How to use the observations to fit an Ising model?

After self-studying for several days, my current guess is:

  • $\theta_{ii} = \log[P(X_{i} = 1)]$

  • $\theta_{ij} = \log[P(X_{i} = 1, X_{j}=1)]$

Or, is it: $\theta_{ij} = \log[P(X_{i} = 1, X_{j}=1)] -\theta_{ii} - \theta_{jj}$ ?

I am just totally confused now.

I will appreciate any suggestion/hint. Thank you.


From measurements over a long time, I estimate:

  • The probability that the $i^{th}$ atom has up spin: $P(X_{i} = 1)$
  • The probability that the $i^{th}$ and $j^{th}$ atoms both have up spin: $P(X_{i} = 1, X_{j} = 1)$


Suppose the random variables $X_{i}$ follow (possibly correlated) Bernoulli distributions.

All parameters of the model are $\Theta$

The probability for an observation $\vec{x}$ to occurr is: $$\log[P(\vec{X} = \vec{x}; \Theta)] = H(\vec{x}; \Theta) - \log[Z(\Theta)]$$


$$H(\vec{x}; \Theta) = \sum_{i} \theta_{ii} x_{i} + \sum_{i < j} \theta_{ij}x_{i}x_{j}$$

$$Z(\Theta) = \sum_{\vec{y}}\exp(H(\vec{y})$$

  • $\begingroup$ I am not sure what is the ground state. I hope the normalization constant $Z(\Theta)$ would fix that. All I want is the probability distribution of the observations. $\endgroup$ – hamster on wheels Dec 30 '18 at 22:49

If I understood your question, you are asking about solutions to the inverse Ising problem. In other words, given a set of measurements of the average magnetization at each site (or, equivalently, the probability of the spin having a value $+1$), and the average correlation between spins (or joint probability function) at each pair of sites, you would like to determine the most reliable estimates of the applied magnetic field (at each site) and the coupling coefficients (between each pair of sites) that defined the original model.

This is a nontrivial problem, and the answer is certainly not given in general by explicit equations of the kind you quote at the start of your question. However, a set of implicit equations which constitute a formal route to the solution has been proposed, see e.g. Albert and Swendsen Physics Procedia, 57, 99 (2014) (open access), and there is also a set of slides by Ricci-Tersenghi which outlines suitable numerical brute-force and approximate methods.

Searching for inverse Ising problem on the web gives further information on the topic.


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