def fsps_filter_indices(bands):
# Find the filter indices
import fsps
flist = fsps.list_filters()
findex = [flist.index(filt) for b in bands for filt in fsps.find_filter(b)]
if len(findex) != len(bands):
raise ValueError("Your band names {} do not give "
"a one-to-one mapping to FSPS filter "
"names {}".format(bands, [flist[i] for i in findex]))
return [flist[i] for i in findex], findex
def test_single_model(model,
spop,
sdss_data,
wave_min=3700,
wave_max=7400,
em_list=SDSS_EMLINES):
"""Simulate the spectrum and SED of a SF dwarf galaxy."""
# Get SDSS filters
sdss_bands = fsps.find_filter('SDSS')
# Generate model spectrum and SED
model = simulate_dwarf_sed(model,
spop,
wave_min=3700,
wave_max=7400,
filters=sdss_bands,
spec_no_emline=True,
peraa=True)
# Measure EWs of key emission lines
model['ew_halpha'] = measure_ew_emission_line(model,
em_list['H_alpha'],
wave_margin=200,
redshift=0.0)
model['ew_hbeta'] = measure_ew_emission_line(model,
em_list['H_beta'],
wave_margin=200,
redshift=0.0)
model['ew_oiii_5007'] = measure_ew_emission_line(model,
em_list['OIII_5007'],
wave_margin=200,
redshift=0.0)
# Get magnitudes
model['ur_color'] = model['sed'][0] - model['sed'][2]
model['ug_color'] = model['sed'][0] - model['sed'][1]
model['gr_color'] = model['sed'][1] - model['sed'][2]
model['gi_color'] = model['sed'][1] - model['sed'][3]
# Print a few important parameters
print("# Model u-r color : %7.3f" % model['ur_color'])
print("# Model u-g color : %7.3f" % model['ug_color'])
print("# Model g-r color : %7.3f" % model['gr_color'])
print("# Model EW(Halpha) : %7.3f" % model['ew_halpha'])
print("# Model EW([OIII]) : %7.3f" % model['ew_oiii_5007'])
print("# Current stellar mass : %8.5f" % model['mstar'])
print("# log(SFR) : %8.3f" % np.log10(model['sfr']))
# ---------------------------------------------------------------------------------------------------- #
# make a summary plot
fig = plt.figure(figsize=(14, 20))
# Grids
gs = fig.add_gridspec(3, 2)
# ---------------------------------------------------------------------------------------------------- #
# Plot the spectrum
ax1 = fig.add_subplot(gs[0, :])
ax1.grid(linestyle='--')
flag_use = (model['wave'] >= 3708) & (model['wave'] <= 7225)
ax1.step(model['wave'][flag_use], model['spec_em'][flag_use], alpha=0.8)
ax1.step(model['wave'][flag_use], model['spec_ne'][flag_use], alpha=0.8)
_ = ax1.set_xlabel(r'$\mathrm{Wavelength}\ [\AA]$')
# ---------------------------------------------------------------------------------------------------- #
# Color-Color plot: u-g v.s. g-r
ax2 = fig.add_subplot(gs[1, 0])
_ = ax2.hist2d(sdss_data['M_u'] - sdss_data['M_g'],
sdss_data['M_g'] - sdss_data['M_r'],
range=[[-0.05, 1.79], [-0.05, 0.79]],
bins=[40, 35],
cmap='viridis',
cmin=5,
alpha=0.8)
ax2.scatter(np.asarray(model['ur_color']) - np.asarray(model['gr_color']),
model['gr_color'],
s=125,
alpha=0.8,
edgecolor='k',
facecolor='r')
ax2.grid(linestyle='--')
_ = ax2.set_xlabel(r'$u-g\ [\mathrm{mag}]$')
_ = ax2.set_ylabel(r'$g-r\ [\mathrm{mag}]$')
# ---------------------------------------------------------------------------------------------------- #
# (g-r) v.s. EW(Halpha)
ax3 = fig.add_subplot(gs[1, 1])
