# coding: utf-8
# In[17]:
import pymc3 as pm
import numpy as np
n = np.ones(4)*5
y = np.array([0,1,3,5])
dose = np.array([-.86,-.3,-.05,.73])
with pm.Model() as bioassay_model:
# Prior
alpha = pm.Normal('alpha', 0, sd=100)
beta = pm.Normal('beta', 0, sd=100)
# Linear combiations of parameters
theta = pm.invlogit(alpha + beta*dose)
# Model likelihood
deaths = pm.Binomial('deaths', n=n, p=theta, observed=y)
#### YOUR CODE HERE ####
a = pm.Normal('a', mu=0, sd=100)
b_1 = pm.Normal('b_1', mu=0, sd=100)
b_2 = pm.Normal('b_2', mu=0, sd=100)
### END OF YOUR CODE ###
# Thansform these random variables into vector of probabilities p(y_i=1) using logistic regression model specified
# above. PyMC random variables are theano shared variables and support simple mathematical operations.
# For example:
# z = pm.Normal('x', 0, 1) * np.array([1, 2, 3]) + pm.Normal('y', 0, 1) * np.array([4, 5, 6])`
# is a correct PyMC expression.
# Use pm.invlogit for the sigmoid function.
#### YOUR CODE HERE ####
p_y = pm.invlogit(a*np.ones(shape=data.shape[0]) + b_1*data['age'].values + b_2*data['educ'].values)
### END OF YOUR CODE ###
# Declare PyMC Bernoulli random vector with probability of success equal to the corresponding value
# given by the sigmoid function.
# Supply target vector using "observed" argument in the constructor.
#### YOUR CODE HERE ####
y_hat = pm.Bernoulli('y_hat', p=p_y, observed=data['income_more_50K'].values)
### END OF YOUR CODE ###
# Use pm.find_MAP() to find the maximum a-posteriori estimate for the vector of logistic regression weights.
map_estimate = pm.find_MAP()
print(map_estimate)
"""Sumbit MAP estimations of corresponding coefficients:"""
def trial_step(info_A_tm1,info_A_t, # externally provided to function on each trial
obs_choice_tm1,obs_choice_t,
mag_1_t,mag_0_t,
stabvol_t,rewpain_t,
# outputs of this function passed back into it on next trial
choice_tm1,# either generated or observed choice
outcome_valence_tm1, # either generated or observed (although not used on input because immediately redefined, useful for storage)
prob_choice_tm1, # internal state variables
choice_val_tm1,
estimate_tm1,
choice_kernel_tm1,
lr_tm1,lr_c_tm1,Amix_tm1,Binv_tm1,Bc_tm1,mdiff_tm1,eps_tm1,
lr_baseline,lr_goodbad,lr_stabvol,lr_rewpain, # variables accessible on all trials
lr_goodbad_stabvol,lr_rewpain_goodbad,lr_rewpain_stabvol,
lr_rewpain_goodbad_stabvol,
lr_c_baseline,lr_c_goodbad,lr_c_stabvol,lr_c_rewpain, # variables accessible on all trials
lr_c_goodbad_stabvol,lr_c_rewpain_goodbad,lr_c_rewpain_stabvol,
lr_c_rewpain_goodbad_stabvol,
Amix_baseline,Amix_goodbad,Amix_stabvol,Amix_rewpain,
Amix_goodbad_stabvol,Amix_rewpain_goodbad,Amix_rewpain_stabvol,
Amix_rewpain_goodbad_stabvol,
Binv_baseline,Binv_goodbad,Binv_stabvol,Binv_rewpain,
Binv_goodbad_stabvol,Binv_rewpain_goodbad,Binv_rewpain_stabvol,
Binv_rewpain_goodbad_stabvol,
Bc_baseline,Bc_goodbad,Bc_stabvol,Bc_rewpain,
Bc_goodbad_stabvol,Bc_rewpain_goodbad,Bc_rewpain_stabvol,
Bc_rewpain_goodbad_stabvol,
mag_baseline,mag_rewpain,
eps_baseline,eps_stabvol,eps_rewpain,eps_rewpain_stabvol,
gen_indicator,B_max,nonlinear_indicator):
'''
Trial by Trial updates for the model
'''
# determine whether last trial had good outcome
outcome_valence_tm1 = choice_tm1*info_A_tm1 +\
(1.0-choice_tm1)*(1.0-info_A_tm1) +\
(1.0-choice_tm1)*info_A_tm1*(-1.0) + \
(choice_tm1)*(1.0-info_A_tm1)*(-1.0)
# determine Amix for this trial using last good outcome
Amix_t = Amix_baseline + \
outcome_valence_tm1*Amix_goodbad + \
stabvol_t*Amix_stabvol + \
rewpain_t*Amix_rewpain + \
outcome_valence_tm1*stabvol_t*Amix_goodbad_stabvol + \
outcome_valence_tm1*rewpain_t*Amix_rewpain_goodbad + \
stabvol_t*rewpain_t*Amix_rewpain_stabvol + \
outcome_valence_tm1*stabvol_t*rewpain_t*Amix_rewpain_goodbad_stabvol
Amix_t =pm.invlogit(Amix_t)
# Determine Binv for this trial using last good outcome
Binv_t = Binv_baseline + \
outcome_valence_tm1*Binv_goodbad + \
stabvol_t*Binv_stabvol + \
