In [1]:
from straightline_utils import *
%matplotlib inline
from matplotlib import rcParams
rcParams['savefig.dpi'] = 100
In [2]:
(x,y,sigmay) = get_data_no_outliers()
plot_yerr(x, y, sigmay)
In [3]:
# Linear algebra: weighted least squares
N = len(x)
A = np.zeros((N,2))
A[:,0] = 1. / sigmay
A[:,1] = x / sigmay
b = y / sigmay
theta,nil,nil,nil = np.linalg.lstsq(A, b)
plot_yerr(x, y, sigmay)
b_ls,m_ls = theta
print 'Least Squares (maximum likelihood) estimator:', b_ls,m_ls
plot_line(m_ls, b_ls);
This procedure will get us the Bayesian solution to the problem - not an estimator, but a probability distribution for the parameters m and b. This PDF captures the uncertainty in the model parameters given the data. For simple, 2-dimensional parameter spaces like this one, evaluating on a grid is not a bad way to go. We'll see that the least squares solution lies at the peak of the posterior PDF - for a certain set of assumptions about the data and the model.
In [4]:
def straight_line_log_likelihood(x, y, sigmay, m, b):
'''
Returns the log-likelihood of drawing data values *y* at
known values *x* given Gaussian measurement noise with standard
deviation with known *sigmay*, where the "true" y values are
*y_t = m * x + b*
x: list of x coordinates
y: list of y coordinates
sigmay: list of y uncertainties
m: scalar slope
b: scalar line intercept
Returns: scalar log likelihood
'''
return (np.sum(np.log(1./(np.sqrt(2.*np.pi) * sigmay))) +
np.sum(-0.5 * (y - (m*x + b))**2 / sigmay**2))
def straight_line_log_prior(m, b):
return 0.
def straight_line_log_posterior(x,y,sigmay, m,b):
return (straight_line_log_likelihood(x,y,sigmay, m,b) +
straight_line_log_prior(m, b))
In [5]:
# Evaluate log P(m,b | x,y,sigmay) on a grid.
# Set up grid
mgrid = np.linspace(mlo, mhi, 100)
bgrid = np.linspace(blo, bhi, 101)
log_posterior = np.zeros((len(mgrid),len(bgrid)))
# Evaluate log probability on grid
for im,m in enumerate(mgrid):
for ib,b in enumerate(bgrid):
log_posterior[im,ib] = straight_line_log_posterior(x, y, sigmay, m, b)
# Convert to probability density and plot
posterior = np.exp(log_posterior - log_posterior.max())
plt.imshow(posterior, extent=[blo,bhi, mlo,mhi],cmap='Blues',
interpolation='nearest', origin='lower', aspect=(bhi-blo)/(mhi-mlo),
vmin=0, vmax=1)
plt.contour(bgrid, mgrid, posterior, pdf_contour_levels(posterior), colors='k')
i = np.argmax(posterior)
i,j = np.unravel_index(i, posterior.shape)
print 'Grid maximum posterior values:', bgrid[i], mgrid[j]
plt.title('Straight line: posterior PDF for parameters');
plt.plot(b_ls, m_ls, 'w+', ms=12, mew=4);
plot_mb_setup();
In problems with higher dimensional parameter spaces, we need a more efficient way of approximating the posterior PDF - both when characterizing it in the first place, and then when doing integrals over that PDF (to get the marginalized PDFs for the parameters, or to compress them in to single numbers with uncertainties that can be easily reported). In most applications it's sufficient to approximate a PDF with a (relatively) small number of samples drawn from it; MCMC is a procedure for drawing samples from PDFs.
In [6]:
def straight_line_posterior(x, y, sigmay, m, b):
return np.exp(straight_line_log_posterior(x, y, sigmay, m, b))
In [7]:
# initial m, b
m,b = 2, 0
# step sizes
mstep, bstep = 0.1, 10.
