def __init__(self,
d,
alpha,
x0,
N,
StepSize,
CovScaling,
PowerOfTwo,
Stream='cud'):
"""
Implements the Bayesian Linear Regression based on
Data set "Data.txt" by using multiple proposal quasi MCMC with
Importance Sampling (IS-MP-QMCMC)
Inputs:
-------
d - int
dimension of posterior
alpha - float
Standard deviation for Observation noise
x0 - array_like
d-dimensional array; starting value
N - int
number of proposals per iteration
StepSize - float
step size for proposed jump in mean
CovScaling - float
scaling of proposal covariance
PowerOfTwo - int
defines size S of seed by S=2**PowerOfTwo-1
Stream - string
either 'cud' or 'iid'; defining what seed is used
"""
#################
# Generate Data #
#################
Data = DataGen(alpha, d)
X = Data.GetDesignMatrix()
Obs = Data.GetObservations()
NumOfSamples = Data.GetNumOfSamples()
##################################
# Choose stream for Markoc Chain #
##################################
xs = SeedGen(d + 1, PowerOfTwo, Stream)
###########################################
# Compute prior and likelihood quantities #
###########################################
# Compute covariance of g-prior
g = 1. / NumOfSamples
sigmaSq = 1. / alpha
G_prior = sigmaSq / g * np.linalg.inv(np.dot(X.T, X))
InvG_prior = np.linalg.inv(G_prior)
# Fisher Information as constant metric tensor
FisherInfo = InvG_prior + alpha * np.dot(X.T, X)
InvFisherInfo = np.linalg.inv(FisherInfo)
##################
# Initialisation #
##################
# List of samples to be collected
self.xVals = list()
self.xVals.append(x0)
# Iteration number
# NumOfIter = int(int((2**PowerOfTwo-1)/d)*d/(N+1))
NumOfIter = int(int((2**PowerOfTwo - 1) / (d + 1)) * (d + 1) / (N + 1))
print('Total number of Iterations = ', NumOfIter)
# set up acceptance rate array
self.AcceptVals = list()
# initialise
xI = self.xVals[0]
I = 0
# Weighted Sum and Covariance Arrays
self.WeightedSum = np.zeros((NumOfIter, d))
####################
# Start Simulation #
####################
for n in range(NumOfIter):
######################
# Generate proposals #
######################
# Load stream of points in [0,1]^d
# U = xs[n*(N+1):(n+1)*(N+1),:]
U = xs[n * (N):(n + 1) * (N), :]
# Compute proposal mean according to Langevin
GradLog_xI = -np.dot(InvG_prior, xI) + alpha * np.dot(
X.T, (Obs - np.dot(X, xI)))
Mean_xI = xI + StepSize**2 / 2. * np.dot(InvFisherInfo, GradLog_xI)
# Generate auxiliary proposal state according to MALA
# (facilitates computation of proposing probabilities)
z = Mean_xI + np.dot(norm.ppf(U[0,:d], loc=np.zeros(d), scale=1.), \
np.linalg.cholesky(CovScaling**2*InvFisherInfo).T)
# Compute mean of auxiliary proposal state according to MALA
GradLog_z = -np.dot(InvG_prior, z) + alpha * np.dot(
X.T, (Obs - np.dot(X, z)))
Mean_z = z + StepSize**2 / 2. * np.dot(InvFisherInfo, GradLog_z)
# Generate proposals via inverse CDF transformation
y = Mean_z + np.dot(norm.ppf(U[1:,:d], loc=np.zeros(d), scale=1.), \
np.linalg.cholesky(CovScaling**2*InvFisherInfo).T)
# Add current state xI to proposals
Proposals = np.insert(y, I, xI, axis=0)
########################################################
# Compute probability ratios = weights of IS-estimator #
########################################################
# Compute Log-posterior probabilities
LogPriors = -0.5 * np.dot(np.dot(Proposals, InvG_prior),
Proposals.T).diagonal(
0) # Zellner's g-prior
fs = np.dot(X, Proposals.T)
LogLikelihoods = -0.5 * alpha * np.dot(Obs - fs.T,
(Obs - fs.T).T).diagonal(0)
LogPosteriors = LogPriors + LogLikelihoods
# Compute Log of transition probabilities
GradLog_states = - np.dot(InvG_prior,Proposals.T) \
