%matplotlib inline
import numpy as np
import matplotlib.pyplot as plt
from scipy import stats

# use seaborn plotting defaults
# If this causes an error, you can comment it out.
import seaborn as sns
sns.set()

from fig_code import linear_data_sample
x, y, dy = linear_data_sample()



def model(theta, x):
    # the `theta` argument is a list of parameter values, e.g., theta = [m, b] for a line
    pass

def ln_likelihood(theta, x, y, dy):
    # we will pass the parameters (theta) to the model function
    # the other arguments are the data
    pass 

def ln_prior(theta):
    pass

def ln_posterior(theta, x, y, dy):
    return ln_prior(theta) + ln_likelihood(theta, x, y, dy)

def run_mcmc(ln_posterior, nsteps, ndim, theta0, stepsize, args=()):
    """
    Run a Markov Chain Monte Carlo
    
    Parameters
    ----------
    ln_posterior: callable
        our function to compute the posterior
    nsteps: int
        the number of steps in the chain
    theta0: list
        the starting guess for theta
    stepsize: float
        a parameter controlling the size of the random step
        e.g. it could be the width of the Gaussian distribution
    args: tuple (optional)
        additional arguments passed to ln_posterior
    """
    # Create the array of size (nsteps, ndims) to hold the chain
    # Initialize the first row of this with theta0
    
    # Create the array of size nsteps to hold the log-likelihoods for each point
    # Initialize the first entry of this with the log likelihood at theta0
    
    # Loop for nsteps
    for i in range(1, nsteps):
        # Randomly draw a new theta from the proposal distribution.
        # for example, you can do a normally-distributed step by utilizing
        # the np.random.randn() function
        
        # Calculate the probability for the new state
        
        # Compare it to the probability of the old state
        # Using the acceptance probability function
        # (remember that you've computed the log probability, not the probability!)
        
        # Chose a random number r between 0 and 1 to compare with p_accept
        
        # If p_accept>1 or p_accept>r, accept the step
            # Save the position to the i^th row of the chain
            
            # Save the probability to the i^th entry of the array
            
        # Else, do not accept the step
            # Set the position and probability are equal to the previous values
        pass
            
    # Return the chain and probabilities