_ = ax3.hist2d(sdss_data['M_g'] - sdss_data['M_r'],
np.log10(-1.0 * (sdss_data['H_ALPHA_EQW'])),
range=[[-0.05, 0.79], [0.05, 2.9]],
bins=[40, 35],
cmap='viridis',
cmin=5,
alpha=0.8)
ax3.scatter(model['gr_color'],
np.log10(model['ew_halpha']),
s=125,
alpha=0.8,
edgecolor='k',
facecolor='r')
ax3.grid(linestyle='--')
_ = ax3.set_xlabel(r'$g-r\ [\mathrm{mag}]$')
_ = ax3.set_ylabel(r'$\log\ \mathrm{EW(H}\alpha)\ \AA$')
# ---------------------------------------------------------------------------------------------------- #
# (u-r) v.s. EW([OIII])
ax4 = fig.add_subplot(gs[2, 0])
_ = ax4.hist2d(sdss_use['M_u'] - sdss_use['M_r'],
np.log10(-1.0 * (sdss_use['OIII_5007_EQW'])),
range=[[0.05, 2.49], [-0.9, 2.9]],
bins=[40, 35],
cmap='viridis',
cmin=5,
alpha=0.8)
ax4.scatter(model['ur_color'],
np.log10(model['ew_oiii_5007']),
s=125,
alpha=0.8,
edgecolor='k',
facecolor='r')
ax4.grid(linestyle='--')
_ = ax4.set_xlabel(r'$u-r\ [\mathrm{mag}]$')
_ = ax4.set_ylabel(r'$\log\ \mathrm{EW([OIII]\ 5007)}\ \AA$')
# ---------------------------------------------------------------------------------------------------- #
# (g-i) v.s. EW(Hbeta)
ax5 = fig.add_subplot(gs[2, 1])
_ = ax5.hist2d(sdss_use['M_g'] - sdss_use['M_i'],
np.log10(-1.0 * (sdss_use['H_BETA_EQW'])),
range=[[-0.05, 0.99], [0.01, 1.9]],
bins=[40, 35],
cmap='viridis',
cmin=5,
alpha=0.8)
ax5.scatter(model['gi_color'],
np.log10(model['ew_hbeta']),
s=125,
alpha=0.8,
edgecolor='k',
facecolor='r')
ax5.grid(linestyle='--')
_ = ax5.set_xlabel(r'$g-i\ [\mathrm{mag}]$')
_ = ax5.set_ylabel(r'$\log\ \mathrm{EW(H}\beta)\ \AA$')
return model, fig
def loss_function(args):
tau_mean, const_mean, tage_mean, fburst_mean, tburst_mean, logzsol_mean, gas_logz_mean, gas_logu_mean,\
tau_sig, const_sig, tage_sig, fburst_sig, tburst_sig, logzsol_sig, gas_logz_sig, gas_logu_sig = args
set_size = 3000
tau_arr = [
float(
priors.ClippedNormal(mean=abs(tau_mean),
sigma=abs(tau_sig),
mini=1.0,
maxi=8.0).sample()) for _ in range(set_size)
]
const_arr = [
float(
priors.ClippedNormal(mean=abs(const_mean),
sigma=abs(const_sig),
mini=0.0,
maxi=0.5).sample()) for _ in range(set_size)
]
tage_arr = [
float(
priors.ClippedNormal(mean=abs(tage_mean),
sigma=abs(tage_sig),
mini=1.0,
maxi=11.0).sample()) for _ in range(set_size)
]
fburst_arr = [
float(
priors.ClippedNormal(mean=abs(fburst_mean),
sigma=abs(fburst_sig),
mini=0.0,
maxi=0.8).sample()) for _ in range(set_size)
]
tburst_arr = [
float(
priors.ClippedNormal(mean=abs(tburst_mean),
sigma=abs(tburst_sig),
mini=0.0,
maxi=abs(tage_mean)).sample())
for _ in range(set_size)
]
logzsol_arr = [
float(
priors.ClippedNormal(mean=-1 * abs(logzsol_mean),
sigma=abs(logzsol_sig),
mini=-1.5,
maxi=0.0).sample()) for _ in range(set_size)
]
gas_logz_arr = [
float(
priors.ClippedNormal(mean=-1 * abs(gas_logz_mean),
sigma=abs(gas_logz_sig),
mini=-1.5,
maxi=0.0).sample()) for _ in range(set_size)
]
gas_logu_arr = [
float(
priors.ClippedNormal(mean=-1 * abs(gas_logu_mean),
sigma=abs(gas_logu_sig),
mini=-4.0,
maxi=-1.0).sample()) for _ in range(set_size)
]
# Fix the fburst + const > 1 issue
for ii in np.arange(len(const_arr)):