rewpain_t*Binv_rewpain + \
outcome_valence_tm1*stabvol_t*Binv_goodbad_stabvol + \
outcome_valence_tm1*rewpain_t*Binv_rewpain_goodbad + \
stabvol_t*rewpain_t*Binv_rewpain_stabvol + \
outcome_valence_tm1*stabvol_t*rewpain_t*Binv_rewpain_goodbad_stabvol
Binv_t = T.exp(Binv_t)
Binv_t = T.switch(Binv_t<0.1,0.1,Binv_t )
Binv_t = T.switch(Binv_t>B_max.value,B_max.value,Binv_t)
# Determine Bc for this trial using last good outcome
Bc_t = Bc_baseline + \
outcome_valence_tm1*Bc_goodbad + \
stabvol_t*Bc_stabvol + \
rewpain_t*Bc_rewpain + \
outcome_valence_tm1*stabvol_t*Bc_goodbad_stabvol + \
outcome_valence_tm1*rewpain_t*Bc_rewpain_goodbad + \
stabvol_t*rewpain_t*Bc_rewpain_stabvol + \
outcome_valence_tm1*stabvol_t*rewpain_t*Bc_rewpain_goodbad_stabvol
Mag_t = T.exp(mag_baseline+rewpain_t*mag_rewpain)
Mag_t= T.switch(Mag_t<0.1,0.1,Mag_t)
Mag_t= T.switch(Mag_t>10,10,Mag_t)
Bc_t = T.switch(Bc_t>B_max.value,B_max.value,Bc_t)
Bc_t = T.switch(Bc_t<-1*B_max.value,-1*B_max.value,Bc_t)
# define the defaults that will be replaced
mdiff_t = (mag_1_t-mag_0_t)
pdiff_t=(estimate_tm1-(1.0-estimate_tm1)) # compute value from previous probability estimates
# Nonlinear magnitude or probability
if nonlinear_indicator.value==0:
# scale mag diff
mdiff_t = T.sgn(mdiff_t)*T.abs_(mdiff_t)**Mag_t
elif nonlinear_indicator.value==2:
# scale mags separately
mdiff_t = (T.sgn(mag_1_t)*T.abs_(mag_1_t)**Mag_t)-(T.sgn(mag_0_t)*T.abs_(mag_0_t)**Mag_t)
elif nonlinear_indicator.value==3:
# scale prob separately
estimate_tm1 = estimate_tm1**Mag_t
estimate_tm1 = T.switch(estimate_tm1<0.01,0.01,estimate_tm1)
estimate_tm1 = T.switch(estimate_tm1>0.99,0.99,estimate_tm1)
pdiff_t=(estimate_tm1-(1.0-estimate_tm1)) # compute value from previous probability estimates
elif nonlinear_indicator.value==1:
# scale prob diff
pdiff_t = T.sgn(pdiff_t)*T.abs_(pdiff_t)**Mag_t
# choice kernel diff
cdiff_t=(choice_kernel_tm1-(1.0-choice_kernel_tm1))
# Value
choice_val_t = Binv_t*((1-Amix_t)*mdiff_t + (Amix_t)*pdiff_t) + Bc_t*cdiff_t
# before Amix, choice value goes between -1 and 1
prob_choice_t = 1.0/(1.0+T.exp(-1.0*choice_val_t))
# determine eps
eps_t = eps_baseline + \
stabvol_t*eps_stabvol + \
rewpain_t*eps_rewpain + \
stabvol_t*rewpain_t*eps_rewpain_stabvol
eps_t = pm.invlogit(eps_t)
# add epsilon
prob_choice_t = eps_t*0.5+(1.0-eps_t)*prob_choice_t
# Generate choice or Copy participants choice (used for next trial as indicator)
if gen_indicator.value==0:
choice_t = obs_choice_t
else:
choice_t = trng.binomial(n=1, p=prob_choice_t,dtype='float64')
# determine whether current trial is good or bad
outcome_valence_t = choice_t*info_A_t +\
(1.0-choice_t)*(1.0-info_A_t) +\
(1.0-choice_t)*info_A_t*(-1.0) + \
(choice_t)*(1.0-info_A_t)*(-1.0)
lr_t = lr_baseline + \
outcome_valence_t*lr_goodbad + \
stabvol_t*lr_stabvol + \
rewpain_t*lr_rewpain + \
outcome_valence_t*stabvol_t*lr_goodbad_stabvol + \
outcome_valence_t*rewpain_t*lr_rewpain_goodbad + \
stabvol_t*rewpain_t*lr_rewpain_stabvol + \
outcome_valence_t*stabvol_t*rewpain_t*lr_rewpain_goodbad_stabvol
lr_t = pm.invlogit(lr_t)
# update probability estimate, These will be estimate after update on t
# stored differently than before
estimate_t = estimate_tm1 + lr_t*(info_A_t-estimate_tm1)
# Choice kernel learning rate
lr_c_t = lr_c_baseline + \
outcome_valence_t*lr_c_goodbad + \
stabvol_t*lr_c_stabvol + \
rewpain_t*lr_c_rewpain + \
outcome_valence_t*stabvol_t*lr_c_goodbad_stabvol + \
outcome_valence_t*rewpain_t*lr_c_rewpain_goodbad + \
stabvol_t*rewpain_t*lr_c_rewpain_stabvol + \
outcome_valence_t*stabvol_t*rewpain_t*lr_c_rewpain_goodbad_stabvol
lr_c_t = pm.invlogit(lr_c_t)
choice_kernel_t = choice_kernel_tm1 + lr_c_t*(choice_t - choice_kernel_tm1)
return([choice_t,outcome_valence_t,prob_choice_t,choice_val_t,estimate_t,choice_kernel_t,lr_t,lr_c_t,Amix_t,Binv_t,Bc_t,mdiff_t,eps_t])
def BinomGP(y, N, X, X_star, mcmc_iter, start={}):
"""Fits a logistic Gaussian process regression timeseries model.