+ alpha * np.dot(X.T, (Obs - np.dot(X, Proposals.T).T).T)
Mean_Proposals = Proposals + StepSize**2 / 2. * np.dot(
InvFisherInfo, GradLog_states).T
LogKiz = -0.5*np.dot(np.dot(Mean_Proposals-z, FisherInfo/(CovScaling**2)), \
(Mean_Proposals - z).T).diagonal(0) # from any state to z
LogKzi = -0.5*np.dot(np.dot(Proposals-Mean_z, FisherInfo/(CovScaling**2)), \
(Proposals - Mean_z).T).diagonal(0) # from z to any state
LogKs = LogKiz + np.sum(LogKzi) - LogKzi
# Compute weights
LogPstates = LogPosteriors + LogKs
Sorted_LogPstates = np.sort(LogPstates)
LogPstates = LogPstates - (Sorted_LogPstates[0] + \
np.log(1 + np.sum(np.exp(Sorted_LogPstates[1:] - Sorted_LogPstates[0]))))
Pstates = np.exp(LogPstates)
#######################
# Compute IS-estimate #
#######################
# Compute weighted sum as posterior mean estimate
WeightedStates = np.tile(Pstates, (d, 1)) * Proposals.T
self.WeightedSum[n, :] = np.sum(WeightedStates, axis=1).copy()
##################################
# Sample according to IS-weights #
##################################
# Sample N new states
# Replace Is sampling with QMC sampling step
PstatesSum = np.cumsum(Pstates)
Is = np.searchsorted(PstatesSum, U[:, d:].flatten())
xvals_new = Proposals[Is]
self.xVals.append(xvals_new)
# Compute approximate acceptance rate
AcceptValsNew = 1. - Pstates[Is]
self.AcceptVals.append(AcceptValsNew)
# Update current state
I = Is[-1] #rv_discrete(values=(range(N+1),Pstates)).rvs(size=1)
xI = Proposals[I, :]
def __init__(self, d, alpha, x0, N, StepSize, PowerOfTwo, \
InitMean, InitCov, Stream, WeightIn=0):
"""
Implements the Bayesian Linear Regression based on
Data set "Data.txt" by using multiple proposal quasi MCMC with
Importance Sampling (IS-MP-QMCMC)
Inputs:
-------
d - int
dimension of posterior
alpha - float
Standard deviation for Observation noise
x0 - array_like
d-dimensional array; starting value
N - int
number of proposals per iteration
StepSize - float
step size for proposed jump in mean
CovScaling - float
scaling of proposal covariance
PowerOfTwo - int
defines size S of seed by S=2**PowerOfTwo-1
Stream - string
either 'cud' or 'iid'; defining what seed is used
"""
#################
# Generate Data #
#################
Data = DataGen(alpha, d)
X = Data.GetDesignMatrix()
Obs = Data.GetObservations()
NumOfSamples = Data.GetNumOfSamples()
##################################
# Choose stream for Markoc Chain #
##################################
xs = SeedGen(d+1, PowerOfTwo, Stream)
###########################################
# Compute prior and likelihood quantities #
###########################################
# Compute covariance of g-prior
g = 1./NumOfSamples
sigmaSq = 1./alpha
G_prior = sigmaSq / g * np.linalg.inv(np.dot(X.T,X))
InvG_prior = np.linalg.inv(G_prior)
##################
# Initialisation #
##################
# List of samples to be collected
self.xVals = list()
self.xVals.append(x0)
# Iteration number
NumOfIter = int(int((2**PowerOfTwo-1)/(d+1))*(d+1)/(N))
print ('Total number of Iterations = ', NumOfIter)
# set up acceptance rate array
self.AcceptVals = list()
# initialise
xI = self.xVals[0]
I = 0
# Number of iterations used for initial approximated posterior mean
M = int(WeightIn/N)+1
# Weighted Sum and Covariance Arrays
self.WeightedSum = np.zeros((NumOfIter+M,d))
self.WeightedCov = np.zeros((NumOfIter+M,d,d))
self.WeightedFunSum = np.zeros((NumOfIter+M,d))
self.WeightedSum[0:M,:] = InitMean
self.WeightedCov[0:M,:] = InitCov
# Approximate Posterior Mean and Covariance as initial estimates
self.ApprPostMean = InitMean
self.ApprPostCov = InitCov