if const_arr[ii] + fburst_arr[ii] >= 0.95:
f_over = (const_arr[ii] + fburst_arr[ii]) - 0.95
if fburst_arr[ii] >= (f_over + 0.01):
fburst_arr[ii] = fburst_arr[ii] - (f_over + 0.01)
else:
const_arr[ii] = const_arr[ii] - (f_over + 0.01)
# Fixed the rest
dust1_arr = np.full(set_size, 0.1)
dust2_arr = np.full(set_size, 0.0)
sf_trunc_arr = np.full(set_size, 0.0)
# List of model parameters
dwarf_sample_parameters = [{
'dust1': dust1_arr[ii],
'dust2': dust2_arr[ii],
'logzsol': logzsol_arr[ii],
'gas_logz': gas_logz_arr[ii],
'gas_logu': gas_logu_arr[ii],
'const': const_arr[ii],
'tau': tau_arr[ii],
'tage': tage_arr[ii],
'sf_trunc': sf_trunc_arr[ii],
'fburst': fburst_arr[ii],
'tburst': tburst_arr[ii]
} for ii in np.arange(set_size)]
# Double check
for ii, model in enumerate(dwarf_sample_parameters):
if model['fburst'] + model['const'] >= 0.99:
print(ii, model['fburst'], model['const'])
# Initialize the spop model
spop_tau = setup_fsps_spop(zcontinuous=1,
imf_type=2,
sfh=1,
dust_type=0,
dust_index=-1.3,
dust1_index=-1.0)
# Get the SDSS filters
sdss_bands = fsps.find_filter('SDSS')
dwarf_sample_gaussian = generate_dwarf_population(spop_tau,
dwarf_sample_parameters,
filters=sdss_bands,
n_jobs=6)
# Measure colors and emission line EWs
# - SDSS_EMLINES is a pre-defined dict of emission lines center wavelength and the
# wavelength window for measuring EW.
# - You can save the results in a numpy array
dwarf_sample_table = measure_color_ew(dwarf_sample_gaussian,
em_list=SDSS_EMLINES,
output=None)
bin_size = 200
ur_size = np.linspace(0.0, 2.5, bin_size)
ug_size = np.linspace(0.0, 2.0, bin_size)
gr_size = np.linspace(-0.1, 0.8, bin_size)
gi_size = np.linspace(-0.2, 1.3, bin_size)
ha_size = np.linspace(0.0, 3.0, bin_size)
hb_size = np.linspace(-0.5, 2.5, bin_size)
oiii_size = np.linspace(-1.0, 3.0, bin_size)
obs_ur = data_to_distribution(
np.asarray(sdss_use['M_u'] - sdss_use['M_r']), ur_size)
obs_ug = data_to_distribution(
np.asarray(sdss_use['M_u'] - sdss_use['M_g']), ug_size)
obs_gr = data_to_distribution(
np.asarray(sdss_use['M_g'] - sdss_use['M_r']), gr_size)
obs_gi = data_to_distribution(
np.asarray(sdss_use['M_g'] - sdss_use['M_i']), gi_size)
obs_ha = data_to_distribution(np.log10(abs(sdss_use['H_ALPHA_EQW'])),
ha_size)
obs_hb = data_to_distribution(np.log10(abs(sdss_use['H_BETA_EQW'])),
hb_size)
obs_oiii = data_to_distribution(np.log10(abs(sdss_use['OIII_5007_EQW'])),
oiii_size)
model_ur = data_to_distribution(dwarf_sample_table['ur_color'], ur_size)
model_ug = data_to_distribution(dwarf_sample_table['ug_color'], ug_size)
model_gr = data_to_distribution(dwarf_sample_table['gr_color'], gr_size)
model_gi = data_to_distribution(dwarf_sample_table['gi_color'], gi_size)
model_ha = data_to_distribution(
np.log10(abs(dwarf_sample_table['ew_halpha'])), ha_size)
model_hb = data_to_distribution(
np.log10(abs(dwarf_sample_table['ew_hbeta'])), hb_size)
model_oiii = data_to_distribution(
np.log10(abs(dwarf_sample_table['ew_oiii_5007'])), oiii_size)
obs_stack = np.transpose(
np.vstack([obs_ur, obs_ug, obs_gr, obs_gi, obs_ha, obs_hb, obs_oiii]))
model_stack = np.transpose(
np.vstack([
model_ur, model_ug, model_gr, model_gi, model_ha, model_hb,