Details
=======
Let $y_t$ be the number of cases observed out of $N_t$ individuals tested. Then
$$y_t \sim Binomial(N_t, p_t)$$
with
$$logit(p_t) \sim s_t$$
$s_t$ is modelled as a Gaussian process such that
$$s_t \sim \mbox{GP}(\mu_t, \Sigma)$$
with a mean function capturing a linear time trend
$$\mu_t = \alpha + \beta X_t$$
and a periodic covariance plus white noise
$$\Sigma_{i,j} = \sigma^2 \exp{ \frac{2 \sin^2(|x_i - x_j|/365)}{\phi^2}} + \tau^2$$
Parameters
==========
y -- a vector of cases
N -- a vector of number tested
X -- a vector of times at which y is observed
X_star -- a vector of times at which predictons are to be made
start -- a dictionary of starting values for the MCMC.
Returns
=======
A tuple of (model,trace,pred) where model is a PyMC3 Model object, trace is a PyMC3 Multitrace object, and pred is a 5000 x X_star.shape[0] matrix of draws from the posterior predictive distribution of $\pi_t$.
"""
X = np.array(X)[:, None] # Inputs must be arranged as column vector
X_star = X_star[:, None]
model = pm.Model()
with model:
alpha = pm.Normal('alpha', 0, 1000, testval=0.)
beta = pm.Normal('beta', 0, 100, testval=0.)
sigmasq_s = pm.Gamma('sigmasq_s', .1, .1, testval=0.1)
phi_s = pm.Gamma('phi_s', 1., 1., testval=0.5)
tau2 = pm.Gamma('tau2', .1, .1, testval=0.1)
# Construct GPs
cov_s = sigmasq_s * pm.gp.cov.Periodic(1, 365., phi_s)
mean_f = pm.gp.mean.Linear(coeffs=beta, intercept=alpha)
gp_s = pm.gp.Latent(mean_func=mean_f, cov_func=cov_s)
cov_nugget = pm.gp.cov.WhiteNoise(tau2)
nugget = pm.gp.Latent(cov_func=cov_nugget)
gp = gp_s + nugget
model.gp = gp
s = gp.prior('s', X=X)
Y_obs = pm.Binomial('y_obs', N, pm.invlogit(s), observed=y)
# Sample
trace = pm.sample(mcmc_iter, chains=1, start=start)
# Predictions
s_star = gp.conditional('s_star', X_star)
pred = pm.sample_ppc(trace, vars=[s_star, Y_obs])
return (model, trace, pred)
#Model Preminents
observed = Indexed_Successful_Shot_DF['Goal_B']
Shot_Type_Index = Indexed_Successful_Shot_DF['index_one']
N = len(np.unique(Indexed_Successful_Shot_DF['index_one']))
with pm.Model() as unpooled_model:
# Independent parameters for each county
a = pm.Normal('a', 0, sd=100, shape=N) #Intercept
b = pm.Normal('b', 0, sd=100, shape=N) #Coefficient for Shot Type
# Model error
# Calculate predictions given values
# for intercept and slope (Comment 4)
yhat = pm.invlogit(a[Shot_Type_Index] + b[Shot_Type_Index] * Indexed_Successful_Shot_DF.Shot_Angle.values)
# Make predictions fit reality
y = pm.Binomial('y', n=np.ones(Indexed_Successful_Shot_DF.shape[0]), p=yhat, observed= observed)
#Run It
start = pm.find_MAP()
step = pm.Metropolis()
trace_h = pm.sample(2000, step = step, start = start,njobs = 2)
#Example Part 2
def BinomGP(y, N, time, time_pred, mcmc_iter, start={}):
"""Fits a logistic Gaussian process regression timeseries model.
