Related
I'm trying to learn more about pyhf and my understanding of what the goals are might be limited. I would love to fit my HEP data outside of ROOT, but I could be imposing expectations on pyhf which are not what the authors intended for it's use.
I'd like to write myself a hello-world example, but I might just not know what I'm doing. My misunderstanding could also be gaps in my statistical knowledge.
With that preface, let me explain what I'm trying to explore.
I have some observed set of events for which I calculate some observable and make a binned histogram of that data. I hypothesize that there are two contributing physics processes, which I call signal and background. I generate some Monte Carlo samples for these processes and the theorized total number of events is close to, but not exactly what I observe.
I would like to:
Fit the data to this two process hypothesis
Get from the fit the optimal values for the number of events for each process
Get the uncertainties on these fitted values
If appropriate, calculate an upper limit on the number of signal events.
My starter code is below, where all I'm doing is an ML fit but I'm not sure where to go. I know it's not set up to do what I want, but I'm getting lost in the examples I find on RTD. I'm sure it's me, this is not a criticism of the documentation.
import pyhf
import numpy as np
import matplotlib.pyplot as plt
nbins = 15
# Generate a background and signal MC sample`
MC_signal_events = np.random.normal(5,1.0,200)
MC_background_events = 10*np.random.random(1000)
signal_data = np.histogram(MC_signal_events,bins=nbins)[0]
bkg_data = np.histogram(MC_background_events,bins=nbins)[0]
# Generate an observed dataset with a slightly different
# number of events
signal_events = np.random.normal(5,1.0,180)
background_events = 10*np.random.random(1050)
observed_events = np.array(signal_events.tolist() + background_events.tolist())
observed_sample = np.histogram(observed_events,bins=nbins)[0]
# Plot these samples, if you like
plt.figure(figsize=(12,4))
plt.subplot(1,3,1)
plt.hist(observed_events,bins=nbins,label='Observations')
plt.legend()
plt.subplot(1,3,2)
plt.hist(MC_signal_events,bins=nbins,label='MC signal')
plt.legend()
plt.subplot(1,3,3)
plt.hist(MC_background_events,bins=nbins,label='MC background')
plt.legend()
# Use a very naive estimate of the background
# uncertainties
bkg_uncerts = np.sqrt(bkg_data)
print("Defining the PDF.......")
pdf = pyhf.simplemodels.hepdata_like(signal_data=signal_data.tolist(), \
bkg_data=bkg_data.tolist(), \
bkg_uncerts=bkg_uncerts.tolist())
print("Fit.......")
data = pyhf.tensorlib.astensor(observed_sample.tolist() + pdf.config.auxdata)
bestfit_pars, twice_nll = pyhf.infer.mle.fit(data, pdf, return_fitted_val=True)
print(bestfit_pars)
print(twice_nll)
plt.show()
Note: this answer is based on pyhf v0.5.2.
Alright, so it looks like you've managed to figure most of the big pieces for sure. However, there's two different ways to do this depending on how you prefer to set things up. In both cases, I assume you want an unconstrained fit and you want to...
fit your signal+background model to observed data
fit your background model to observed data
First, let's discuss uncertainties briefly. At the moment, we default to numpy for the tensor background and scipy for the optimizer. See documentation:
numpy backend
scipy optimizer
However, one unfortunate drawback right now with the scipy optimizer is that it cannot return the uncertainties. What you need to do anywhere in your code before the fit (although we generally recommend as early as possible) is to use the minuit optimizer instead:
pyhf.set_backend('numpy', 'minuit')
This will get you the nice features of being able to get the correlation matrix, the uncertainties on the fitted parameters, and the hessian -- amongst other things. We're working to make this consistent for scipy as well, but this is not ready right now.
All optimizations go through our optimizer API which you can currently view through the mixin here in our documentation. Specifically, the signature is
minimize(
objective,
data,
pdf,
init_pars,
par_bounds,
fixed_vals=None,
return_fitted_val=False,
return_result_obj=False,
do_grad=None,
do_stitch=False,
**kwargs)
There are a lot of options here. Let's just focus on the fact that one of the keyword arguments we can pass through is return_uncertainties which will change the bestfit parameters by adding a column for the fitted parameter uncertainty which you want.