model_oiii
]))
total_loss = entropy(obs=obs_stack, model=model_stack)
return total_loss
#plt.rc('text', usetex=True)
mlim = 8.7
mmax = 12.5
boxsize = int(MODEL[1:3])
size_comp = True
plotvar = 'sdss_r'
#if len(sys.argv)==5: plotvar=sys.argv[4]
#plotvar = 'sfr'
plot_dir = '/home/sapple/simba_sizes/plots/'
data_dir = '/home/sapple/simba_sizes/data/'
if plotvar not in ['mstar', 'sfr']:
try:
sdss_bands = fsps.find_filter(plotvar)
except:
sys.exit('Filter %s not found' % plotvar)
print('Doing rhalf in band %s, generating FSPS stellar pop...' %
plotvar)
spop = fsps.StellarPopulation(zcontinuous=1,
sfh=0,
logzsol=0.0,
dust_type=2,
dust2=0.2)
print spop.ssp_ages
# load in input file
fig, ax = plt.subplots()
for iwind in range(0, len(WIND)):
snapfile = '/home/rad/data/%s/%s/snap_%s_%03d.hdf5' % (
"""
import fsps
import matplotlib.pyplot as plt
import numpy as np
import seaborn as sns
sns.set_context("poster")
## PROBLEM 1
Gal_age = 12.5 # Age of Galaxy in Gyrs
Uni_age = 13.8 # Age of the Universe in Gyr
sf_start = Uni_age - Gal_age
sp = fsps.StellarPopulation(compute_vega_mags=False, sfh=1, zmet=20,
dust_type=2, dust2=0.2, sf_start=sf_start)
sdss_filters = fsps.find_filter('sdss')
tau_grid = np.logspace(-2,2,50)
A = Gal_age
def mean_age(tau):
return A - tau * (1.0-np.exp(-A/tau)*(1.0+A/tau)) / (1.0-np.exp(-A/tau))
mean_age_grid = []
for tau in tau_grid:
mean_age_grid.append(mean_age(tau))
# Find what tau is needed to provide appropriate colors for
# red or blue galaxy
skip_lowmass = 0
low_mass = 10.
want_random = '' #''#'_scatter'
sfh_ap = '_30kpc' #'_30kpc'#'_3rad'#'_30kpc'#''
cam_filt = 'sdss_desi' # should be desi; use des with fsps filters
# No implementation error
if want_random == '_scatter':
print("For now we are not incorporating redshift uncertainties")
exit(0)
# create a list of the wanted color bands
bands = []
filts = cam_filt.split('_')
for i in range(len(filts)):
bands += fsps.find_filter(filts[i])
# If using the sedpy magnitudes with decam filters
bands = [
'sdss_u0', 'sdss_g0', 'sdss_r0', 'sdss_i0', 'sdss_z0', 'decam_g',
'decam_r', 'decam_i', 'decam_z', 'decam_Y'
]
print(bands)
# list of all redshifts we have
redshifts = np.array([
1.74324, 1.5, 1.41131, 1.35539, 1.30228, 1.25173, 1.20355, 1.15755,
1.11358, 1.07147, 1.03111, 1, 0.95514, 0.91932, 0.88482, 0.85156, 0.81947,
0.78847, 0.7585, 0.7295, 0.7, 0.5, 0
])
def loss_function(args):
tau_mean, const_mean, tage_mean, fburst_mean, tburst_mean, logzsol_mean, gas_logz_mean, gas_logu_mean = args
set_size = 3000
tau_arr = [
float(
priors.ClippedNormal(mean=tau_mean,
sigma=tau_sig,
mini=1.0,
maxi=8.0).sample()) for _ in range(set_size)
]
const_arr = [
float(
priors.ClippedNormal(mean=const_mean,
sigma=const_sig,
mini=0.0,
maxi=0.5).sample()) for _ in range(set_size)
]
tage_arr = [
float(
priors.ClippedNormal(mean=tage_mean,
sigma=tage_sig,
mini=1.0,
maxi=11.0).sample()) for _ in range(set_size)
]
fburst_arr = [
float(
priors.ClippedNormal(mean=fburst_mean,
sigma=fbust_sig,
mini=0.0,
maxi=0.8).sample()) for _ in range(set_size)
]
tburst_arr = [
float(
priors.ClippedNormal(mean=tburst_mean,
sigma=tburst_sig,
mini=0.0,