# Cholesky decomposition of initial Approximate Posterior Covariance
CholApprPostCov = np.linalg.cholesky(self.ApprPostCov)
InvApprPostCov = np.linalg.inv(self.ApprPostCov)
####################
# Start Simulation #
####################
for n in range(NumOfIter):
######################
# Generate proposals #
######################
# Load stream of points in [0,1]^d
U = xs[n*(N):(n+1)*(N),:]
# Sample new proposed States according to multivariate t-distribution
y = self.ApprPostMean + np.dot(norm.ppf(U[:,:d], loc=np.zeros(d), \
scale=StepSize), CholApprPostCov)
# Add current state xI to proposals
Proposals = np.insert(y, 0, xI, axis=0)
########################################################
# Compute probability ratios = weights of IS-estimator #
########################################################
# Compute Log-posterior probabilities
LogPriors = -0.5*np.dot(np.dot(Proposals, InvG_prior), Proposals.T).diagonal(0) # Zellner's g-prior
fs = np.dot(X,Proposals.T)
LogLikelihoods = -0.5*alpha*np.dot(Obs-fs.T, (Obs-fs.T).T).diagonal(0)
LogPosteriors = LogPriors + LogLikelihoods
# Compute Log of transition probabilities
LogK_ni = -0.5*np.dot(np.dot(Proposals-self.ApprPostMean, InvApprPostCov/(StepSize**2)), \
(Proposals - self.ApprPostMean).T).diagonal(0)
LogKs = np.sum(LogK_ni) - LogK_ni # from any state to all others
# Compute weights
LogPstates = LogPosteriors + LogKs
Sorted_LogPstates = np.sort(LogPstates)
LogPstates = LogPstates - (Sorted_LogPstates[0] + \
np.log(1 + np.sum(np.exp(Sorted_LogPstates[1:] - Sorted_LogPstates[0]))))
Pstates = np.exp(LogPstates)
#######################
# Compute IS-estimate #
#######################
# Compute weighted sum as posterior mean estimate
WeightedStates = np.tile(Pstates, (d,1)) * Proposals.T
self.WeightedSum[n+M,:] = np.sum(WeightedStates, axis=1).copy()
# Update Approximate Posterior Mean
self.ApprPostMean = np.mean(self.WeightedSum[:n+M+1,:], axis=0)
# Compute weighted sum as posterior covariance estimate
B1 = (Proposals - self.ApprPostMean).reshape(N+1,d,1)
B2 = np.transpose(B1,(0,2,1))
A = np.matmul(B1, B2)
self.WeightedCov[n+M,:,:] = np.sum((np.tile(Pstates, (d,d,1)) * A.T).T, axis=0)
InvApprPostCov = np.linalg.inv(self.ApprPostCov)
if n> 2*d/N: # makes sure NumOfSamples > d for covariance estimate
self.ApprPostCov = np.mean(self.WeightedCov[:n+M+1,:,:], axis=0)
CholApprPostCov = np.linalg.cholesky(self.ApprPostCov)
InvApprPostCov = np.linalg.inv(self.ApprPostCov)
##################################
# Sample according to IS-weights #
##################################
# Sample N new states
PstatesSum = np.cumsum(Pstates)
Is = np.searchsorted(PstatesSum, U[:,d:].flatten())
xvals_new = Proposals[Is]
self.xVals.append(xvals_new)
# Compute approximate acceptance rate
AcceptValsNew = 1. - Pstates[Is]
self.AcceptVals.append(AcceptValsNew)
# Update current state
I = Is[-1]
xI = Proposals[I,:]
# Proposal step size
StepSize = np.sqrt(2)
# Proposal covariance scaling
CovScaling = 1
# Dimension
d = 1
# Obervation noise scaling
alpha = 0.5
#################
# Generate Data #
#################
Data = DataGen(alpha, d)
X = Data.GetDesignMatrix()
Obs = Data.GetObservations()
NumOfSamples = Data.GetNumOfSamples()
######################################################
# Compute prior, likelihood and posterior quantities #
######################################################
# Compute covariance of g-prior
g = 1. / NumOfSamples
sigmaSq = 1. / alpha
G_prior = sigmaSq / g * np.linalg.inv(np.dot(X.T, X))
InvG_prior = np.linalg.inv(G_prior)
Lambda0 = sigmaSq * InvG_prior
# Fisher Information as constant metric tensor
FisherInfo = InvG_prior + alpha * np.dot(X.T, X)