1. Signal+Background
In this case, we want to just use the default model
result, twice_nll = pyhf.infer.mle.fit(
data,
pdf,
return_uncertainties=True,
return_fitted_val=True
)
bestfit_pars, errors = result.T
2. Background-Only
In this case, we need to turn off the signal. The way we do this is by setting the parameter of interest (POI) fixed to 0.0. Then we can get the fitted parameters for the background-only model in a similar way, but using fixed_poi_fit instead of an unconstrained fit:
result, twice_nll = pyhf.infer.mle.fixed_poi_fit(
0.0,
data,
pdf,
return_uncertainties=True,
return_fitted_val=True
)
bestfit_pars, errors = result.T
Note that this is quite simply a quick way of doing the following unconstrained fit
bkg_params = pdf.config.suggested_init()
fixed_params = pdf.config.suggested_fixed()
bkg_params[pdf.config.poi_index] = 0.0
fixed_params[pdf.config.poi_index] = True
result, twice_nll = pyhf.infer.mle.fit(
data,
pdf,
init_pars=bkg_params,
fixed_params=fixed_params,
return_uncertainties=True,
return_fitted_val=True
)
bestfit_pars, errors = result.T
Hopefully that clarifies things up more!
Giordon's solution should answer all of your question, but I thought I'd also write out the code to basically address everything we can.
I also take the liberty of changing some of your values a bit so that the signal isn't so strong that the observed CLs value isn't far off to the right of the Brazil band (the results aren't wrong obviously, but it probably makes more sense to be talking about using the discovery test statistic at that point then setting limits. :))
Environment
For this example I'm going to setup a clean Python 3 virtual environment and then install the dependencies (here we're going to be using pyhf v0.5.2)
$ python3 -m venv "${HOME}/.venvs/question"
$ . "${HOME}/.venvs/question/bin/activate"
(question) $ cat requirements.txt
pyhf[minuit,contrib]~=0.5.2
black
(question) $ python -m pip install -r requirements.txt
Code
While we can't easily get the best fit value for both the number of signal events as well as the background events we definitely can do inference to get the best fit value for the signal strength.
The following chunk of code (which is long only because of the visualization) should address all of the points of your question.
# answer.py
import numpy as np
import pyhf
import matplotlib.pyplot as plt
import pyhf.contrib.viz.brazil
# Goals:
# - Fit the model to the observed data
# - Infer the best fit signal strength given the model
# - Get the uncertainties on the best fit signal strength
# - Calculate an 95% CL upper limit on the signal strength
def plot_hist(ax, bins, data, bottom=0, color=None, label=None):
bin_width = bins[1] - bins[0]
bin_leftedges = bins[:-1]
bin_centers = [edge + bin_width / 2.0 for edge in bin_leftedges]
ax.bar(
bin_centers, data, bin_width, bottom=bottom, alpha=0.5, color=color, label=label
)
def plot_data(ax, bins, data, label="Data"):
bin_width = bins[1] - bins[0]
bin_leftedges = bins[:-1]
bin_centers = [edge + bin_width / 2.0 for edge in bin_leftedges]
ax.scatter(bin_centers, data, color="black", label=label)
def invert_interval(test_mus, hypo_tests, test_size=0.05):
# This will be taken care of in v0.5.3
cls_obs = np.array([test[0] for test in hypo_tests]).flatten()
cls_exp = [
np.array([test[1][idx] for test in hypo_tests]).flatten() for idx in range(5)
]
crossing_test_stats = {"exp": [], "obs": None}
for cls_exp_sigma in cls_exp:
crossing_test_stats["exp"].append(
np.interp(
test_size, list(reversed(cls_exp_sigma)), list(reversed(test_mus))
)
)
crossing_test_stats["obs"] = np.interp(
test_size, list(reversed(cls_obs)), list(reversed(test_mus))
)
return crossing_test_stats
def main():
np.random.seed(0)
pyhf.set_backend("numpy", "minuit")
