To solve Problem 5, follow these steps:

### Step 1: Simulate Data

### Step 2: Stan Model Code
Write the Stan model (`bayesian_regression.stan`):
```stan
data {
  int<lower=0> N;
  vector[N] x;
  vector[N] y;
}
parameters {
  real alpha;
  real beta;
  real<lower=0> sigma_sq;
}
transformed parameters {
  real<lower=0> sigma = sqrt(sigma_sq);
}
model {
  sigma_sq ~ inv_gamma(1, 1); // Prior on variance
  alpha ~ normal(0, 10);
  beta ~ normal(0, 10);
  y ~ normal(alpha + beta * x, sigma); // Likelihood
}
```

### Step 3: Fit the Model and Check Diagnostics
Use `pystan` or `cmdstanpy` to run the model. Check Rhat (≈1) and ESS (sufficiently large). For example:
```python
import cmdstanpy

model = cmdstanpy.CmdStanModel(stan_file="bayesian_regression.stan")
data = {"N": N, "x": x, "y": y}
fit = model.sample(data=data, chains=4, iter_sampling=2000)

# Check diagnostics
print(fit.diagnose())
```

### Step 4: Analyze Results for N=100
Posterior summaries:
- **Posterior means** should be close to true values (α=2.3, β=4.0, σ=2.0).
- **Uncertainty**: Compute 95% credible intervals. Example output:
  - α: 2.1 ± 0.4 (1.7 to 2.5)
  - β: 3.8 ± 0.5 (3.3 to 4.3)
  - σ: 1.9 ± 0.2 (1.7 to 2.1)

### Step 5: Repeat with N=1000
Increase sample size and rerun:
```python
N_large = 1000
x_large = np.random.normal(size=N_large)
y_large = alpha_true + beta_true * x_large + sigma_true * np.random.normal(size=N_large)
```
Fit the model again. Results will show:
- **Tighter credible intervals** (e.g., β: 3.95 ± 0.1).
- Reduced posterior variance, indicating higher precision.

### Key Observations:
1. **Accuracy**: Posterior means align closely with true parameters.
2. **Uncertainty**: Credible intervals narrow as \(N\) increases, reflecting reduced uncertainty.
3. **Diagnostics**: Ensure Rhat ≈1 and sufficient ESS for reliable inferences.

**Visualization**: Plot prior vs. posterior histograms for parameters (using tools like `arviz` or `seaborn`), showing posterior concentration around true values, especially for \(N=1000\).

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**Answer for LMS Submission**  
Implement the steps above, ensuring your write-up includes code snippets, diagnostic results, and graphical comparisons. Highlight the reduction in posterior variance when increasing \(N\), demonstrating the influence of data quantity on Bayesian inference.