How to handle shape of pymc3 Deterministic variables

Question

I've been working on getting a hierarchical model of some psychophysical behavioral data up and running in pymc3. I'm incredibly impressed with things overall, but after trying to get up to speed with Theano and pymc3 I have a model that mostly works, however has a couple problems.

The code is built to fit a parameterized version of a Weibull to seven sets of data. Each trial is modeled as a binary Bernoulli outcome, while the thresholds (output of thact as the y values which are used to fit a Gaussian function for height, width, and elevation (a, c, and d on a typical Gaussian).

Using the parameterized Weibull seems to be working nicely, and is now hierarchical for the slope of the Weibull while the thresholds are fit separately for each chunk of data. However - the output I'm getting from k and y_est leads me to believe they may not be the correct size, and unlike the probability distributions, it doesn't look like I can specify shape (unless there's a theano way to do this that I haven't found - though from what I've read specifying shape in theano is tricky).

Ultimately, I'd like to use y_est to estimate the gaussian height or width, however the output right now results in an incredible mess that I think originates with size problems in y_est and k. Any help would be fantastic - the code below should simulate some data and is followed by the model. The model does a nice job fitting each individual threshold and getting the slopes, but falls apart when dealing with the rest.

Thanks for having a look - I'm super impressed with pymc3 so far!

EDIT: Okay, so the shape output by y_est.tag.test_value.shape looks like this

y_est.tag.test_value.shape
(101, 7)
k.tag.test_value.shape
(7,)

I think this is where I'm running into trouble, though it may just be poorly constructed on my part. k has the right shape (one k value per unique_xval). y_est is outputting an entire set of data (101x7) instead of a single estimate (one y_est per unique_xval) for each difficulty level. Is there some way to specify that y_est get specific subsets of df_y_vals to control this?

#Import necessary modules and define our weibull function
import numpy as np
import pylab as pl    
from scipy.stats import bernoulli

#x stimulus intensity
#g chance (0.5 for 2AFC)
# m slope
# t threshold
# a performance level defining threshold 
def weib(x,g,a,m,t):
    k=-np.log(((1-a)/(1-g))**(1/t))
    return 1- (1-g)*np.exp(- (k*x/t)**m);

#Output values from weibull function
xit=101
xvals=np.linspace(0.05,1,xit)
out_weib=weib(xvals, 0.5, 0.8, 3, 0.6)

#Okay, fitting the perfect output of a Weibull should be easy, contaminate         with some noise
#Slope of 3, threshold of 0.6


#How about 5% noise!

noise=0.05*np.random.randn(np.size(out_weib))
out=out_weib+noise

#Let's make this more like a typical experiment - 
#i.e. no percent correct, just one or zero
#Randomly pick based on the probability at each point whether they got the trial right or wrong
trial=np.zeros_like(out)
for i in np.arange(out.size):
    p=out_weib[i]
    trial[i] = bernoulli.rvs(p)

#Iterate for 6 sets of data, similar slope (from a normal dist), different thresh (output from gaussian)
#Gauss parameters=

true_gauss_height = 0.3
true_gauss_width = 0.01
true_gauss_elevation = 0.2

#What thresholds will we get then? 6 discrete points along that gaussian, from 0 to 180 degree mask

x_points=[0, 30, 60, 90, 120, 150, 180]

x_points=np.asarray(x_points)
gauss_points=true_gauss_height*np.exp(-    ((x_points**2)/2*true_gauss_width**2))+true_gauss_elevation

import pymc as pm2
import pymc3 as pm
import pandas as pd

slopes=pm2.rnormal(3, 3, size=7)
out_weib=np.zeros([xvals.size,x_points.size])

for i in np.arange(x_points.size):
    out_weib[:,i]=weib(xvals, 0.5, 0.8, slopes[i], gauss_points[i])

#Let's make this more like a typical experiment - i.e. no percent correct, just one or zero
#Randomly pick based on the probability at each point whether they got the trial right or wrong
trials=np.zeros_like(out_weib)

for i in np.arange(len(trials)):
    for ii in np.arange(gauss_points.size):
        p=out_weib[i,ii]
        trials[i,ii] = bernoulli.rvs(p)

#Let's make that data into a DataFrame for pymc3
y_vals=np.tile(xvals, [7, 1])

df_correct = pd.DataFrame(trials, columns=x_points)
df_y_vals = pd.DataFrame(y_vals.T, columns=x_points)
unique_xvals=x_points

import theano as th

with pm.Model() as hierarchical_model:
    # Hyperpriors for group node
    mu_slope = pm.Normal('mu_slope', mu=3, sd=1)
    sigma_slope = pm.Uniform('sigma_slope', lower=0.1, upper=2)

#Priors for the overall gaussian function - 3 params, the height of the gaussian
#Width, and elevation

gauss_width = pm.HalfNormal('gauss_width', sd=1)
gauss_elevation = pm.HalfNormal('gauss_elevation', sd=1)

slope = pm.Normal('slope', mu=mu_slope, sd=sigma_slope,     shape=unique_xvals.size)

thresh=pm.Uniform('thresh', upper=1, lower=0.1, shape=unique_xvals.size)

k = -th.tensor.log(((1-0.8)/(1-0.5))**(1/thresh))
y_est=1-(1-0.5)*th.tensor.exp(-(k*df_y_vals/thresh)**slope)

#We want our model to predict either height or width...height would be easier.
#Our Gaussian function has y values estimated by y_est as the 82% thresholds
#and Xvals based on where each of those psychometrics were taken.
#height_est=pm.Deterministic('height_est', (y_est/(th.tensor.exp((-unique_xvals**2)/2*gauss_width)))+gauss_elevation)
height_est = pm.Deterministic('height_est', (y_est-gauss_elevation)*th.tensor.exp((unique_xvals**2)/2*gauss_width**2))

#Define likelihood as Bernoulli for each binary trial
likelihood = pm.Bernoulli('likelihood',p=y_est, shape=unique_xvals.size, observed=df_correct)

#Find start
start=pm.find_MAP()
step=pm.NUTS(state=start)
#Do MCMC
trace = pm.sample(5000, step, njobs=1, progressbar=True) # draw 5000 posterior samples using NUTS sampling

Answer 1

I'm not sure exactly what you want to do when you say "Is there some way to specify that y_est get specific subsets of df_y_vals to control this". Can you describe for each y_est value what values of df_y_vals are you supposed to use? What's the shape of df_y_vals? What's the shape of y_est supposed to be? (7,)?

I suspect what you want is to index into df_y_vals using numpy advanced indexing , which works the same in PyMC as in numpy. Its hard to say exactly without more information.