observable_range = [0.0, 10.0]
bin_width = 0.5
_bins = np.arange(observable_range[0], observable_range[1] + bin_width, bin_width)
n_bkg = 2000
n_signal = int(np.sqrt(n_bkg))
# Generate simulation
bkg_simulation = 10 * np.random.random(n_bkg)
signal_simulation = np.random.normal(5, 1.0, n_signal)
bkg_sample, _ = np.histogram(bkg_simulation, bins=_bins)
signal_sample, _ = np.histogram(signal_simulation, bins=_bins)
# Generate observations
signal_events = np.random.normal(5, 1.0, int(n_signal * 0.8))
bkg_events = 10 * np.random.random(int(n_bkg + np.sqrt(n_bkg)))
observed_events = np.array(signal_events.tolist() + bkg_events.tolist())
observed_sample, _ = np.histogram(observed_events, bins=_bins)
# Visualize the simulation and observations
fig, ax = plt.subplots()
fig.set_size_inches(7, 5)
plot_hist(ax, _bins, bkg_sample, label="Background")
plot_hist(ax, _bins, signal_sample, bottom=bkg_sample, label="Signal")
plot_data(ax, _bins, observed_sample)
ax.legend(loc="best")
ax.set_ylim(top=np.max(observed_sample) * 1.4)
ax.set_xlabel("Observable")
ax.set_ylabel("Count")
fig.savefig("components.png")
# Build the model
bkg_uncerts = np.sqrt(bkg_sample)
model = pyhf.simplemodels.hepdata_like(
signal_data=signal_sample.tolist(),
bkg_data=bkg_sample.tolist(),
bkg_uncerts=bkg_uncerts.tolist(),
)
data = pyhf.tensorlib.astensor(observed_sample.tolist() + model.config.auxdata)
# Perform inference
fit_result = pyhf.infer.mle.fit(data, model, return_uncertainties=True)
bestfit_pars, par_uncerts = fit_result.T
print(
f"best fit parameters:\
\n * signal strength: {bestfit_pars[0]} +/- {par_uncerts[0]}\
\n * nuisance parameters: {bestfit_pars[1:]}\
\n * nuisance parameter uncertainties: {par_uncerts[1:]}"
)
# Perform hypothesis test scan
_start = 0.0
_stop = 5
_step = 0.1
poi_tests = np.arange(_start, _stop + _step, _step)
print("\nPerforming hypothesis tests\n")
hypo_tests = [
pyhf.infer.hypotest(
mu_test,
data,
model,
return_expected_set=True,
return_test_statistics=True,
qtilde=True,
)
for mu_test in poi_tests
]
# Upper limits on signal strength
results = invert_interval(poi_tests, hypo_tests)
print(f"Observed Limit on µ: {results['obs']:.2f}")
print("-----")
for idx, n_sigma in enumerate(np.arange(-2, 3)):
print(
"Expected {}Limit on µ: {:.3f}".format(
" " if n_sigma == 0 else "({} σ) ".format(n_sigma),
results["exp"][idx],
)
)
# Visualize the "Brazil band"
fig, ax = plt.subplots()
fig.set_size_inches(7, 5)
ax.set_title("Hypothesis Tests")
ax.set_ylabel(r"$\mathrm{CL}_{s}$")
ax.set_xlabel(r"$\mu$")
pyhf.contrib.viz.brazil.plot_results(ax, poi_tests, hypo_tests)
fig.savefig("brazil_band.png")
if __name__ == "__main__":
main()
which when run gives
(question) $ python answer.py
best fit parameters:
* signal strength: 1.5884737977889158 +/- 0.7803435235862329
* nuisance parameters: [0.99020988 1.06040191 0.90488207 1.03531383 1.09093327 1.00942088
1.07789316 1.01125627 1.06202964 0.95780043 0.94990993 1.04893286
1.0560711 0.9758487 0.93692481 1.04683181 1.05785515 0.92381263
0.93812855 0.96751869]
* nuisance parameter uncertainties: [0.06966439 0.07632218 0.0611428 0.07230328 0.07872258 0.06899675
0.07472849 0.07403246 0.07613661 0.08606657 0.08002775 0.08655314
0.07564512 0.07308117 0.06743479 0.07383134 0.07460864 0.06632003
0.06683251 0.06270965]
Performing hypothesis tests
/home/stackoverflow/.venvs/question/lib/python3.7/site-packages/pyhf/infer/calculators.py:229: RuntimeWarning: invalid value encountered in double_scalars
teststat = (qmu - qmu_A) / (2 * self.sqrtqmuA_v)
Observed Limit on µ: 2.89
-----
Expected (-2 σ) Limit on µ: 0.829
Expected (-1 σ) Limit on µ: 1.110
Expected Limit on µ: 1.542
Expected (1 σ) Limit on µ: 2.147
Expected (2 σ) Limit on µ: 2.882
Let us know if you have any further questions!
