11.2 One-Dimensional Models

The model we first look at is the coin flip model: given 17 heads in 25 flips, what is the probability distribution of the the measure of the coin's bent-ness, $\theta$ . We know the solution is of the form of a beta distribution, but we perform the same analysis with the MCMC technique.

h,N=data=17,25

def P_data(data,theta):
    h,N=data
    distribution=Bernoulli(h,N)
    return distribution(theta)

model=MCMCModel(data,P_data,
                theta=Uniform(0,1))

About a dozen overlaid MCMC chains for θ plotted against time step, 0 to 500. The traces start spread across the whole 0-1 range, then within roughly 25 steps settle into a dense band between about 0.4 and 0.9 centered near 0.7. — **Figure 11.1.** So-called MCMC “chains” for parameter $\theta$ versus time. Observe that the values of $\theta$ start spread evenly from 0 to 1 at the beginning and then thin down to a range of about 0.5-0.8 with the middle around 0.7 (17/25 = 0.68).

model.run_mcmc(500)
model.plot_chains()

Reading the Output

We can now plot the distributions of the parameters, just $\theta$ in this case, yielding best-fits, uncertainties, etc...

model.plot_distributions()

(see Figure 11.2)

Jagged sampled histogram of the posterior for θ (blue dots) with two smooth fitted curves over it, titled with the estimate θ=0.669 (+0.157/-0.187 for the 2.5-97.5 percentile range). The hump peaks near θ=0.67. — **Figure 11.2.** Distribution of $\theta$ , and the 95% credible interval.

We can further perform some simple calculations on the probabilities for the parameters, such as

model.P('theta>0.5')

Show expected output

0.96173333333333333

model.P('(0.2<theta) & (theta<.5)')

Show expected output

0.038266666666666664

Adapted from Statistical Inference for Everyone, by Brian Blais (Bryant University), licensed under CC BY-SA 4.0 (dual-licensed under the GNU FDL 1.2 or later; this adaptation uses the CC BY-SA grant). Changes were made; this adaptation is distributed under the same license. License: CC-BY-SA-4.0.