# how many steps?
nsteps = 10000
chain = []
probs = []
naccept = 0
print 'Running MH for', nsteps, 'steps'
# First point:
L_old = straight_line_log_likelihood(x, y, sigmay, m, b)
p_old = straight_line_log_prior(m, b)
prob_old = np.exp(L_old + p_old)
for i in range(nsteps):
# step
mnew = m + np.random.normal() * mstep
bnew = b + np.random.normal() * bstep
# evaluate probabilities
# prob_new = straight_line_posterior(x, y, sigmay, mnew, bnew)
L_new = straight_line_log_likelihood(x, y, sigmay, mnew, bnew)
p_new = straight_line_log_prior(mnew, bnew)
prob_new = np.exp(L_new + p_new)
if (prob_new / prob_old > np.random.uniform()):
# accept
m = mnew
b = bnew
L_old = L_new
p_old = p_new
prob_old = prob_new
naccept += 1
else:
# Stay where we are; m,b stay the same, and we append them
# to the chain below.
pass
chain.append((b,m))
probs.append((L_old,p_old))
print 'Acceptance fraction:', naccept/float(nsteps)
In [8]:
# Pull m and b arrays out of the Markov chain and plot them:
mm = [m for b,m in chain]
bb = [b for b,m in chain]
# Scatterplot of m,b posterior samples
plt.clf()
plt.contour(bgrid, mgrid, posterior, pdf_contour_levels(posterior), colors='k')
plt.gca().set_aspect((bhi-blo)/(mhi-mlo))
plt.plot(bb, mm, 'b.', alpha=0.1)
plot_mb_setup()
plt.show()
In [9]:
# 1 and 2D marginalised distributions:
import triangle
triangle.corner(chain, labels=['b','m'], range=[(blo,bhi),(mlo,mhi)],quantiles=[0.16,0.5,0.84],
show_titles=True, title_args={"fontsize": 12},
plot_datapoints=True, fill_contours=True, levels=[0.68, 0.95], color='b', bins=40, smooth=1.0);
plt.show()
# Traces, for convergence inspection:
plt.clf()
plt.subplot(2,1,1)
plt.plot(mm, 'k-')
plt.ylim(mlo,mhi)
plt.ylabel('m')
plt.subplot(2,1,2)
plt.plot(bb, 'k-')
plt.ylabel('b')
plt.ylim(blo,bhi)
plt.show()
How do we know if our model is any good? There are two properties that "good" models have: the first is accuracy, and the second is efficiency. Accurate models generate data that is like the observed data. What does this mean? First we have to define what similarity is, in this context. Visual impression is one very important way. Test statistics that capture relevant features of the data are another. Let's look at the posterior predictive distributions for the datapoints, and for a particularly interesting test statistic, the reduced chi-squared.
In [10]:
# Posterior predictive check, in data space
X = np.array(xlimits)
for i in (np.random.rand(100)*len(chain)).astype(int):
b,m = chain[i]
plt.plot(X, b+X*m, 'b-', alpha=0.1)
plot_line(m_ls, b_ls);
plot_yerr(x, y, sigmay)
In [11]:
def test_statistic(x,y,sigmay,b_ls,m_ls):
return np.sum((y - m_ls*x - b_ls)**2.0/sigmay**2.0)/(len(y)-2)
def mean_test_stat(x,y,sigmay):
pass
T = np.zeros(len(chain))
for k,(b,m) in enumerate(chain):
yp = b + m*x + np.random.randn(len(x)) * sigmay
T[k] = test_statistic(x,yp,sigmay,b_ls,m_ls)
Td = test_statistic(x,y, sigmay, b_ls, m_ls)
plt.hist(T, 100, histtype='step', color='b', lw=2, range=(0,4))
plt.axvline(Td, color='k', linestyle='--', lw=2)
plt.xlabel('Test statistic')
plt.ylabel('Posterior predictive distribution');
Note that we did not have to look up the "chi squared distribution" - we can simply compute the posterior predictive distribution given our generative model.
Now let's add an extra parameter to the model: intrinisc scatter. What does this mean? We imagine the model y values to be drawn from a PDF that is conditional on m, b and also s, the intrinsic scatter of the population. This scatter parameter can be inferred from the data as well: in this simple case we can introduce it, along with a "true" y value for each data point, and analytically marginalize over the "true" y's. This yields a new likelihood function, which looks as though it has an additional source of uncertainty in the y values - which is what scatter is.
In [12]:
def straight_line_with_scatter_log_likelihood(x, y, sigmay, m, b, log_s):
'''
Returns the log-likelihood of drawing data values *y* at
known values *x* given Gaussian measurement noise with standard
deviation with known *sigmay*, where the "true" y values have
been drawn from N(mean=m * x + b, variance=(s^2)).
x: list of x coordinates
y: list of y coordinates
sigmay: list of y uncertainties
m: scalar slope
b: scalar line intercept
s: intrinsic scatter, Gaussian std.dev
Returns: scalar log likelihood
'''
s = np.exp(log_s)
V = sigmay**2 + s**2
return (np.sum(np.log(1./(np.sqrt(2.*np.pi*V)))) +
np.sum(-0.5 * (y - (m*x + b))**2 / V))
def straight_line_with_scatter_log_prior(m, b, log_s):
if log_s < np.log(slo) or log_s > np.log(shi):
return -np.inf
return 0.
