pymc-fundamentals

# PyMC 5 Fundamentals

## When to Use This Skill

- Writing new PyMC models in Python
- Understanding PyMC syntax and API
- Converting models from Stan/JAGS to PyMC
- Diagnosing sampling issues with ArviZ

## Model Structure

```python
import pymc as pm
import numpy as np
import arviz as az

with pm.Model() as model:
    # 1. Priors
    mu = pm.Normal("mu", mu=0, sigma=10)
    sigma = pm.HalfNormal("sigma", sigma=1)

    # 2. Likelihood
    y_obs = pm.Normal("y_obs", mu=mu, sigma=sigma, observed=y_data)

    # 3. Sample
    trace = pm.sample(1000, tune=1000, return_inferencedata=True)

# 4. Diagnostics
az.summary(trace)
```

## CRITICAL: SD Parameterization

**PyMC uses SD (like Stan), NOT precision (like BUGS):**

```python
# PyMC (SD)
pm.Normal("x", mu=0, sigma=1)      # sigma is SD

# BUGS equivalent would be tau = 1/sigma² = 1
```

## Distribution Quick Reference

### Continuous
```python
pm.Normal("x", mu=0, sigma=1)           # Normal
pm.HalfNormal("x", sigma=1)             # Half-normal (>0)
pm.HalfCauchy("x", beta=2.5)            # Half-Cauchy (>0)
pm.Exponential("x", lam=1)              # Exponential
pm.Uniform("x", lower=0, upper=1)       # Uniform
pm.Beta("x", alpha=1, beta=1)           # Beta
pm.Gamma("x", alpha=2, beta=1)          # Gamma
pm.StudentT("x", nu=3, mu=0, sigma=1)   # Student-t
pm.LogNormal("x", mu=0, sigma=1)        # Log-normal
pm.TruncatedNormal("x", mu=0, sigma=1, lower=0)  # Truncated
```

### Discrete
```python
pm.Bernoulli("x", p=0.5)                # Bernoulli
pm.Binomial("x", n=10, p=0.5)           # Binomial
pm.Poisson("x", mu=5)                   # Poisson
pm.NegativeBinomial("x", mu=5, alpha=1) # Negative binomial
pm.Categorical("x", p=[0.3, 0.5, 0.2])  # Categorical
```

### Multivariate
```python
pm.MvNormal("x", mu=np.zeros(K), cov=np.eye(K))
pm.Dirichlet("x", a=np.ones(K))
pm.LKJCholeskyCov("chol", n=K, eta=2, sd_dist=pm.Exponential.dist(1))
```

## Sampling

```python
# Standard NUTS
trace = pm.sample(
    draws=1000,