PyMC Sampling and Inference

import pymc as pm

def sample(draws=1000, tune=1000, chains=None, cores=None, 
           random_seed=None, progressbar=True, step=None, 
           nuts_sampler='nutpie', initvals=None, init='auto',
           jitter_max_retries=10, n_init=200000, trace=None,
           discard_tuned_samples=True, compute_convergence_checks=True,
           keep_warning_stat=False, return_inferencedata=True,
           idata_kwargs=None, callback=None, mp_ctx=None, **kwargs):
    """
    Draw samples from the posterior distribution using MCMC.
    
    Parameters:
    - draws (int): Number of samples to draw (default: 1000)
    - tune (int): Number of tuning samples (default: 1000)  
    - chains (int): Number of chains (default: auto)
    - cores (int): Number of parallel processes (default: auto)
    - random_seed (int): Random seed for reproducibility
    - step (step method): Custom step method (default: auto-select)
    - init (str): Initialization method ('auto', 'adapt_diag', 'jitter+adapt_diag')
    - nuts_sampler (str): NUTS implementation ('nutpie', 'pymc')
    - return_inferencedata (bool): Return ArviZ InferenceData object
    
    Returns:
    - trace: MCMC samples as InferenceData or MultiTrace
    """
    
# Basic sampling
with pm.Model() as model:
    # Define model...
    trace = pm.sample()

# Custom sampling configuration
trace = pm.sample(
    draws=2000,           # More samples
    tune=1500,            # More tuning
    chains=4,             # Parallel chains
    cores=4,              # Use 4 cores
    target_accept=0.9,    # Higher acceptance rate
    max_treedepth=12      # Deeper trees for complex models
)

def init_nuts(init='auto', chains=None, n_init=200000, model=None,
              random_seed=None, progressbar=True, jitter_max_retries=10,
              tune=None, initvals=None, **kwargs):
    """
    Initialize NUTS sampler with optimal step size and mass matrix.
    
    Parameters:
    - init (str): Initialization strategy
        - 'auto': Automatic initialization
        - 'adapt_diag': Adapt diagonal mass matrix  
        - 'jitter+adapt_diag': Add jitter + adapt diagonal
        - 'adapt_full': Adapt full mass matrix
    - chains (int): Number of chains to initialize
    - n_init (int): Number of initialization samples
    - initvals (dict): Initial parameter values
    
    Returns:
    - step: Initialized NUTS step method
    - potential: Model potential function
    """

# Initialize NUTS with custom settings
step, potential = pm.init_nuts(
    init='adapt_full',     # Full mass matrix adaptation
    n_init=500000,         # More initialization samples
    chains=4
)

# Use initialized sampler
trace = pm.sample(step=step)

from pymc.step_methods import NUTS

class NUTS:
    """
    No-U-Turn Sampler for continuous variables.
    
    Parameters:
    - vars (list): Variables to sample (default: all continuous)
    - target_accept (float): Target acceptance probability (default: 0.8)
    - max_treedepth (int): Maximum tree depth (default: 10)
    - step_scale (float): Initial step size scaling
    - is_cov (array): Covariance matrix for mass matrix
    """
    
    def __init__(self, vars=None, target_accept=0.8, max_treedepth=10, **kwargs):
        pass

# Custom NUTS configuration
nuts_step = pm.NUTS(
    vars=[param1, param2],    # Specific variables
    target_accept=0.95,       # High acceptance rate
    max_treedepth=15          # Deep trees for complex geometry
)

trace = pm.sample(step=nuts_step)

from pymc.step_methods import HamiltonianMC

class HamiltonianMC:
    """
    Hamiltonian Monte Carlo sampler.
    