I am trying to replicate the resnet18 paper. Before running this on the full Image Net dataset on disk, I'm doing some evaluation runs with the publicly available imagenette/320px dataset from TFDS (much much smaller subset of imagenet with 10 classes, already in .tfrecord format._
Note: the full notebook to do training and tracing is available here: resnet18_baseline.ipynb Just switch to a GPU runtime and run all the cells. It's already set-up with tensorboard profiling on the second batch. (You can use TPU as well, but some keras.layers.experimental.preprocessing layers do not support TPU ops yet and you have to enable soft device placement. Please use a GPU).
Input Operations
read images from the input dataset. These images usually have got different dimensions and we need some crop function because input tensors can not have different dimensions for batching. Therefore, for training I use random crop and for testing/validation datasets a center crop.
random_crop_layer = keras.layers.experimental.preprocessing.RandomCrop(224, 224)
center_crop_layer = keras.layers.experimental.preprocessing.CenterCrop(224, 224)
#tf.function(experimental_relax_shapes=True) # avoid retracing
def train_crop_fn(x, y):
return random_crop_layer(x), y
#tf.function(experimental_relax_shapes=True)
def eval_crop_fn(x, y):
return center_crop_layer(x), y
Perform some simple preprocessing/augmentations to the input data. These include rescaling to 0-1 and also scaling based on mean and stdev of the rgb colours on imagenet. Also, random
rescaling_layer = keras.layers.experimental.preprocessing.Rescaling(1./255)
train_preproc = keras.Sequential([
rescaling_layer
])
# from https://github.com/tensorflow/models/blob/master/official/vision/image_classification/preprocessing.py
# Calculated from the ImageNet training set
MEAN_RGB = (0.485 , 0.456, 0.406)
STDDEV_RGB = (0.229, 0.224, 0.225)
#tf.function
def z_score_scale(x):
return (x - MEAN_RGB) / STDDEV_RGB
#tf.function
def train_preproc_fn(x, y):
return z_score_scale(train_preproc(x)), y
#tf.function
def eval_preproc_fn(x, y):
return z_score_scale(eval_preproc(x)), y
Input Pipeline
def get_input_pipeline(input_ds, bs, crop_fn, augmentation_fn):
ret_ds = (
input_ds
.batch(1) # pre-crop are different dimensions and can't be batched
.map(crop_fn,
num_parallel_calls=tf.data.experimental.AUTOTUNE)
.unbatch()
.batch(bs)
.map(augmentation_fn, # augmentations can be batched though.
num_parallel_calls=tf.data.experimental.AUTOTUNE)
)
return ret_ds
# dataset loading
def load_imagenette():
train_ds, ds_info = tfds.load('imagenette/320px', split='train', as_supervised=True, with_info=True)
valid_ds = tfds.load('imagenette/320px', split='validation', as_supervised=True)
return train_ds, valid_ds, valid_ds, ds_info.features['label'].num_classes
# pipeline construction
train_ds, valid_ds, test_ds, num_classes = load_imagenette()
# datasets used for training (notice that I use prefetch here)
train_samples = get_input_pipeline(train_ds, BS, train_crop_fn, train_preproc_fn).prefetch(tf.data.experimental.AUTOTUNE)
valid_samples = get_input_pipeline(valid_ds, BS, eval_crop_fn, eval_preproc_fn).prefetch(tf.data.experimental.AUTOTUNE)
test_samples = get_input_pipeline(test_ds, BS, eval_crop_fn, eval_preproc_fn).prefetch(tf.data.experimental.AUTOTUNE)
Training and Profiling
I use tensorboard profiler to check the second batch size and I get a warning that this is highly input bound, with about 40% of processing wasted on inputs.
