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71 lines
2.3 KiB
Plaintext
71 lines
2.3 KiB
Plaintext
To solve Problem 5, follow these steps:
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### Step 1: Simulate Data
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### Step 2: Stan Model Code
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Write the Stan model (`bayesian_regression.stan`):
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```stan
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data {
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int<lower=0> N;
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vector[N] x;
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vector[N] y;
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}
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parameters {
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real alpha;
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real beta;
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real<lower=0> sigma_sq;
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}
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transformed parameters {
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real<lower=0> sigma = sqrt(sigma_sq);
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}
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model {
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sigma_sq ~ inv_gamma(1, 1); // Prior on variance
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alpha ~ normal(0, 10);
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beta ~ normal(0, 10);
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y ~ normal(alpha + beta * x, sigma); // Likelihood
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}
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```
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### Step 3: Fit the Model and Check Diagnostics
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Use `pystan` or `cmdstanpy` to run the model. Check Rhat (≈1) and ESS (sufficiently large). For example:
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```python
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import cmdstanpy
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model = cmdstanpy.CmdStanModel(stan_file="bayesian_regression.stan")
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data = {"N": N, "x": x, "y": y}
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fit = model.sample(data=data, chains=4, iter_sampling=2000)
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# Check diagnostics
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print(fit.diagnose())
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```
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### Step 4: Analyze Results for N=100
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Posterior summaries:
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- **Posterior means** should be close to true values (α=2.3, β=4.0, σ=2.0).
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- **Uncertainty**: Compute 95% credible intervals. Example output:
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- α: 2.1 ± 0.4 (1.7 to 2.5)
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- β: 3.8 ± 0.5 (3.3 to 4.3)
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- σ: 1.9 ± 0.2 (1.7 to 2.1)
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### Step 5: Repeat with N=1000
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Increase sample size and rerun:
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```python
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N_large = 1000
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x_large = np.random.normal(size=N_large)
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y_large = alpha_true + beta_true * x_large + sigma_true * np.random.normal(size=N_large)
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```
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Fit the model again. Results will show:
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- **Tighter credible intervals** (e.g., β: 3.95 ± 0.1).
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- Reduced posterior variance, indicating higher precision.
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### Key Observations:
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1. **Accuracy**: Posterior means align closely with true parameters.
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2. **Uncertainty**: Credible intervals narrow as \(N\) increases, reflecting reduced uncertainty.
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3. **Diagnostics**: Ensure Rhat ≈1 and sufficient ESS for reliable inferences.
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**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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---
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**Answer for LMS Submission**
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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. |