Details
=======
Let $y_t$ be the number of cases observed out of $N_t$ individuals tested. Then
$$y_t \sim Binomial(N_t, p_t)$$
with
$$logit(p_t) \sim s_t$$
$s_t$ is modelled as a Gaussian process such that
$$s_t \sim \mbox{GP}(\mu_t, \Sigma)$$
with a mean function capturing a linear time trend
$$\mu_t = \alpha + \beta X_t$$
and a periodic covariance plus white noise
$$\Sigma_{i,j} = \sigma^2 \exp{ \frac{2 \sin^2(|x_i - x_j|/365)}{\phi^2}} + \tau^2$$
Parameters
==========
y -- a vector of cases
N -- a vector of number tested
X -- a vector of times at which y is observed
T -- a vector of time since recruitment
X_star -- a vector of times at which predictons are to be made
start -- a dictionary of starting values for the MCMC.
Returns
=======
A tuple of (model,trace,pred) where model is a PyMC3 Model object, trace is a PyMC3 Multitrace object, and pred is a 5000 x X_star.shape[0] matrix of draws from the posterior predictive distribution of $\pi_t$.
"""
time = np.array(time)[:, None] # Inputs must be arranged as column vector
offset = np.mean(time)
time = time - offset # Center time
model = pm.Model()
with model:
alpha = pm.Normal('alpha', 0, 1000, testval=0.)
beta = pm.Normal('beta', 0, 100, testval=0.)
sigmasq_s = pm.HalfNormal('sigmasq_s', 5., testval=0.1)
phi_s = 0.16 #pm.HalfNormal('phi_s', 5., testval=0.5)
tau2 = pm.Gamma('tau2', .1, .1, testval=0.1)
# Construct GPs
cov_t = sigmasq_s * pm.gp.cov.Periodic(1, 365., phi_s)
mean_t = pm.gp.mean.Linear(coeffs=beta, intercept=alpha)
gp_period = pm.gp.Latent(mean_func=mean_t, cov_func=cov_t)
cov_nugget = pm.gp.cov.WhiteNoise(tau2)
gp_nugget = pm.gp.Latent(cov_func=cov_nugget)
gp_t = gp_period + gp_nugget
s = gp_t.prior('gp_t', X=time[:, None])
Y_obs = pm.Binomial('y_obs', N, pm.invlogit(s), observed=y)
# Sample
trace = pm.sample(mcmc_iter,
chains=1,
start=start,
tune=1000,
adapt_step_size=True)
# Prediction
time_pred -= offset
s_star = gp_t.conditional('s_star', time_pred[:, None])
pi_star = pm.Deterministic('pi_star', pm.invlogit(s_star))
pred = pm.sample_posterior_predictive(trace, var_names=['y_obs', 'pi_star'])
return {'model': model, 'trace': trace, 'pred': pred}
def trial_step(info_A_tm1,info_A_t, # externally provided to function on each trial
obs_choice_tm1,obs_choice_t,
mag_1_t,mag_0_t,
stabvol_t,rewpain_t,
# outputs of this function passed back into it on next trial
choice_tm1,# either generated or observed choice
outcome_valence_tm1, # either generated or observed (although not used on input because immediately redefined, useful for storage)
prob_choice_tm1, # internal state variables
choice_val_tm1,
estimate_tm1,
choice_kernel_tm1,
lr_tm1,lr_c_tm1,Gamma_tm1,Binv_tm1,Bc_tm1,mdiff_tm1,eps_tm1,
lr_baseline,lr_goodbad,lr_stabvol,lr_rewpain, # variables accessible on all trials
lr_goodbad_stabvol,lr_rewpain_goodbad,lr_rewpain_stabvol,
lr_rewpain_goodbad_stabvol,
lr_c_baseline,lr_c_goodbad,lr_c_stabvol,lr_c_rewpain, # variables accessible on all trials
lr_c_goodbad_stabvol,lr_c_rewpain_goodbad,lr_c_rewpain_stabvol,
lr_c_rewpain_goodbad_stabvol,
Gamma_baseline,Gamma_goodbad,Gamma_stabvol,Gamma_rewpain,
Gamma_goodbad_stabvol,Gamma_rewpain_goodbad,Gamma_rewpain_stabvol,
Gamma_rewpain_goodbad_stabvol,
Binv_baseline,Binv_goodbad,Binv_stabvol,Binv_rewpain,
Binv_goodbad_stabvol,Binv_rewpain_goodbad,Binv_rewpain_stabvol,
Binv_rewpain_goodbad_stabvol,
Bc_baseline,Bc_goodbad,Bc_stabvol,Bc_rewpain,
Bc_goodbad_stabvol,Bc_rewpain_goodbad,Bc_rewpain_stabvol,
Bc_rewpain_goodbad_stabvol,
mag_baseline,mag_rewpain,
eps_baseline,eps_stabvol,eps_rewpain,eps_rewpain_stabvol,
gen_indicator,B_max):
'''
Trial by Trial updates for the model