For a classic resnet18 model, you can drive the batch size up to 768 without getting a OOM error, which is what I use. A single step with bs 256 takes about 2-3 seconds.
I also get a warning that on_train_batch_size_end is slow, at around ~1.5 seconds, compared to the 1s batch time.
The model training code is very simple keras:
model.fit(
train_samples,
validation_data=valid_samples,
epochs=100,
batch_size=BS,
use_multiprocessing=True
callbacks=[tensorboard_callback, model_checkpoint_callback, early_stop_callback, reduce_lr_callback]
)
and the callbacks are specified as:
log_dir = os.path.join(os.getcwd(), 'logs')
tensorboard_callback = TensorBoard(log_dir=log_dir, update_freq="epoch", profile_batch=2)
reduce_lr_callback = tf.keras.callbacks.ReduceLROnPlateau(monitor='val_loss', factor=0.1, patience=5, min_lr=0.001, verbose=1)
model_checkpoint_callback = tf.keras.callbacks.ModelCheckpoint(filepath='model.{epoch:02d}-{val_loss:.4f}.h5',
monitor='val_loss',
verbose=1,
save_best_only=True)
early_stop_callback = keras.callbacks.EarlyStopping(monitor='val_loss', patience=15)
Lastly, here are some sample tensorboard profiling screenshots. I can't figure out how to make this run faster:
I have a binary image of 650x650 size. I want to create patches of 50x50. This means that I need 169 patches. I want to examine if there is in every patch is at least "ONE" element.
I need also the result to be pairs of every patch.
Here there is an example of 2d:
2d example
So far so good.When I implement the view_as_blocks function from skimage.util.shape it returns a list of (13,13,50,50).
The way i search for "ONES" will be numpy.where but i am lost in dimensions...
here is my code:
def distinguish_patches_for_label(image_path):
im1=cv2.imread(image_path)
im2 = cv2.cvtColor(im1, cv2.COLOR_BGR2GRAY)
im3=(im2>100).astype(np.uint8)
im4 = cv2.resize(im3, (650,650))
patches=view_as_blocks(im4, block_shape=(50,50))
for i in range(patches.shape[0]):
for j in range(patches.shape[0]):
indexes_normal=list(zip(*np.where(patches[i,j,:,:]== 0)))
for i in range(patches.shape[0]):
for j in range(patches.shape[0]):
indexes_XDs=list(zip(*np.where(patches[i,j,:,:]== 1)))
list1=[]
list2=[]
for i in range(len(indexes_normal)):
list1.append(indexes_normal[i][0])
list2.append(indexes_normal[i][1])
zipped_indexes_normal=list(zip(list1,list2))
list3=[]
list4=[]
for i in range(len(indexes_XDs)):
list3.append(indexes_XDs[i][0])
list4.append(indexes_XDs[i][1])
zipped_indexes_XDs=list(zip(list3,list4))
return zipped_indexes_normal,zipped_indexes_XDs
I'm not exactly sure what you are trying to get as your output, but the below will find the number of 1s in each patch:
import numpy as np
from skimage import io, util
def count_ones_in_patches(image_path):
image = io.imread(image_path)
patches = util.view_as_blocks(image, (50, 50))
ones_per_patch = np.sum(patches, axis=(2, 3))
return ones_per_patch
Using the numpy axis= keyword argument is a very good idea in general. =)
I have set up a Bayes net with 3 states per node as below, and can read logp's for particular states from it (as in the code).
