqrt{\frac{\text{Var}^+}{\text{Within-chain variance}}}
$$

| $\hat{R}$ Value | Interpretation |
|-----------------|----------------|
| ≈ 1.0 | Chains have converged |
| > 1.1 | Chains have NOT mixed — run longer |

---

## Proposal Tuning

### Acceptance Rate

**Optimal acceptance rate** (for random walk Metropolis):

| Dimension | Optimal Rate |
|-----------|--------------|
| 1D | 44% |
| High-D | 23.4% |
| Practical | 20–40% |

**Tuning guidance:**

| Acceptance Rate | Problem | Fix |
|-----------------|---------|-----|
| Too high (> 50%) | Proposals too small | Increase `proposal_std` |
| Too low (< 10%) | Proposals too large | Decrease `proposal_std` |

### Adaptive Tuning

During burn-in, automatically adjust proposal variance:

```python
# Start with initial guess, let Rust backend tune
samples = mcmc_sample(
    log_likelihood_fn=log_likelihood,
    data=observations,
    initial_params=initial,
    param_bounds=bounds,
    n_samples=10000,
    burn_in=2000,         # longer burn-in for adaptation
    proposal_std=0.5,     # initial value, will be adjusted
    adaptive=True,        # enable adaptive tuning
)
```

### Optimal Scaling

Roberts and Rosenthal (2001): For Gaussian targets in $d$ dimensions:

$$
\sigma^2_{\text{optimal}} = \frac{2.38^2}{d} \cdot \Sigma
$$

where $\Sigma$ is the posterior covariance.

---

## Applications

### 1. Bayesian Regression

```python
import numpy as np
from optimizr import mcmc_sample

def log_posterior(params, data):
    X, y = data
    beta = params[:-1]
    sigma = params[-1]
    
    if sigma <= 0:
        return -np.inf
    
    # Likelihood
    y_pred = X @ beta
    residuals = (y - y_pred) / sigma
    ll = -0.5 * np.sum(residuals**2) - len(y) * np.log(sigma)
    
    # Prior: N(0, 10) for beta, InvGamma for sigma
    log_prior = -0.5 * np.sum(beta**2) / 100
    
    return ll + log_prior

# Fit Bayesian linear regression
X = np.column_stack([np.ones(100), np.random.randn(100)])
y = 2 + 3 * X[:, 1] + np.random.randn(100) * 0.5

samples = mcmc_sample(
    log_likelihood_fn=log_posterior,
    data=(X, y),
    initial_params=np.array([0.0, 0.0, 1.0]),
    param_bounds=[(-10, 10), (-10, 10), (0.01, 5)],
    n_samples=5000,
    burn_in=500,
)

print("Intercept:", samples[:, 0].mean(), "±", samples[:, 0].std())
print("Slope:", samples[:, 1].mean(), "±", samples[:, 1].std())
print("Sigma:", samples[:, 2].mean(), "±", samples[:, 2].std())
```

### 2. Stochastic Volatility

```python
def log_posterior_sv(params, returns):
    mu, phi, sigma_v = params
    
    if not (0 < phi < 1) or sigma_v <= 0:
        return -np.inf
    
    # Autoregressive volatility model
    T = len(returns)
    log_var = np.zeros(T)
    log_var[0] = mu / (1 - phi)
    
    for t in range(1, T):
        log_var[t] = mu + phi * (log_var[t-1] - mu)
    
    # Likelihood
    ll = -0.5 * np.sum(returns**2 / np.exp(log_var) + log_var)
    
    return ll

samples = mcmc_sample(
    log_likelihood_fn=log_posterior_sv,
    data=daily_returns,
    initial_params=np.array([-1.0, 0.9, 0.2]),
    param_bounds=[(-5, 0), (0.01, 0.99), (0.01, 1.0)],
    n_samples=10000,
    burn_in=2000,
)
```

### 3. Portfolio Optimization with Uncertainty

```python
# Sample from posterior of expected returns
posterior_means = samples[:, :n_assets]

# For each posterior sample, compute optimal weights
optimal_weights = []
for mu_sample in posterior_means[::10]:  # thin for speed
    w = optimize_portfolio(mu_sample, cov_matrix)
    optimal_weights.append(w)

# Report posterior distribution of weights
weights_mean = np.mean(optimal_weights, axis=0)
weights_std = np.std(optimal_weights, axis=0)
```

---

## Performance

Benchmarks on Apple M1:

| Parameters | Samples | Time | Samples/sec |
|------------|---------|------|-------------|
| 2 | 10,000 | 0.8 s | 12,500 |
| 5 | 10,000 | 1.2 s | 8,333 |
| 10 | 10,000 | 2.1 s | 4,762 |
| 20 | 10,000 | 4.8 s | 2,083 |

Performance scales approximately linearly with the number of parameters.

---

## Troubleshooting

| Symptom | Cause | Fix |
|---------|-------|-----|
| Acceptance rate ~0% | `proposal_std` too large | Decrease by 50% |
| Acceptance rate ~100% | `proposal_std` too small | Increase by 50–100% |
| Chains stuck | Local mode | Use multiple chains, different starts |
| Poor mixing | Strong correlations | Reparameterize or increase samples |
| `log_likelihood` returns `-inf` | Invalid parameters | Check bounds, add guards |

---

## Tips

### 1. Keep `proposal_std` Modest

Start with 0.1–0.5 of the expected posterior standard deviation. Adjust to 
achieve 20–40% acceptance rate.

### 2. Use Adequate Burn-in

`burn_in` should be at least 5–10% of total samples for stable chains.

### 3. Provide Tight Bounds

Specify `param_bounds` to avoid exploring invalid regions (negative variances, etc.).

### 4. Monitor Convergence

Always check trace plots and autocorrelation before using posterior samples.

### 5. Multiple Chains

Run 2–4 chains from different starting points. Compare posteriors and compute $\hat{R}$.

---

## MCMC vs. Alternatives

| Method | Pros | Cons |
|--------|------|------|
| **MCMC** | General, exact (asymptotically) | Slow convergence, diagnostics needed |
| Variational Inference | Fast, scalable | Approximate, may be biased |
| Importance Sampling | Simple, independent samples | Requires good proposal |
| Grid/Quadrature | Deterministic | Exponential in dimension |

---

## References

1. Metropolis, N. et al. (1953). "Equation of state calculations by fast computing machines." *J. Chem. Phys.*, 21(6):1087–1092.

2. Hastings, W.K. (1970). "Monte Carlo sampling methods using Markov chains and their applications." *Biometrika*, 57(1):97–109.

3. Gelfand, A.E. & Smith, A.F.M. (1990). "Sampling-based approaches to calculating marginal densities." *JASA*, 85(410):398–409.

4. Roberts, G.O. & Rosenthal, J.S. (2001). "Optimal scaling for various Metropolis-Hastings algorithms." *Statistical Science*, 16(4):351–367.

5. Brooks, S. et al. (2011). *Handbook of Markov Chain Monte Carlo*. CRC Press.

---

## Related Topics

- [HMM](hmm.md) – Sequential latent variable models with EM learning
- [Differential Evolution](differential_evolution.md) – Global optimization for finding MAP estimates
- [Mean Field Games](mean_field_games.md) – Population dynamics with coupled PDEs