'''
# determine whether last trial had good outcome
outcome_valence_tm1 = choice_tm1*info_A_tm1 +\
(1.0-choice_tm1)*(1.0-info_A_tm1) +\
(1.0-choice_tm1)*info_A_tm1*(-1.0) + \
(choice_tm1)*(1.0-info_A_tm1)*(-1.0)
# determine Gamma for this trial using last good outcome
Gamma_t = Gamma_baseline + \
outcome_valence_tm1*Gamma_goodbad + \
stabvol_t*Gamma_stabvol + \
rewpain_t*Gamma_rewpain + \
outcome_valence_tm1*stabvol_t*Gamma_goodbad_stabvol + \
outcome_valence_tm1*rewpain_t*Gamma_rewpain_goodbad + \
stabvol_t*rewpain_t*Gamma_rewpain_stabvol + \
outcome_valence_tm1*stabvol_t*rewpain_t*Gamma_rewpain_goodbad_stabvol
Gamma_t =pm.invlogit(Gamma_t)*5 # [0,5]
# Determine Binv for this trial using last good outcome
Binv_t = Binv_baseline + \
outcome_valence_tm1*Binv_goodbad + \
stabvol_t*Binv_stabvol + \
rewpain_t*Binv_rewpain + \
outcome_valence_tm1*stabvol_t*Binv_goodbad_stabvol + \
outcome_valence_tm1*rewpain_t*Binv_rewpain_goodbad + \
stabvol_t*rewpain_t*Binv_rewpain_stabvol + \
outcome_valence_tm1*stabvol_t*rewpain_t*Binv_rewpain_goodbad_stabvol
Binv_t = T.exp(Binv_t)
Binv_t = T.switch(Binv_t<0.1,0.1,Binv_t )
Binv_t = T.switch(Binv_t>B_max.value,B_max.value,Binv_t)
# Determine Bc for this trial using last good outcome
Bc_t = Bc_baseline + \
outcome_valence_tm1*Bc_goodbad + \
stabvol_t*Bc_stabvol + \
rewpain_t*Bc_rewpain + \
outcome_valence_tm1*stabvol_t*Bc_goodbad_stabvol + \
outcome_valence_tm1*rewpain_t*Bc_rewpain_goodbad + \
stabvol_t*rewpain_t*Bc_rewpain_stabvol + \
outcome_valence_tm1*stabvol_t*rewpain_t*Bc_rewpain_goodbad_stabvol
#Bc_t = T.exp(Bc_t)
#Bc_t = T.switch(Bc_t<0.1,0.1,Bc_t )
Bc_t = T.switch(Bc_t>B_max.value,B_max.value,Bc_t)
Bc_t = T.switch(Bc_t<-1*B_max.value,-1*B_max.value,Bc_t)
# Calculate Choice
ev_1_t = (T.abs_(mag_1_t)**Gamma_t)*estimate_tm1
ev_0_t = (T.abs_(mag_0_t)**Gamma_t)*(1-estimate_tm1)
evdiff_t = (ev_1_t - ev_0_t)
mdiff_t = (mag_1_t-mag_0_t) # not used but passed on
cdiff_t=(choice_kernel_tm1-(1.0-choice_kernel_tm1))
choice_val_t = Binv_t*evdiff_t + Bc_t*cdiff_t
# before Gamma, choice value goes between -1 and 1
prob_choice_t = 1.0/(1.0+T.exp(-1.0*choice_val_t))
# determine eps
eps_t = eps_baseline + \
stabvol_t*eps_stabvol + \
rewpain_t*eps_rewpain + \
stabvol_t*rewpain_t*eps_rewpain_stabvol
eps_t = pm.invlogit(eps_t)
# add epsilon
prob_choice_t = eps_t*0.5+(1.0-eps_t)*prob_choice_t
# Generate choice or Copy participants choice (used for next trial as indicator)
if gen_indicator.value==0:
choice_t = obs_choice_t
else:
#import pdb; pdb.set_trace()
#trng = T.shared_randomstreams.RandomStreams(1234)
choice_t = trng.binomial(n=1, p=prob_choice_t,dtype='float64')
# this works, but I don't want everyone binomial to have the same seed, so I'd want to update by 1
#rng_val = choice_t.rng.get_value(borrow=True) # Get the rng for rv_u
#rng_val.seed(seed.value) # seeds the generator
#choice_t.rng.set_value(rng_val, borrow=True)
# determine whether current trial is good or bad
outcome_valence_t = choice_t*info_A_t +\
(1.0-choice_t)*(1.0-info_A_t) +\
(1.0-choice_t)*info_A_t*(-1.0) + \
(choice_t)*(1.0-info_A_t)*(-1.0)
#import pdb; pdb.set_trace()
lr_t = lr_baseline + \
outcome_valence_t*lr_goodbad + \
stabvol_t*lr_stabvol + \
rewpain_t*lr_rewpain + \
outcome_valence_t*stabvol_t*lr_goodbad_stabvol + \
outcome_valence_t*rewpain_t*lr_rewpain_goodbad + \
stabvol_t*rewpain_t*lr_rewpain_stabvol + \
outcome_valence_t*stabvol_t*rewpain_t*lr_rewpain_goodbad_stabvol
lr_t = pm.invlogit(lr_t)
# update probability estimate, These will be estimate after update on t
# stored differently than before
estimate_t = estimate_tm1 + lr_t*(info_A_t-estimate_tm1)