Next I would like to sample from it. In the code below, sampling runs but I don't see distributions over the three states in the outputs; rather, I see a mean and variance as if they were continuous nodes. How do I get the posterior over the three states?
import numpy as np
import pymc3 as mc
import pylab, math
model = mc.Model()
with model:
rain = mc.Categorical('rain', p = np.array([0.5, 0. ,0.5]))
sprinkler = mc.Categorical('sprinkler', p=np.array([0.33,0.33,0.34]))
CPT = mc.math.constant(np.array([ [ [.1,.2,.7], [.2,.2,.6], [.3,.3,.4] ],\
[ [.8,.1,.1], [.3,.4,.3], [.1,.1,.8] ],\
[ [.6,.2,.2], [.4,.4,.2], [.2,.2,.6] ] ]))
p_wetgrass = CPT[rain, sprinkler]
wetgrass = mc.Categorical('wetgrass', p_wetgrass)
#brute force search (not working)
for val_rain in range(0,3):
for val_sprinkler in range(0,3):
for val_wetgrass in range(0,3):
lik = model.logp(rain=val_rain, sprinkler=val_sprinkler, wetgrass=val_wetgrass )
print([val_rain, val_sprinkler, val_wetgrass, lik])
#sampling (runs but don't understand output)
if 1:
niter = 10000 # 10000
tune = 5000 # 5000
print("SAMPLING:")
#trace = mc.sample(20000, step=[mc.BinaryGibbsMetropolis([rain, sprinkler])], tune=tune, random_seed=124)
trace = mc.sample(20000, tune=tune, random_seed=124)
print("trace summary")
mc.summary(trace)
answering own question: the trace does contain the discrete values but the mc.summary(trace) function is set up to compute continuous mean and variance stats. To make a histogram of the discrete states, use h = hist(trace.get_values(sprinkler)) :-)
I have this problem: I have a cohort of individuals grouped in 5 age groups. Initially all of them are susceptible and then they develop disease and finally they have cancers. I have information about the age group distribution of the susceptible and then the cancer carrier. Between the susceptible state and the cancer they pass through 7 stages , with same transition rate.
I'm trying to create a model that simulate each transition as a binomial extraction and fit the data I have.
I tried something but in the moment of analysing the traces , nothing work
You can see the code
Where am I getting wrong?
Thanks for any help
from pylab import *
from pymc import *
from pymc.Matplot import plot as plt
#susceptible_data = array([647,1814,8838,9949,1920])
susceptible_data = array([130,398,1415,1303,206])
infected_data_100000 = array([0,197,302,776,927])
infected_data = array([0,7,38,90,17])
prior_values=np.zeros(len(infected_data))
for i in range(0,len(infected_data)):
prior_values[i]=infected_data[i]/susceptible_data[i]
# stochastic priors
beta1 = Uniform('beta1', 0., 1.)
lambda_0_temp=susceptible_data[0]
lambda_0_1=pymc.Binomial("lambda_0_1",lambda_0_temp,pow(beta1,1))
lambda_0_2=pymc.Binomial("lambda_0_2",lambda_0_1.value,pow(beta1,1))
lambda_0_3=pymc.Binomial("lambda_0_3",lambda_0_2.value,pow(beta1,1))
lambda_0_4=pymc.Binomial("lambda_0_4",lambda_0_3.value,pow(beta1,1))
lambda_0_5=pymc.Binomial("lambda_0_5",lambda_0_4.value,pow(beta1,1))
lambda_0_6=pymc.Binomial("lambda_0_6",lambda_0_5.value,pow(beta1,1))
