1. ERGM conditional form
Much easier to calculate delta (change statistics)

2. Metropolis algorithm
Generates a random sample from a given distribution (any distribution!)
Given a current state of the network, randomly propose a new state, then randomly accept the change or stay put.
Probability of acceptance is:
If proposal is a single edge toggle, then

3. MCMC diagnostics
What’s the difference between an MCMC sample and a random i.i.d. sample?
Burn-in, autocorrelation, mixing
Effective sample size
“these diagnostics only find obvious, gross, embarrassing problems that jump out of simple plots.”
Increasing "mcmc.interval" will decrease autocorrelation and solve most of your problems. I usually use 5x the number of edges as the interval.

4. MCMC diagnostics
Good:
Bad:

5. Markov chain maximum likelihood estimation
Want to maximize the likelihood function with respect to theta
Equivalent to maximizing the log-likelihood function:
Equivalent to maximizing the log-likelihood minus a constant. We’re free to choose l(θ0) as the constant

6. Approximating the log-likelihood
Augment it with an arbitrary θ0
Approximate this using an MCMC sample
We can now maximize this w.r.t. θ

7. Approximating the normalizing constant
Algebra happens...
Approximate the expectation with a sample average, where the sample uses the known parameter θ0

8. Approximating the normalizing constant
The algebra:
pull the summation out
multiply top and bottom by the same thing
to get prob given theta0
write sum as expectation

9. MCMLE algorithm Initialize θ0 with some value Repeat until converge:
Get MCMC sample of m networks with θ0
Maximize this function w.r.t. θ
Update θ0 to the maximized θ

10. MCMLE algorithm How do we initialize θ0 ?
How big of an MCMC sample do we need?
How can we tell when it converges?
Since we are using an approximate likelihood, how does that affect our standard error?