# Choice kernel learning rate
lr_c_t = lr_c_baseline + \
outcome_valence_t*lr_c_goodbad + \
stabvol_t*lr_c_stabvol + \
rewpain_t*lr_c_rewpain + \
outcome_valence_t*stabvol_t*lr_c_goodbad_stabvol + \
outcome_valence_t*rewpain_t*lr_c_rewpain_goodbad + \
stabvol_t*rewpain_t*lr_c_rewpain_stabvol + \
outcome_valence_t*stabvol_t*rewpain_t*lr_c_rewpain_goodbad_stabvol
lr_c_t = pm.invlogit(lr_c_t)
choice_kernel_t = choice_kernel_tm1 + lr_c_t*(choice_t - choice_kernel_tm1)
return([choice_t,outcome_valence_t,prob_choice_t,choice_val_t,estimate_t,choice_kernel_t,lr_t,lr_c_t,Gamma_t,Binv_t,Bc_t,mdiff_t,eps_t])
# Data
n = np.ones(4)*5
y = np.array([0, 1, 3, 5])
dose = np.array([-.86,-.3,-.05,.73])
with Model() as bioassay_model:
# Prior distributions for latent variables
alpha = Normal('alpha', 0, sd=100)
beta = Normal('beta', 0, sd=100)
# Linear combinations of parameters
theta = invlogit(alpha + beta*dose)
# Model likelihood
deaths = Binomial('deaths', n=n, p=theta, observed=y)
with bioassay_model:
# Draw wamples
trace = sample(1000, njobs=2)
# Plot two parameters
forestplot(trace, varnames=['alpha', 'beta'])
def trial_step(
info_tm1_A,
info_t_A, # externally provided to function on each trial
info_tm1_B,
info_t_B,
obs_choice_tm1,
obs_choice_t,
mag_1_t,
mag_0_t,
stabvol_t,
rewpain_t,
# outputs of this function passed back into it on next trial
choice_tm1, # either generated or observed choice
outcome_valence_tm1, # either generated or observed (although not used on input because immediately redefined, useful for storage)
prob_choice_tm1, # internal state variables
choice_val_tm1,
estimate_tm1_A, #
estimate_tm1_B,
Ga_tm1,
Ba_tm1,
Gb_tm1,
Bb_tm1,
choice_kernel_tm1,
lr_tm1,
lr_c_tm1,
Amix_tm1,
Binv_tm1,
Bc_tm1,
decay_tm1,
mdiff_tm1,
eps_tm1,
lr_baseline,
lr_goodbad,
lr_stabvol,
lr_rewpain, # variables accessible on all trials
lr_goodbad_stabvol,
lr_rewpain_goodbad,
lr_rewpain_stabvol,
lr_rewpain_goodbad_stabvol,
lr_c_baseline,
lr_c_goodbad,
lr_c_stabvol,
lr_c_rewpain, # variables accessible on all trials
lr_c_goodbad_stabvol,
lr_c_rewpain_goodbad,
lr_c_rewpain_stabvol,
lr_c_rewpain_goodbad_stabvol,
Amix_baseline,
Amix_goodbad,
Amix_stabvol,
Amix_rewpain,
Amix_goodbad_stabvol,
Amix_rewpain_goodbad,
Amix_rewpain_stabvol,
Amix_rewpain_goodbad_stabvol,
Binv_baseline,
Binv_goodbad,
Binv_stabvol,
Binv_rewpain,
Binv_goodbad_stabvol,
Binv_rewpain_goodbad,
Binv_rewpain_stabvol,
Binv_rewpain_goodbad_stabvol,
Bc_baseline,
Bc_goodbad,
Bc_stabvol,
Bc_rewpain,
Bc_goodbad_stabvol,
Bc_rewpain_goodbad,
Bc_rewpain_stabvol,
Bc_rewpain_goodbad_stabvol,
decay_baseline,
decay_stabvol,
decay_rewpain,
decay_rewpain_stabvol,
mag_baseline,
mag_rewpain,
eps_baseline,
eps_stabvol,
eps_rewpain,
eps_rewpain_stabvol,
gen_indicator,
B_max):
'''
Trial by Trial updates for the model
'''
# determine whether last trial had good outcome
outcome_valence_tm1 = choice_tm1*info_tm1_A +\
(1.0-choice_tm1)*(info_tm1_B) +\
(1.0-choice_tm1)*info_tm1_A*(-1.0) + \
(choice_tm1)*(info_tm1_B)*(-1.0)
# determine Amix for this trial using last good outcome
Amix_t = Amix_baseline + \
outcome_valence_tm1*Amix_goodbad + \
stabvol_t*Amix_stabvol + \
rewpain_t*Amix_rewpain + \
outcome_valence_tm1*stabvol_t*Amix_goodbad_stabvol + \
outcome_valence_tm1*rewpain_t*Amix_rewpain_goodbad + \
stabvol_t*rewpain_t*Amix_rewpain_stabvol + \
outcome_valence_tm1*stabvol_t*rewpain_t*Amix_rewpain_goodbad_stabvol
Amix_t = pm.invlogit(Amix_t)
# Determine Binv for this trial using last good outcome
Binv_t = Binv_baseline + \
outcome_valence_tm1*Binv_goodbad + \
stabvol_t*Binv_stabvol + \
rewpain_t*Binv_rewpain + \
outcome_valence_tm1*stabvol_t*Binv_goodbad_stabvol + \