maxi=8.0).sample()) for _ in range(set_size)
]
logzsol_arr = [
float(
priors.ClippedNormal(mean=logzsol_mean,
sigma=logzsol_sig,
mini=-1.5,
maxi=0.0).sample()) for _ in range(set_size)
]
gas_logz_arr = [
float(
priors.ClippedNormal(mean=gas_logz_mean,
sigma=gas_logz_sig,
mini=-1.5,
maxi=0.0).sample()) for _ in range(set_size)
]
gas_logu_arr = [
float(
priors.ClippedNormal(mean=gas_logu_mean,
sigma=gas_logu_sig,
mini=-4.0,
maxi=-1.0).sample()) for _ in range(set_size)
]
# Fix the fburst + const > 1 issue
for ii in np.arange(len(const_arr)):
if const_arr[ii] + fburst_arr[ii] >= 0.95:
f_over = (const_arr[ii] + fburst_arr[ii]) - 0.95
if fburst_arr[ii] >= (f_over + 0.01):
fburst_arr[ii] = fburst_arr[ii] - (f_over + 0.01)
else:
const_arr[ii] = const_arr[ii] - (f_over + 0.01)
# Fixed the rest
dust1_arr = np.full(set_size, 0.1)
dust2_arr = np.full(set_size, 0.0)
sf_trunc_arr = np.full(set_size, 0.0)
# List of model parameters
dwarf_sample_parameters = [{
'dust1': dust1_arr[ii],
'dust2': dust2_arr[ii],
'logzsol': logzsol_arr[ii],
'gas_logz': gas_logz_arr[ii],
'gas_logu': gas_logu_arr[ii],
'const': const_arr[ii],
'tau': tau_arr[ii],
'tage': tage_arr[ii],
'sf_trunc': sf_trunc_arr[ii],
'fburst': fburst_arr[ii],
'tburst': tburst_arr[ii]
} for ii in np.arange(set_size)]
# Double check
for ii, model in enumerate(dwarf_sample_parameters):
if model['fburst'] + model['const'] >= 0.99:
print(ii, model['fburst'], model['const'])
# Initialize the spop model
spop_tau = setup_fsps_spop(zcontinuous=1,
imf_type=2,
sfh=1,
dust_type=0,
dust_index=-1.3,
dust1_index=-1.0)
# Get the SDSS filters
sdss_bands = fsps.find_filter('SDSS')
dwarf_sample_gaussian = generate_dwarf_population(spop_tau,
dwarf_sample_parameters,
filters=sdss_bands,
n_jobs=4)
# Measure colors and emission line EWs
# - SDSS_EMLINES is a pre-defined dict of emission lines center wavelength and the
# wavelength window for measuring EW.
# - You can save the results in a numpy array
dwarf_sample_table = measure_color_ew(dwarf_sample_gaussian,
em_list=SDSS_EMLINES,
output=None)
bin_size = 200
ur_loss = loss(dwarf_sample_table['ur_color'],
np.asarray(sdss_use['M_u'] - sdss_use['M_r']),
np.linspace(0, 2.5, bin_size))
ug_loss = loss(dwarf_sample_table['ug_color'],
np.asarray(sdss_use['M_u'] - sdss_use['M_g']),
np.linspace(0, 1.75, bin_size))
gr_loss = loss(dwarf_sample_table['gr_color'],
np.asarray(sdss_use['M_g'] - sdss_use['M_r']),
np.linspace(-0.1, 0.8, bin_size))
gi_loss = loss(dwarf_sample_table['gi_color'],
np.asarray(sdss_use['M_g'] - sdss_use['M_i']),
np.linspace(-0.2, 1.2, bin_size))
OIII_loss = loss(np.log10(dwarf_sample_table['ew_oiii_5007']),
np.log10(-1.0 * (sdss_use['OIII_5007_EQW'])),
np.linspace(-1, 3, bin_size))
Ha_loss = loss(np.log10(np.log10(dwarf_sample_table['ew_halpha'])),
np.log10(-1.0 * sdss_use['H_ALPHA_EQW']),
np.linspace(0, 3, bin_size))
Hb_loss = loss(np.log10(dwarf_sample_table['ew_hbeta']),
np.log10(-1.0 * sdss_use['H_BETA_EQW']),
np.linspace(-0.5, 2.5, bin_size))
total_loss = (ur_loss + ug_loss + gr_loss + gi_loss + OIII_loss + Ha_loss +
Hb_loss) / 7
part_loss = [
ur_loss, ug_loss, gr_loss, gi_loss, OIII_loss, Ha_loss, Hb_loss
]