def straight_line_with_scatter_log_posterior(x,y,sigmay, m,b,log_s):
return (straight_line_with_scatter_log_likelihood(x,y,sigmay,m,b,log_s) +
straight_line_with_scatter_log_prior(m,b,log_s))
def straight_line_with_scatter_posterior(x,y,sigmay,m,b,log_s):
return np.exp(straight_line_with_scatter_log_posterior(x,y,sigmay,m,b,log_s))
In [13]:
# initial m, b, s
m,b,log_s = 2, 20, 0.
# step sizes
mstep, bstep, log_sstep = 1., 10., 1.
# how many steps?
nsteps = 30000
schain = []
sprobs = []
naccept = 0
print 'Running MH for', nsteps, 'steps'
L_old = straight_line_with_scatter_log_likelihood(x, y, sigmay, m, b, log_s)
p_old = straight_line_with_scatter_log_prior(m, b, log_s)
prob_old = np.exp(L_old + p_old)
for i in range(nsteps):
# step
mnew = m + np.random.normal() * mstep
bnew = b + np.random.normal() * bstep
log_snew = log_s + np.random.normal() * log_sstep
# evaluate probabilities
# prob_new = straight_line_with_scatter_posterior(x, y, sigmay, mnew, bnew, log_snew)
L_new = straight_line_with_scatter_log_likelihood(x, y, sigmay, mnew, bnew, log_snew)
p_new = straight_line_with_scatter_log_prior(mnew, bnew, log_snew)
prob_new = np.exp(L_new + p_new)
if (prob_new / prob_old > np.random.uniform()):
# accept
m = mnew
b = bnew
log_s = log_snew
L_old = L_new
p_old = p_new
prob_old = prob_new
naccept += 1
else:
# Stay where we are; m,b stay the same, and we append them
# to the chain below.
pass
schain.append((b,m,np.exp(log_s)))
sprobs.append((L_old,p_old))
print 'Acceptance fraction:', naccept/float(nsteps)
In [14]:
# Histograms:
import triangle
slo,shi = [0,10]
triangle.corner(schain, labels=['b','m','s'], range=[(blo,bhi),(mlo,mhi),(slo,shi)],quantiles=[0.16,0.5,0.84],
show_titles=True, title_args={"fontsize": 12},
plot_datapoints=True, fill_contours=True, levels=[0.68, 0.95], color='b', bins=20, smooth=1.0);
plt.show()
# Traces:
plt.clf()
plt.subplot(3,1,1)
plt.plot([b for b,m,s in schain], 'k-')
plt.ylabel('b')
plt.subplot(3,1,2)
plt.plot([m for b,m,s in schain], 'k-')
plt.ylabel('m')
plt.subplot(3,1,3)
plt.plot([s for b,m,s in schain], 'k-')
plt.ylabel('s')
plt.show()
The other virtue we are looking for in a model is efficiency. Not because this is intrinsically a good thing, but rather because the data might prefer it. We can compare two models, given the data, with their relative posterior probabilities. This is not easy, because we have to specify their prior probabilities, and then compute the probability of getting the data given each model (the "Evidence") - but under certain assumptions we can do this.
Evidence computation is famously difficult - the simplest way is by simple Monte Carlo. We draw prior samples, and weight each one by the likelihood function. It's inefficient, but sometimes that doesn't matter.
In [15]:
# Draw a buttload of prior samples and hope for the best
N=50000
mm = np.random.uniform(mlo,mhi, size=N)
bb = np.random.uniform(blo,bhi, size=N)
slo,shi = [0.001,10]
log_slo = np.log(slo)
log_shi = np.log(shi)
log_ss = np.random.uniform(log_slo, log_shi, size=N)
log_likelihood_vanilla = np.zeros(N)
log_likelihood_scatter = np.zeros(N)
for i in range(N):
log_likelihood_vanilla[i] = straight_line_log_likelihood(x, y, sigmay, mm[i], bb[i])
log_likelihood_scatter[i] = straight_line_with_scatter_log_likelihood(x, y, sigmay, mm[i], bb[i], log_ss[i])
def logsum(x):
mx = x.max()
return np.log(np.sum(np.exp(x - mx))) + mx
log_evidence_vanilla = logsum(log_likelihood_vanilla) - np.log(N)
log_evidence_scatter = logsum(log_likelihood_scatter) - np.log(N)
print 'Log evidence vanilla:', log_evidence_vanilla
print 'Log evidence scatter:', log_evidence_scatter
print 'Odds ratio in favour of the vanilla model:', np.exp(log_evidence_vanilla - log_evidence_scatter)
In this case there is very little to choose between the two models. Both provide comparably good fits to the data, so the only thing working against the scatter model is its extra parameter. However, the prior for s is very well -matched to the data (uniform in log s corresponds to a 1/s distribution, favoring small values, and so there is not a very big "Occam's Razor" factor in the evidence. Both models are appropriate for this dataset.