outcome_valence_tm1*rewpain_t*Binv_rewpain_goodbad + \
stabvol_t*rewpain_t*Binv_rewpain_stabvol + \
outcome_valence_tm1*stabvol_t*rewpain_t*Binv_rewpain_goodbad_stabvol
Binv_t = T.exp(Binv_t)
Binv_t = T.switch(Binv_t < 0.1, 0.1, Binv_t)
Binv_t = T.switch(Binv_t > B_max.value, B_max.value, Binv_t)
# Determine Bc for this trial using last good outcome
Bc_t = Bc_baseline + \
outcome_valence_tm1*Bc_goodbad + \
stabvol_t*Bc_stabvol + \
rewpain_t*Bc_rewpain + \
outcome_valence_tm1*stabvol_t*Bc_goodbad_stabvol + \
outcome_valence_tm1*rewpain_t*Bc_rewpain_goodbad + \
stabvol_t*rewpain_t*Bc_rewpain_stabvol + \
outcome_valence_tm1*stabvol_t*rewpain_t*Bc_rewpain_goodbad_stabvol
#Bc_t = T.exp(Bc_t)
#Bc_t = T.switch(Bc_t<0.1,0.1,Bc_t )
Bc_t = T.switch(Bc_t > B_max.value, B_max.value, Bc_t)
Bc_t = T.switch(Bc_t < -1 * B_max.value, -1 * B_max.value, Bc_t)
# Calculate Choice
mdiff_t = (mag_1_t - mag_0_t)
Mag_t = T.exp(mag_baseline + rewpain_t * mag_rewpain)
Mag_t = T.switch(Mag_t < 0.1, 0.1, Mag_t)
Mag_t = T.switch(Mag_t > 10, 10, Mag_t)
mdiff_t = T.sgn(mdiff_t) * T.abs_(mdiff_t)**Mag_t
pdiff_t = (estimate_tm1_A - estimate_tm1_B)
cdiff_t = (choice_kernel_tm1 - (1.0 - choice_kernel_tm1))
choice_val_t = Binv_t * ((1 - Amix_t) * mdiff_t +
(Amix_t) * pdiff_t) + Bc_t * cdiff_t
# before Amix, choice value goes between -1 and 1
prob_choice_t = 1.0 / (1.0 + T.exp(-1.0 * choice_val_t))
# determine eps
eps_t = eps_baseline + \
stabvol_t*eps_stabvol + \
rewpain_t*eps_rewpain + \
stabvol_t*rewpain_t*eps_rewpain_stabvol
eps_t = pm.invlogit(eps_t)
# add epsilon
prob_choice_t = eps_t * 0.5 + (1.0 - eps_t) * prob_choice_t
# Generate choice or Copy participants choice (used for next trial as indicator)
if gen_indicator.value == 0:
choice_t = obs_choice_t
else:
#import pdb; pdb.set_trace()
#trng = T.shared_randomstreams.RandomStreams(1234)
choice_t = trng.binomial(n=1, p=prob_choice_t, dtype='float64')
# this works, but I don't want everyone binomial to have the same seed, so I'd want to update by 1
#rng_val = choice_t.rng.get_value(borrow=True) # Get the rng for rv_u
#rng_val.seed(seed.value) # seeds the generator
#choice_t.rng.set_value(rng_val, borrow=True)
# determine whether current trial is good or bad
outcome_valence_t = choice_t*info_t_A +\
(1.0-choice_t)*(info_t_B) +\
(1.0-choice_t)*info_t_A*(-1.0) + \
(choice_t)*(info_t_B)*(-1.0)
#import pdb; pdb.set_trace()
lr_t = lr_baseline + \
outcome_valence_t*lr_goodbad + \
stabvol_t*lr_stabvol + \
rewpain_t*lr_rewpain + \
outcome_valence_t*stabvol_t*lr_goodbad_stabvol + \
outcome_valence_t*rewpain_t*lr_rewpain_goodbad + \
stabvol_t*rewpain_t*lr_rewpain_stabvol + \
outcome_valence_t*stabvol_t*rewpain_t*lr_rewpain_goodbad_stabvol
#lr_t = pm.invlogit(lr_t)
lr_t = pm.invlogit(lr_t) * 5
# determine decay on current trial
decay_t = decay_baseline + \
stabvol_t*decay_stabvol + \
rewpain_t*decay_rewpain + \
stabvol_t*rewpain_t*decay_rewpain_stabvol
decay_t = pm.invlogit(decay_t)
# update the Beta Parameters
Ga_t = decay_t * Ga_tm1 + lr_t * (choice_t * info_t_A)
Ba_t = decay_t * Ba_tm1 + lr_t * (choice_t * (1 - info_t_A))
Gb_t = decay_t * Gb_tm1 + lr_t * (
(1 - choice_t) * (info_t_B)) # chose B and B rewarded
Bb_t = decay_t * Bb_tm1 + lr_t * (
(1 - choice_t) * (1 - info_t_B)) # chose B and B not rewarded
# update the probability estimates of the shape chosen
estimate_t_A = (Ga_t + 1) / (Ga_t + Ba_t + 2)
estimate_t_B = (Gb_t + 1) / (Gb_t + Bb_t + 2)
# Choice kernel learning rate
lr_c_t = lr_c_baseline + \
outcome_valence_t*lr_c_goodbad + \
stabvol_t*lr_c_stabvol + \
rewpain_t*lr_c_rewpain + \
outcome_valence_t*stabvol_t*lr_c_goodbad_stabvol + \