lambda_0_7=pymc.Binomial("lambda_0_7",n=lambda_0_6.value,p=pow(beta1,1),value=infected_data[0],observed=True)
lambda_1_temp=susceptible_data[1]
lambda_1_1=pymc.Binomial("lambda_1_1",lambda_1_temp,pow(beta1,1))
lambda_1_2=pymc.Binomial("lambda_1_2",lambda_1_1.value,pow(beta1,1))
lambda_1_3=pymc.Binomial("lambda_1_3",lambda_1_2.value,pow(beta1,1))
lambda_1_4=pymc.Binomial("lambda_1_4",lambda_1_3.value,pow(beta1,1))
lambda_1_5=pymc.Binomial("lambda_1_5",lambda_1_4.value,pow(beta1,1))
lambda_1_6=pymc.Binomial("lambda_1_6",lambda_1_5.value,pow(beta1,1))
lambda_1_7=pymc.Binomial("lambda_1_7",n=lambda_1_6.value,p=pow(beta1,1),value=infected_data[1],observed=True)
lambda_2_temp=susceptible_data[2]
lambda_2_1=pymc.Binomial("lambda_2_1",lambda_2_temp,pow(beta1,1))
lambda_2_2=pymc.Binomial("lambda_2_2",lambda_2_1.value,pow(beta1,1))
lambda_2_3=pymc.Binomial("lambda_2_3",lambda_2_2.value,pow(beta1,1))
lambda_2_4=pymc.Binomial("lambda_2_4",lambda_2_3.value,pow(beta1,1))
lambda_2_5=pymc.Binomial("lambda_2_5",lambda_2_4.value,pow(beta1,1))
lambda_2_6=pymc.Binomial("lambda_2_6",lambda_2_5.value,pow(beta1,1))
lambda_2_7=pymc.Binomial("lambda_2_7",n=lambda_2_6.value,p=pow(beta1,1),value=infected_data[2],observed=True)
lambda_3_temp=susceptible_data[3]
lambda_3_1=pymc.Binomial("lambda_3_1",lambda_3_temp,pow(beta1,1))
lambda_3_2=pymc.Binomial("lambda_3_2",lambda_3_1.value,pow(beta1,1))
lambda_3_3=pymc.Binomial("lambda_3_3",lambda_3_2.value,pow(beta1,1))
lambda_3_4=pymc.Binomial("lambda_3_4",lambda_3_3.value,pow(beta1,1))
lambda_3_5=pymc.Binomial("lambda_3_5",lambda_3_4.value,pow(beta1,1))
lambda_3_6=pymc.Binomial("lambda_3_6",lambda_3_5.value,pow(beta1,1))
lambda_3_7=pymc.Binomial("lambda_3_7",n=lambda_3_6.value,p=pow(beta1,1),value=infected_data[3],observed=True)
lambda_4_temp=susceptible_data[4]
lambda_4_1=pymc.Binomial("lambda_4_1",lambda_4_temp,pow(beta1,1))
lambda_4_2=pymc.Binomial("lambda_4_2",lambda_4_1.value,pow(beta1,1))
lambda_4_3=pymc.Binomial("lambda_4_3",lambda_4_2.value,pow(beta1,1))
lambda_4_4=pymc.Binomial("lambda_4_4",lambda_4_3.value,pow(beta1,1))
lambda_4_5=pymc.Binomial("lambda_4_5",lambda_4_4.value,pow(beta1,1))
lambda_4_6=pymc.Binomial("lambda_4_6",lambda_4_5.value,pow(beta1,1))
lambda_4_7=pymc.Binomial("lambda_4_7",n=lambda_4_6.value,p=pow(beta1,1),value=infected_data[4],observed=True)
model=pymc.Model([lambda_0_7,lambda_1_7,lambda_2_7,lambda_3_7,lambda_4_7,beta1])
mcmc =pymc.MCMC(model)
mcmc.sample(iter=100000, burn=50000, thin=10, verbose=1)
lambda_0_samples=mcmc.trace('lambda_0_7')[:]
lambda_1_samples=mcmc.trace('lambda_1_7')[:]
lambda_2_samples=mcmc.trace('lambda_2_7')[:]
lambda_3_samples=mcmc.trace('lambda_3_7')[:]
lambda_4_samples=mcmc.trace('lambda_4_7')[:]
beta1_samples=mcmc.trace('beta1')[:]
What you have implemented above only associates data with the 7th distribution in each set; the others are seemingly-redundant hierarchies on the binomial probability. I would think you want data informing each stage. I'm not sure there is information to inform what the values of p should be at each stage, based on what is provided.