    Parameters:
    - vars (list): Variables to sample
    - path_length (float): Length of Hamiltonian trajectory
    - step_rand (function): Step size randomization function
    - step_scale (float): Step size scaling factor
    """

# HMC with fixed trajectory length
hmc_step = pm.HamiltonianMC(vars=continuous_vars, path_length=2.0)

from pymc.step_methods import (Metropolis, BinaryMetropolis, 
                               CategoricalGibbsMetropolis, DEMetropolis)

# Standard Metropolis-Hastings
metropolis = pm.Metropolis(vars=param, proposal_dist=pm.NormalProposal)

# For binary variables
binary_metropolis = pm.BinaryMetropolis(vars=binary_var)

# For categorical variables  
categorical_gibbs = pm.CategoricalGibbsMetropolis(vars=categorical_var)

# Differential Evolution Metropolis
de_metropolis = pm.DEMetropolis(vars=param, scaling=1e-4)

from pymc.step_methods import Slice

class Slice:
    """
    Slice sampler for univariate distributions.
    
    Parameters:
    - vars (list): Variables to sample
    - w (float): Initial width of slice
    - tune (bool): Tune width during sampling
    """

# Slice sampling for specific parameter
slice_step = pm.Slice(vars=[difficult_param], w=2.0)

from pymc.step_methods import (
    BinaryGibbsMetropolis, BinaryMetropolis, CategoricalGibbsMetropolis,
    DEMetropolis, DEMetropolisZ, CompoundStep, BlockedStep
)

# Binary Gibbs Metropolis for binary variables
binary_gibbs = pm.BinaryGibbsMetropolis('binary_var')

# Binary Metropolis with custom proposal
binary_metro = pm.BinaryMetropolis('binary_var')

# Categorical Gibbs for categorical variables
cat_gibbs = pm.CategoricalGibbsMetropolis('categorical_var')

# Differential Evolution Metropolis
de_metro = pm.DEMetropolis(vars=[var1, var2], 
                          scaling=0.001,    # Scaling factor
                          tune_scaling=True, # Auto-tune scaling
                          tune_interval=100) # Tuning frequency

# DE Metropolis with Z proposals  
de_metro_z = pm.DEMetropolisZ(vars=[var1, var2])

from pymc.step_methods.metropolis import (
    CauchyProposal, LaplaceProposal, MultivariateNormalProposal,
    NormalProposal, PoissonProposal, UniformProposal
)

# Custom Metropolis with different proposals
metro_cauchy = pm.Metropolis('var', proposal_dist=CauchyProposal)
metro_laplace = pm.Metropolis('var', proposal_dist=LaplaceProposal)
metro_uniform = pm.Metropolis('var', proposal_dist=UniformProposal)

# Multivariate normal proposal with custom covariance
mv_proposal = MultivariateNormalProposal(np.eye(3) * 0.1)
metro_mv = pm.Metropolis(vars=[var1, var2, var3], proposal_dist=mv_proposal)

# Poisson proposal for count data
metro_poisson = pm.Metropolis('count_var', proposal_dist=PoissonProposal)

# Compound step combining different samplers
step1 = pm.NUTS([continuous_var1, continuous_var2])
step2 = pm.BinaryGibbsMetropolis([binary_var])
step3 = pm.CategoricalGibbsMetropolis([categorical_var])

compound_step = pm.CompoundStep([step1, step2, step3])

# Use compound step in sampling
trace = pm.sample(step=compound_step)

# Blocked step for grouping variables
blocked_step = pm.BlockedStep(methods=[step1, step2], blocking=[0, 1])

from pymc.step_methods import CompoundStep

# Combine different step methods
compound = pm.CompoundStep([
    pm.NUTS(vars=continuous_vars),
    pm.BinaryMetropolis(vars=binary_vars),
    pm.CategoricalGibbsMetropolis(vars=categorical_vars)
])

trace = pm.sample(step=compound)

from pymc.step_methods.metropolis import (
    NormalProposal, UniformProposal, LaplaceProposal,
    CauchyProposal, PoissonProposal, MultivariateNormalProposal
)

# Normal proposal for Metropolis
normal_prop = pm.NormalProposal(s=0.5)  # Standard deviation

# Uniform proposal
uniform_prop = pm.UniformProposal(s=1.0)  # Half-width

# Multivariate normal proposal  
mv_prop = pm.MultivariateNormalProposal(cov=covariance_matrix)

def sample_smc(draws=1000, kernel='metropolis', n_steps=25, parallel=False,
               start=None, cores=None, compute_convergence_checks=True,
               model=None, random_seed=None, progressbar=True, **kernel_kwargs):
    """
    Sequential Monte Carlo sampling.
    