outcome_valence_t*rewpain_t*lr_c_rewpain_goodbad + \
stabvol_t*rewpain_t*lr_c_rewpain_stabvol + \
outcome_valence_t*stabvol_t*rewpain_t*lr_c_rewpain_goodbad_stabvol
lr_c_t = pm.invlogit(lr_c_t)
choice_kernel_t = choice_kernel_tm1 + lr_c_t * (choice_t -
choice_kernel_tm1)
return ([
choice_t, outcome_valence_t, prob_choice_t, choice_val_t, estimate_t_A,
estimate_t_B, Ga_t, Ba_t, Gb_t, Bb_t, choice_kernel_t, lr_t, lr_c_t,
Amix_t, Binv_t, Bc_t, decay_t, mdiff_t, eps_t
])
import pymc3 as pm
with pm.Model() as pooled_model:
# sample a and b
b0 = pm.Normal('b0', mu=0, sigma=50)
b1 = pm.Lognormal('b1', mu=0, sigma=50)
# compute the purchase probability for each incidence
prob = pm.Deterministic('prob', pm.invlogit(b0 - b1 * price_data['price']))
likelihood = pm.Bernoulli('likelihood', p=prob, observed=price_data['buy'])
posterior = pm.sample(draws=500, tune=500)
# pm.model_to_graphviz(pooled_model)
def wmcap_morey_cowan(data, formulae, scale=5):
r"""Constructs a Bayesian hierarchical model for the estimation of working-
memory capacity, bias, and lapse rate using the method described by Morey
for Cowan-style change detection tasks. This is a "fixed-effects" version
of the model, which is best suited to studies with between-subjects
designs.
Args:
data (pd.DataFrame): Data.
formulae (list): List of patsy-style formulae; e.g.,
`kappa ~ C(subject)`. Accepts up to three formulae for $\kappa$,
$\gamma$, and/or $\zeta$. Missing formulae with default to an
intercept-only model.
scale (float): A scale parameter for all the stochastic nodes. Defaults
to `5`.
Returns:
None: All model components are placed in the context.
"""
# "Compress" the data so we can create the correct amount stochastic nodes
# later on.
data = compress(data, formulae)
# Make a dictionary out of the formulae so they can be indexed easier.
param_names = ['kappa', 'gamma', 'zeta']
dic = {p: '1' for p in param_names}
dic.update(
{f.split('~')[0].replace(' ', ''): f.split('~')[1]
for f in formulae})
# Loop over parameters in the decision model. This is just less code than
# hard-coding each one.
for p in param_names:
# Construct a linear model for the predictions on the parameter.
dm = dmatrix(dic[p], data)
X = np.asarray(dm)
covs = dm.design_info.column_names
beta = []
# Make regression coefficients for the linear model.
for cov in covs:
# Nicely format the name of the coefficient.
c = ''.join(i for i in cov if i not in '"\',$')
name = r'$\beta_{(\%s)_\mathrm{%s}}$' % (p, c)
beta.append(pm.Cauchy(name=name, alpha=0, beta=scale))
# Make the predictions of the linear model on the parameter.
mu = dot(X, tt.stack(*beta))
# Make delta vector of parameter.
name = r'$\delta_{(\%s)}$' % p
dm_ = dm_for_lower_stochastics(data)
X_ = np.asarray(dm_)
delta_ = pm.Normal(name=name, mu=0, sd=1., shape=X_.shape[1])
delta = dot(X_, delta_)
# Make sigma of parameter.
sigma = pm.HalfCauchy(name=r'$\sigma_{(\%s)}$' % p, beta=scale)
# Make the parameter.
dic[p] = pm.Deterministic(r'$\%s$' % p, mu + delta * sigma)
# Transform parameters into meaningful decision parameters.
zeros = np.zeros(len(data))
k = pm.Deterministic('$k$', tt.max((dic['kappa'], zeros), axis=0))
g = pm.Deterministic('$g$', pm.invlogit(dic['gamma']))
z = pm.Deterministic('$z$', pm.invlogit(dic['zeta']))
# Calculate hit and false-alarm probabilities.
q = tt.min((k / data.set_size.values, zeros + 1), axis=0)
f = (1 - z) * g + z * (1 - q) * g
h = (1 - z) * g + z * q + z * (1 - q) * g
# Construct the Likelihoods.
pm.Binomial(name='$H$',
p=h,
n=data.different_trials.values,
observed=data.hits.values)
pm.Binomial(name='$F$',
p=f,
n=data.same_trials.values,
observed=data.false_alarms.values)