return total_loss
# s_m = matplotlib.cm.ScalarMappable(cmap=c_m, norm=norm)
# s_m.set_array([])
# plt.figure(2)
# for i in bump:
# plt.plot(wlinv,attenuation.conroy(wl,f_bump=i),color=s_m.to_rgba(i))
# plt.colorbar(s_m,label='Bump Fraction')
# plt.plot(wlinv,cardelli,'k--' ,label='Cardelli Law',linewidth=1,alpha=0.7)
# plt.legend()
## To select the UV filters plot each one to pick the ones you
## like. Picking manually.
uv = fsps.find_filter('uv')
# for fil in uv:
# plt.figure()
# plt.plot(wlinv,10*attenuation.conroy(wl,f_bump=0.5),'-')
# f = fsps.get_filter(fil)
# plt.plot((f.transmission[0]*1e-4)**-1,f.transmission[1],label=f.name)
# plt.legend()
## Amplify all the HST (WFC3) filters:
wfc3 = fsps.find_filter('wfc3_uvis')
for fil in wfc3:
plt.figure()
plt.plot(wlinv, attenuation.conroy(wl, f_bump=0.5) / 10, '-')
if __name__ == '__main__':
import fsps
sp = fsps.StellarPopulation(compute_vega_mags=False,
zcontinuous=1,
sfh=0,
logzsol=0.0,
dust_type=2,
dust2=0.2)
sdss_bands = fsps.find_filter('sdss')
print(sp.get_mags(tage=13.7, bands=sdss_bands))
# parts a and b
import fsps
sp = fsps.StellarPopulation(compute_vega_mags=False,
sfh=0,
zmet=1,
dust_type=1,
uvb=.08,
sf_start=.1)
hst_bands = fsps.find_filter('wfc_acs')
wanted = [hst_bands[1], hst_bands[-1]]
got_absmags = sp.get_mags(tage=14.1, bands=wanted)
import numpy as np
import emcee
got_noise = np.array([np.random.normal(.1), np.random.normal(.1)])
got_wave = np.array([8.14e-7, 6.06e-7])
distance = 8.2e3
got_appmags = got_absmags + got_noise + 5 * np.log10(distance / 10.)
got_error = np.array([.1, .1])
def model(theta): # met, age, dist, extinction
sp.params['zmet'] = np.round(theta[0])
sp.params['sf_start'] = 14.1 - theta[1]
sp.params['uvb'] = theta[3]
got_absmags1 = sp.get_mags(tage=14.1, bands=wanted)
got_appmags1 = got_absmags1 + 5 * np.log10(theta[2] / 10.)
"""theta = (zmet,age,distance)"""
lp = lnprior(theta)
print(lp)
if (np.isfinite(lp)).all() == False:
return -np.inf
z, age, dist = lp[0], lp[1], lp[2]
mag = ssp.get_mags(zmet=z, tage=age, bands=f)
mag += -5 + 5 * np.log10(dist * 1e3)
d = norm.logpdf(data, loc=mag, scale=sigma)
return np.sum(d)
f = [fsps.find_filter('f606w')[1], fsps.find_filter('f814w')[1]]
#data = gen_data(f,0.1)
data = [20.58183853, 20.37690413]
ssp = fsps.StellarPopulation(dust_type=1)
ndim, nwalkers = 3, 100
sampler = emcee.EnsembleSampler(nwalkers, ndim, lnP, args=(data, ssp, 0.1))
theta0 = np.array([np.random.ranf(ndim) for i in range(nwalkers)])
theta0[:, 0] = (-2.5 + 2) * theta0[:, 0] - 2.00
theta0[:, 1] = (14 - 5) * theta0[:, 1] + 5
theta0[:, 2] = (10 - 8) * theta0[:, 2] + 8
sampler.run_mcmc(theta0, 100)
print("Ran the Markov chain! Plotting now..")
