O.rka - 9 months ago 161

Python Question

I've been trying to implement **Bayesian Linear Regression** models using

`PyMC3`

`sklearn.datasets`

`load_diabetes()`

`(442, 10)`

`442 samples`

`10 attributes`

I believe I got the model working, the posteriors look decent enough to try and predict with to figure out how this stuff works but...I realized I have no idea how to predict with these Bayesian Models! I'm trying to avoid using the

`glm`

`patsy`

I tried following:

Generating predictions from inferred parameters in pymc3

and also http://pymc-devs.github.io/pymc3/posterior_predictive/ but my model is either extremely terrible at predicting or I'm doing it wrong.

If I actually am doing the prediction correctly (which I'm probably not) then can anyone help me

`mean squared error`

`absolute error`

`X_te`

`import pymc3 as pm`

import numpy as np

import pandas as pd

import matplotlib.pyplot as plt

import seaborn as sns; sns.set()

from scipy import stats, optimize

from sklearn.datasets import load_diabetes

from sklearn.cross_validation import train_test_split

from theano import shared

np.random.seed(9)

%matplotlib inline

#Load the Data

diabetes_data = load_diabetes()

X, y_ = diabetes_data.data, diabetes_data.target

#Split Data

X_tr, X_te, y_tr, y_te = train_test_split(X,y_,test_size=0.25, random_state=0)

#Shapes

X.shape, y_.shape, X_tr.shape, X_te.shape

#((442, 10), (442,), (331, 10), (111, 10))

#Preprocess data for Modeling

shA_X = shared(X_tr)

#Generate Model

linear_model = pm.Model()

with linear_model:

# Priors for unknown model parameters

alpha = pm.Normal("alpha", mu=0,sd=10)

betas = pm.Normal("betas", mu=0,#X_tr.mean(),

sd=10,

shape=X.shape[1])

sigma = pm.HalfNormal("sigma", sd=1)

# Expected value of outcome

mu = alpha + np.array([betas[j]*shA_X[:,j] for j in range(X.shape[1])]).sum()

# Likelihood (sampling distribution of observations)

likelihood = pm.Normal("likelihood", mu=mu, sd=sigma, observed=y_tr)

# Obtain starting values via Maximum A Posteriori Estimate

map_estimate = pm.find_MAP(model=linear_model, fmin=optimize.fmin_powell)

# Instantiate Sampler

step = pm.NUTS(scaling=map_estimate)

# MCMC

trace = pm.sample(1000, step, start=map_estimate, progressbar=True, njobs=1)

#Traceplot

pm.traceplot(trace)

`# Prediction`

shA_X.set_value(X_te)

ppc = pm.sample_ppc(trace, model=linear_model, samples=1000)

#What's the shape of this?

list(ppc.items())[0][1].shape #(1000, 111) it looks like 1000 posterior samples for the 111 test samples (X_te) I gave it

#Looks like I need to transpose it to get `X_te` samples on rows and posterior distribution samples on cols

for idx in [0,1,2,3,4,5]:

predicted_yi = list(ppc.items())[0][1].T[idx].mean()

actual_yi = y_te[idx]

print(predicted_yi, actual_yi)

# 158.646772735 321.0

# 160.054730647 215.0

# 149.457889418 127.0

# 139.875149489 64.0

# 146.75090354 175.0

# 156.124314452 275.0

Answer

I think one of the problems with your model is that your data has very different scales, you have ~0.3 range for your "Xs" and ~300 for your "Ys". Hence you should expect larger slopes (and sigma) that your priors are specifying. One logical option is to adjust your priors, like in the following example.

```
#Generate Model
linear_model = pm.Model()
with linear_model:
# Priors for unknown model parameters
alpha = pm.Normal("alpha", mu=y_tr.mean(),sd=10)
betas = pm.Normal("betas", mu=0, sd=1000, shape=X.shape[1])
sigma = pm.HalfNormal("sigma", sd=100) # you could also try with a HalfCauchy that has longer/fatter tails
mu = alpha + pm.dot(betas, X_tr.T)
likelihood = pm.Normal("likelihood", mu=mu, sd=sigma, observed=y_tr)
step = pm.NUTS()
trace = pm.sample(1000, step)
chain = trace[100:]
pm.traceplot(chain);
```

Posterior predictive checks, shows that you have a more or less reasonable model.

```
sns.kdeplot(y_tr, alpha=0.5, lw=4, c='b')
for i in range(100):
sns.kdeplot(ppc['likelihood'][i], alpha=0.1, c='g')
```

Another option will be to put the data in the same scale by standardizing it, doing so you will get that the slope should be around +-1 and in general you can use the same diffuse prior for any data (something useful unless you have informative priors you can use). In fact, many people recommend this practice for Generalized linear models. Your can read more about this in the book doing bayesian data analysis or Statistical Rethinking

If you want to predict values you have several options one is to use the mean of the inferred parameters, like:

```
alpha_pred = chain['alpha'].mean()
betas_pred = chain['betas'].mean(axis=0)
y_pred = alpha_pred + np.dot(betas_pred, X_tr.T)
```

Another option is to use `pm.sample_ppc`

to get samples of predicted values that take into account the uncertainty in you estimates.

The main idea of doing PPC is to compare the predicted values against your data to check where they both agree and where they don't. This information can be used for example to improve the model. Doing

`pm.sample_ppc(trace, model=linear_model, samples=100)`

Will give you 100 samples each one with 331 predicted observation (since in your example `y_tr`

has length 331). Hence you can compare each predicted data point with a sample of size 100 taken from the posterior. You get a distribution of predicted values because the posterior is itself a distribution of possible parameters (the distribution reflects the uncertainty).
Regarding the arguments of `sample_ppc`

: `samples`

specify how many points from the posterior you get, each point is vector of parameters.
`size`

specifies how many times you use that vector of parameters to sample predicted values (by default `size=1`

).

You have more examples of using `sample_ppc`

in this tutorial