    Parameters:
    - draws (int): Number of samples to draw
    - kernel (str): SMC kernel ('metropolis', 'nuts')
    - n_steps (int): Number of SMC steps
    - parallel (bool): Parallel chain execution
    - start (dict): Starting values
    
    Returns:
    - trace: SMC samples as InferenceData
    """

# Basic SMC sampling
with pm.Model() as model:
    # Define model...
    smc_trace = pm.sample_smc(draws=2000, kernel='metropolis')

# Advanced SMC configuration
smc_trace = pm.sample_smc(
    draws=5000,
    kernel='nuts', 
    n_steps=50,
    parallel=True,
    cores=4
)

def sample_prior_predictive(samples=500, var_names=None, model=None,
                           size=None, keep_size=False, random_seed=None,
                           return_inferencedata=True, extend_inferencedata=True,
                           predictions=False, **kwargs):
    """
    Sample from prior distributions.
    
    Parameters:
    - samples (int): Number of prior samples
    - var_names (list): Variables to sample (default: all)
    - size (int): Size for each sample draw
    - predictions (bool): Include deterministic variables
    - return_inferencedata (bool): Return ArviZ InferenceData
    
    Returns:
    - prior_samples: Prior predictive samples
    """

# Sample from priors
with pm.Model() as model:
    # Define priors and likelihood...
    prior_pred = pm.sample_prior_predictive(samples=1000)

# Sample specific variables only
prior_pred = pm.sample_prior_predictive(
    samples=2000,
    var_names=['alpha', 'beta', 'sigma']
)

def sample_posterior_predictive(trace, samples=None, var_names=None,
                               size=None, keep_size=False, model=None,
                               progressbar=True, random_seed=None,
                               extend_inferencedata=True, predictions=False,
                               return_inferencedata=True, **kwargs):
    """
    Sample from posterior predictive distribution.
    
    Parameters:
    - trace: MCMC trace or InferenceData object
    - samples (int): Number of predictive samples per posterior sample
    - var_names (list): Variables to predict
    - size (int): Size for each prediction
    - predictions (bool): Include deterministic predictions
    
    Returns:
    - posterior_pred: Posterior predictive samples
    """

# Basic posterior predictive sampling
with pm.Model() as model:
    # Define model and sample...
    trace = pm.sample()
    post_pred = pm.sample_posterior_predictive(trace)

# Out-of-sample predictions
pm.set_data({'X_new': X_test})  # Update data
post_pred = pm.sample_posterior_predictive(
    trace, 
    var_names=['y_pred'],
    predictions=True
)

def draw(vars, draws=1, random_seed=None, **kwargs):
    """
    Draw samples directly from distributions.
    
    Parameters:
    - vars: Distribution(s) to sample from
    - draws (int): Number of samples
    - random_seed (int): Random seed
    
    Returns:
    - samples: Array of samples
    """

# Draw from distribution objects
normal_samples = pm.draw(pm.Normal.dist(0, 1), draws=1000)

# Draw from multiple distributions
samples = pm.draw([
    pm.Normal.dist(0, 1),
    pm.Beta.dist(2, 3)
], draws=500)

# Draw within model context
with pm.Model() as model:
    mu = pm.Normal('mu', 0, 1)
    sigma = pm.HalfNormal('sigma', 1)
    
    # Draw from prior
    mu_samples = pm.draw(mu, draws=100)

def compile_forward_sampling_function(outputs, inputs, **kwargs):
    """
    Compile fast forward sampling function.
    