Incidentally, let's look at a possible approximation for the evidence - the posterior mean log likelihood from our MCMC chains:
In [16]:
mean_log_L_vanilla = np.average(np.atleast_1d(probs).T[0])
mean_log_L_scatter = np.average(np.atleast_1d(sprobs).T[0])
print "No scatter: Evidence, mean log L, difference: ",log_evidence_vanilla,mean_log_L_vanilla,(mean_log_L_vanilla - log_evidence_vanilla)
print " Scatter: Evidence, mean log L, difference: ",log_evidence_scatter,mean_log_L_scatter,(mean_log_L_scatter - log_evidence_scatter)
The difference between the posterior mean log likelihood and the Evidence is the Shannon information gained when we updated the prior into the posterior. In both cases we gained about 2 bits of information - perhaps corresponding to approximately 2 good measurements (regardless of the number of parameters being inferred)?
In [16]:
In [60]:
def likelihood_outliers((m, b, pbad), (x, y, sigmay, sigmabad)):
return np.prod(pbad * 1./(np.sqrt(2.*np.pi)*sigmabad) *
np.exp(-y**2 / (2.*sigmabad**2))
+ (1.-pbad) * (1./(np.sqrt(2.*np.pi)*sigmay)
* np.exp(-(y-(m*x+b))**2/(2.*sigmay**2))))
In [61]:
def prior_outliers((m, b, pbad)):
if pbad < 0:
return 0
if pbad > 1:
return 0
return 1.
In [62]:
def prob_outliers((m,b,pbad), x,y,sigmay,sigmabad):
return (likelihood_outliers((m,b,pbad), (x,y,sigmay,sigmabad)) *
prior_outliers((m,b,pbad)))
In [65]:
x,y,sigmay = data1.T
sigmabad = np.std(y)
prob_args = (x,y,sigmay,sigmabad)
mstep = 0.1
bstep = 1.
pbadstep = 0.01
proposal_args = ((mstep, bstep,pbadstep),)
m,b,pbad = 2.2, 30, 0.1
mh(prob_outliers, prob_args, gaussian_proposal, proposal_args,
(m,b,pbad), 100000);
In [ ]:
def likelihood_t((m, b, nu), (x, y, sigmay)):
return np.prod(pbad * 1./(np.sqrt(2.*np.pi)*sigmabad) *
np.exp(-y**2 / (2.*sigmabad**2))
+ (1.-pbad) * (1./(np.sqrt(2.*np.pi)*sigmay)
* np.exp(-(y-(m*x+b))**2/(2.*sigmay**2))))
In [97]:
def complexity_brewer_likelihood((m, b, q), (x, y, sigmay)):
# q: quadratic term
if q < 0:
q = 0
else:
k = 0.01
q = -k * np.log(1 - q)
return np.prod(np.exp(-(y-(b+m*x+q*(x - 150)**2))**2/(2.*sigmay**2)))
In [98]:
def complexity_brewer_prior((m,b,q)):
if q < -1:
return 0.
return 1.
In [99]:
def complexity_brewer_prob(params, *args):
return complexity_brewer_prior(params) * complexity_brewer_likelihood(params, args)
In [109]:
x,y,sigmay = get_data_no_outliers()
print 'x', x.min(), x.max()
print 'y', y.min(), y.max()
y = y + (0.001 * (x-150)**2)
prob_args = (x,y,sigmay)
mstep = 0.1
bstep = 1.
qstep = 0.1
proposal_args = ((mstep, bstep, qstep, cstep),)
m,b,q = 2.2, 30, 0.
plt.errorbar(x, y, fmt='.', yerr=sigmay)
plt.show()
mh(complexity_brewer_prob, prob_args, gaussian_proposal, proposal_args,
(m,b,q), 10000, pnames=['m','b','q']);
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