Getting Started with MCMC

Outcomes as distributions

• Average profit?
• How likely to make more than $90k?
• How long is runway?

Outcomes as distributions

Outcomes as distributions

Outcomes as distributions

Expectations

$$ \mathbb{E}_p[f] = \int f(x) p(x)~dx $$

Expectations

$$ \mathbb{E}_p[f] = \int f(x) p(x)~dx \approx \frac{1}{k}\sum_{j=1}^k f(x_j) $$ where $$ x_j \text{ is distributed like } p $$

Do the math

$$ u \sim \operatorname{Uniform}(0, 1) $$ $$ \text{solve } \int_{-\infty}^x p(t)~dt = u $$

Do some simulations

Do some simulations

Do some simulations

MCMC!

MCMC!

def metropolis_hastings(n_samples, pdf):
    position = 0.
    samples = []
    for _ in range(n_samples):
	proposal = np.random.normal(loc=position)
        accept_prob = pdf(proposal) / pdf(position)
        if np.random.rand() < accept_prob:
            position = proposal
        samples.append(position)
    return samples
				

Traceplot Quiz!

Traceplot Quiz!

Traceplot Quiz!

Traceplot Quiz!

Traceplot Quiz!

Traceplot Quiz!

Traceplot Quiz!

Traceplot Quiz!

Traceplot Quiz!

Traceplot Quiz!

Traceplot Quiz!

Traceplot Quiz!

Traceplot Quiz!

Traceplot Quiz!

Traceplot Quiz!

Traceplot Quiz!

Traceplot Quiz!

Traceplot Quiz!

Traceplot Quiz!

Traceplot Quiz!

Traceplot Quiz!

Traceplot Quiz!

Traceplot Quiz!

Traceplot Quiz!

Traceplot Quiz!

Traceplot Quiz!

Traceplot Quiz!

Traceplot Quiz!

Traceplot Quiz!

Traceplot Quiz!

Traceplot Quiz!

Traceplot Quiz!

Traceplot Quiz!

Traceplot Quiz!

Traceplot Quiz!

Traceplot Quiz!

Traceplot Quiz!

Traceplot Quiz!

Traceplot Quiz!

Traceplot Quiz!

Traceplot Quiz!

Traceplot Quiz!

Traceplot Quiz!

Traceplot Quiz!

Traceplot Quiz!

Traceplot Quiz!

Traceplot Quiz!

Traceplot Quiz!

Traceplot Quiz!

Traceplot Quiz!

Traceplot Quiz!

Traceplot Quiz!

Traceplot Quiz!

Traceplot Quiz!

Traceplot Quiz!

Traceplot Quiz!

Traceplot Quiz!

Traceplot Quiz!

Traceplot Quiz!

Traceplot Quiz!

Traceplot Quiz!

Traceplot Quiz!

Traceplot Quiz!

Traceplot Quiz!

Traceplot Quiz!

Traceplot Quiz!

Traceplot Quiz!

Traceplot Quiz!

Traceplot Quiz!

Traceplot Quiz!

Traceplot Quiz!

Traceplot Quiz!

Real libraries

import arviz as az
import pymc3 as pm

with pm.Model():
    x = pm.Normal('x')
    trace = pm.sample(step=pm.Metropolis())

az.plot_trace(trace)