    Parameters:
    - outputs (list): Output variables to sample
    - inputs (list): Input parameter variables
    - **kwargs: Additional compilation options
    
    Returns:
    - compiled_fn: Compiled sampling function
    """

# Compile sampling function for predictions
with pm.Model() as model:
    # Define model...
    sampling_fn = pm.compile_forward_sampling_function(
        outputs=[y_pred],
        inputs=[X, theta]
    )
    
# Use compiled function
predictions = sampling_fn(X_new, theta_samples)

def vectorize_over_posterior(fn, trace, var_names=None, **kwargs):
    """
    Vectorize function over posterior samples.
    
    Parameters:
    - fn (callable): Function to vectorize
    - trace: Posterior samples
    - var_names (list): Variables to pass to function
    
    Returns:
    - results: Vectorized function results
    """

# Define custom function
def custom_prediction(alpha, beta, X):
    return alpha + beta * X

# Vectorize over posterior
results = pm.vectorize_over_posterior(
    custom_prediction,
    trace,
    var_names=['alpha', 'beta'],
    X=X_new
)

from pymc.step_methods import BlockedStep

class CustomStep(BlockedStep):
    """
    Template for custom step methods.
    """
    
    def __init__(self, vars, model=None, **kwargs):
        self.vars = vars
        self.model = model
        super().__init__(vars, [model.compile_logp(vars)])
    
    def step(self, point):
        """Implement custom step logic."""
        # Custom sampling logic here
        return new_point, stats

# Parallel sampling with custom configuration
import multiprocessing as mp

# Set up parallel context
with mp.Pool(processes=4) as pool:
    trace = pm.sample(
        chains=4,
        cores=4,
        mp_ctx=pool
    )

# Control parallelization
trace = pm.sample(
    chains=8,           # More chains than cores
    cores=4,            # Limit parallel processes
    chain_method='sequential'  # Sequential within process
)

# Use Zarr backend for large traces
trace = pm.sample(
    draws=100000,
    trace=pm.backends.ZarrTrace('large_trace.zarr')
)

# Streaming inference for very large models
with pm.Model() as streaming_model:
    # Large model definition...
    
    # Sample in batches
    for batch in range(10):
        batch_trace = pm.sample(
            draws=1000,
            trace=pm.backends.NDArray(),
            compute_convergence_checks=False
        )
        # Process batch_trace...

# Sample with full diagnostics
trace = pm.sample(
    compute_convergence_checks=True,  # Compute R-hat, ESS
    keep_warning_stat=True,           # Keep warning statistics
    progressbar=True                  # Show progress
)

# Access sampling statistics
print(trace.get_sampler_stats('diverging').sum())  # Divergences
print(trace.get_sampler_stats('energy'))           # Energy statistics

def custom_callback(trace, draw):
    """Custom callback function called during sampling."""
    if draw % 100 == 0:
        print(f"Completed draw {draw}")
        # Custom diagnostics or early stopping logic

# Sample with callback
trace = pm.sample(callback=custom_callback)

# Compare models using sampling
models = [model1, model2, model3]
traces = []

for model in models:
    with model:
        trace = pm.sample()
        traces.append(trace)

# Compare using information criteria
comparison = pm.compare({f'model_{i}': trace 
                        for i, trace in enumerate(traces)})

# Robust sampling for difficult models
with pm.Model() as difficult_model:
    # Complex model definition...
    
    # Start with MAP estimate
    map_estimate = pm.find_MAP()
    
    # Initialize sampler carefully
    step = pm.NUTS(
        target_accept=0.99,      # Very high acceptance
        max_treedepth=15,        # Deep trees
        step_scale=0.1           # Small initial steps
    )
    
    # Sample with more tuning
    trace = pm.sample(
        draws=2000,
        tune=3000,               # Extra tuning
        init='adapt_full',       # Full mass matrix
        initvals=map_estimate,   # Start from MAP
        step=step
    )