--- title: Markov Random Fields author: 36-467/36-667 output: slidy_presentation bibliography: locusts.bib --- \[ \newcommand{\Prob}[1]{\mathbb{P}\left( #1 \right)} \newcommand{\Neighbors}{\mathcal{N}} \] ## In our previous episodes - Markov processes in time: $\Prob{X(t+1)|X(0), X(1), \ldots X(t)} = \Prob{X(t+1)|X(t)}$ + Markov chains: $X(t)$ is discrete-valued - Asymptotic behavior of Markov chains - Likelihood-based inference for Markov models - What about space? What about space and time? ## Markov Random Fields - $X(r) =$ value of the random field at site $r$ - $X$ or $X(\cdot) =$ value of the field at all sites, the **configuration** - $X(-r) =$ value of the random field everywhere other than $r$ - $X(\Neighbors(r)) =$ value of the random field at the neighbors of $r$ + the **neighborhood configuration** - $X$ is a **Markov random field** when \[ \Prob{X(r)|X(-r)} = \Prob{X(r)|X(\Neighbors(r))} \] - $p_r(x, y) \equiv \Prob{X(r)=x|X(\Neighbors(r))=y}$ is the **local characteristic** for site $r$ + People also write $q_{r}(x,y)$ + Can vary with parameters, $q_{r}(x,y;\theta)$ ## What Does a Markov Random Field Look Like? - Working out joint $\Prob{X}$ from $\Prob{X(r)|X(\Neighbors(r))}$ is not easy! + No natural order to use in factorizing - Approach I: Heroic mathematics + Truly heroic in some cases... - Approach II: Gibbs-Markov equivalence + Upshot: For the right functions $V_r$, $V_{rq}$, \[ \log{\Prob{X=x}} = \text{(constant)} + \sum_{r}{V_r(x(r))} + \sum_{r}{\sum_{q\in\Neighbors(r)}{V_{rq}(x(r), x(q))}} \] + Getting the constant is hard + See backup - Approach III: Gibbs sampler + Same Gibbs, different idea ## The Gibbs Sampler (reprise) - Assumes space $r$ is discrete, but state $X(r)$ can be continuous - Start with _some_ initial value of $X(r)$ for all $r$ - Fix an order on the sites $r$, and then: + $X(r) \sim \Prob{X(r)|X(\Neighbors(r)) = X(r)}$ + $=$ draw a new value for $X(r)$ from the conditional distribution / local characteristic and replace the old value of $X(r)$ + N.B., later sites get conditioned on the updated value for $X(r)$ - Sweep through all sites at least once, and generally a multiple times - Each sweep is one step of a big Markov chain $\Rightarrow$ converges on an invariant _joint_ distribution - See @Kaplan-conclique-Gibbs-sampler for a clever idea for speeding this up by simultaneous updating ## Inference: Basics - If we have multiple, independent copies of the field + Can find $p_r(x,y)$ for each $r$ + Use conditional density estimation or even regression - If $p_r(x,y)$ is the same for all $r$ + Could do conditional density estimation or even regression + Usual error statistics aren't valid because of dependence - Can also do simulation-based inference + [Conditional auto-regressions](http://www.stat.cmu.edu/~cshalizi/dst/18/lectures/13/lecture-13.html#(11)) make good auxiliaries for indirect inference ## Inference: Likelihood - Full likelihood inference is hard + Need the joint distribution $\Prob{X=x}$ + Often need to do Monte Carlo just to get the likelihood - **Pseudolikelihood**: Product of all the local characteristics $p_r$ \[ \prod_{r}{\Prob{X(r)=x(r)|X(\Neighbors(r))=x(\Neighbors(r))}} \] + $\neq$ the joint probability + Also called a **composite likelihood** + Generally, consistent, but not as statistically efficient as likelihood ## Inference: Uncertainty - If you can get the likelihood, use the Hessian of the log-likelihood - Simulation-based methods work as usual - Parametric bootstrap works as usual - Nonparametric bootstrap for stationary fields + Use rectangular blocks ## Adding Time Back In - Now have $X(r,t)$ + Write $X(\cdot, t)$ for the whole configuration at time $t$ - Two choices for Markov property - Option I: $X(r,t)$ should be conditioned on $X(\Neighbors(r), t-1)$ and $X(r,t-1)$ + So $X(r,t)$ and $X(q,t)$ are independent given $X(\cdot, t-1)$ - Option II: Condition on $X(\Neighbors(r), t-1)$, $X(r,t-1)$ _and_ $X(\Neighbors(r), t)$ + Neighbors at time $t$ are still dependent given $X(\cdot, t-1)$ - Sometimes called **recursive** vs. **simultaneous** ## Spatio-temporal Markov Random Fields - Option I (recursive) fields are _much easier_ + $\Prob{X(\cdot, t)|X(\cdot, t-1)} = \prod_{r}{\Prob{X(r,t)|X(\Neighbors(r), t-1), X(r,t-1)}}$ + Can actually calculate the likelihood + Can estimate $\Prob{X(r,t)|X(\Neighbors(r), t-1), X(r,t-1)}$ the same way as in a Markov chain - Option II (simultaneous) fields are not as nice + Basically, back to spatial models with an extra coordinate + Composite likelihood often the best bet ## Cellular Automata - $X(r,t)$ discrete (usually finite), $r$ on a regular lattice, $t$ discrete - Recursive (option I) spatio-temporal Markov model + Sometimes conditionally deterministic - These are _very_ expressive models + Some examples: [I](http://www.stat.cmu.edu/~cshalizi/462/lectures/10/10.pdf), [II](http://www.stat.cmu.edu/~cshalizi/462/lectures/11/11.pdf) ## Summary - In a (spatial) Markov random field, $X(r)$ is "screened off" from the rest of the field by its neighbors + Conditional distribution of $X(r)$ given neighbors = local characteristic of site $r$ - Local characteristics determine the joint distribution, but actually _solving_ for the joint distribution is hard + Gibbs-Markov equivalence gives a general result, but not always an easy thing to calculate with - Estimation via + Local conditional model (under homogeneity or multiple realizations) + Likelihood (if you can calculate it) + Likelihood-ish objective functions + Simulation-based inference ## Backup: Gibbs-Markov Theorem - $X=$ the whole random field, the vector of $X(r)$ for all $r$ - $X$ has a **Gibbs distribution** when \[ \Prob{X=x} = \frac{1}{Z}\exp{-\left(\sum_{A}{V_A(x)}\right)} \] + $A =$ subsets of sites + $V_A=$ **potential function**, depending only on the state of sites in A + $Z \equiv \sum_{x}{exp{-\left(\sum_{A}{V_A(x)}\right)}} =$ **partition function** - A set of sites is a **clique** if every site in $A$ is a neighbor of every other site - A potential is a **nearest-neighbor potential** if $V_A = 0$ whenever $A$ isn't a clique ## Backup: Gibbs-Markov Theorem - On a finite set of sites, if $\Prob{X=x}> 0$ for all configurations $x$, then $X$ is a Markov random field if and only if $X$ has a Gibbs distribution with a nearest-neighbor potential - Gibbs $\Rightarrow$ Markov: direct calculation - Markov $\Rightarrow$ Gibbs: complicated! + Our readings from Guttorp sketch a proof due to @Griffeath-on-random-fields - $Z$ is hard to calculate either way - Sampling from Gibbs distributions: + Use Gibbs sampler (since it's Markov) + Use MCMC, since that doesn't need $Z$ ## Backup: Gibbs-Markov Theorem - Statisticians usually call this the "Hammersley-Clifford" theorem - It was proved simultaneously and independent by H&C, Besag, Griffeath, Grimmett, ... - Priority is a mess, and Griffeath was one of my thesis advisers... - "Gibbs-Markov" expresses the content better, and is equally slighting to everyone still alive ## Backup: A Conditional Likelihood for a Fraction of the Data - On a square lattice, color the sites alternately black and white - Each white site has only black neighbors - White sites are independent given the black sites (by Markov property) - Multiply the local characteristics of the white sites to get a conditional likelihood \[ \Prob{X(\text{white})=x(\text{white})|X(\text{black})=x(\text{black})} = \prod_{r \in \text{white}}{\Prob{X(r)=x(r)|X(\Neighbors(r)) = x(\Neighbors(r))}} \] - @Bartlett-spatial-pattern, pp. 27--28, calls this "coding" + $=$ the "concliques" of @Kaplan-conclique-Gibbs-sampler + Adapts to other geometries (see Bartlett again) + Efficiency compared to pseudolikelihood is unclear ## References