--- title: Markov Random Fields author: 36-467/36-667 date: 17 November 2020 (Lecture 20) output: slidy_presentation bibliography: locusts.bib --- ```{r, include=FALSE} # General set-up options library(knitr) opts_chunk$set(size="small", background="white", cache=TRUE, autodep=TRUE, tidy=FALSE, warning=FALSE, message=FALSE, echo=FALSE) ``` \[ \newcommand{\Prob}[1]{\mathbb{P}\left( #1 \right)} \newcommand{\Neighbors}{\mathcal{N}} \] ## 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 called the **local characteristic** for site $r$ + People also write $q_{r}(x,y)$ + Can vary with parameters, $p_{r}(x,y;\theta)$ ## 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 $V_r$, $V_{rq}$, the **potentials**, \[ \log{\Prob{X=x}} = \text{(constant)} + \sum_{r}{V_r(x(r))} + \sum_{r}{\sum_{q\in\Neighbors(r)}{V_{rq}(x(r), x(q))}} \] + Getting the constant is hard + See backup - Approach III: Gibbs sampler + Same Gibbs, different idea ## An Example: The Ising Model {.smaller} - Space is 2D square lattice - Each $X(r)$ is binary, say $+1$ or $-1$ - Potentials: \[ V_{rq}(x(r), x(q)) = \theta x(r) x(q) \] + "Homogeneous potentials" - Add up potentials across neighbors: \[ U_r(x|y) = \sum_{q \in \Neighbors(r)}{\theta x y(q)} \] - Local characteristics: \[ p_r(x,y) = \Prob{X(r) = x|X(\Neighbors(r))=y} = \frac{e^{U_r(x|y)}}{e^{U_r(-1|y)} + e^{U_r(+1|y)}} \] + "Autologistic" model ## The Gibbs Sampler Reprise from [Lecture 12](http://www.stat.cmu.edu/~cshalizi/dst/20/lectures/12/lecture-12.html): - 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: + $X(r) \sim \Prob{X(r)|X(\Neighbors(r))}$ + $=$ 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 $X(r)$ - Cycle through all sites at least once, and generally multiple times - Each cycle is one step of a big Markov chain $\Rightarrow$ converges on an invariant _joint_ distribution; that invariant distribution is what we want - See @Kaplan-conclique-Gibbs-sampler for a clever idea for speeding this up by simultaneous updating ## The Gibbs Sampler (actually running) - The Ising model from before, run on a $150\times 150$ lattice, using the `mrf2d` package + 4 independent runs, all with $\theta=1$ ```{r, echo=FALSE} library(mrf2d) # After the example in help(rmrf2d) ising.theta <- array(rep(c(0,-1,-1,0),2), dim=c(2,2,2)) par(pty="s", mfrow=c(2,2)) image(rmrf2d(c(150,150), mrfi(1), ising.theta)) image(rmrf2d(c(150,150), mrfi(1), ising.theta)) image(rmrf2d(c(150,150), mrfi(1), ising.theta)) image(rmrf2d(c(150,150), mrfi(1), ising.theta)) ``` ## Changing dependence strength ```{r, echo=FALSE} par(pty="s", mfrow=c(2,2)) image(rmrf2d(c(150,150), mrfi(1), 0.1*ising.theta), main=expression(theta==0.1)) image(rmrf2d(c(150,150), mrfi(1), 0.5*ising.theta), main=expression(theta==0.5)) image(rmrf2d(c(150,150), mrfi(1), 2*ising.theta), main=expression(theta==2)) image(rmrf2d(c(150,150), mrfi(1), 10*ising.theta), main=expression(theta==10)) ``` ## 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/20/lectures/20/lecture-12.html) make good auxiliaries for indirect inference ## Inference: Likelihood (1) - Full likelihood inference is _hard_ + Need the joint distribution $\Prob{X=x}$ + Which means needing the normalizing factor as a function of the parameters + Often need to do Monte Carlo just to get the likelihood [@Geyer-on-MCMCMLE;@Geyer-Thompson-MCMCMLE] - What are our alternatives which stay close to likelihood? ## Inference: Likelihood vs. Matching Sufficient Statistics - Summary statistics are **sufficient** when we can write the likelihood in terms of them and not the full data - For Markov chains, the sufficient statistics were the counts of pairs of successive states ($N_{ij}$) - For MRFs, the sufficient statistics are counts of clique configurations - E.g., in Ising model, counts of $(-1,-1)$ for (self, neighbor), of $(-1,+1)$, of $(+1,-1)$, of $(+1,+1)$ - We can get these easily from the data - Handy fact about exponential families: the MLE is the value of $\theta$ where the _expected_ value of the sufficient statistics equals the observed value - Pick an initial value of $\theta$; simulate multiple times to get $\approx$ expected sufficient statistics; adjust to align these with the observed values (**stochastic approximation**) ## Inference: Likelihood (3) - **Pseudo-likelihood**: Product of all the local characteristics $p_r$ \[ \prod_{r}{p_r(x(r)|x(\Neighbors(r)))} = \prod_{r}{\Prob{X(r)=x(r)|X(\Neighbors(r))=x(\Neighbors(r))}} \] + $\neq$ the joint probability + Also called a **composite likelihood** (or pseudolikelihood with no hyphen...) + Maximizing this is generally a consistent estimator, but not as statistically efficient as likelihood * See @Geyer-Thompson-MCMCMLE for head-to-head comparisons on the Ising model; may be unsurprising that pseudo-likelihood works better when interactions are weaker ## Inference: Uncertainty - If you can get the likelihood, use the Hessian of the log-likelihood - [Sandwich-covariance methods](http://www.stat.cmu.edu/~cshalizi/dst/20/lectures/14/lecture-14.html) work as usual (if we can calculate gradients and second derivatives) - Parametric bootstrap works as usual - Nonparametric bootstrap for stationary fields + Use rectangular blocks [@Lahiri-resampling-for-dependent] + Or maybe weirder shapes [@Levina-Bickel-resampling-random-fields] ## Adding Time Back In - Now have $X(r,t)$ + Write $X(\cdot, t)$ for the whole configuration at time $t$ - Two choices for how we think about the Markov property: 1. $X(r,t)$ 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 $X(\cdot, t-1)$ 2. 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 - Recursive (first option) 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 + Similar to a vector-valued Markov process, but with more structure to the transition probabilities (because they have to products of local characteristics) + Can estimate $\Prob{X(r,t)|X(\Neighbors(r), t-1), X(r,t-1)}$ the same way as in a Markov chain - Simultaneous (second option) fields are not as nice + Basically, back to spatial models with an extra coordinate + Pseudo-likelihood often the best bet + Simultaneous dependence may be necessary if the time between measurements is slow compared to the time-scale of the actual dynamics ## Cellular Automata {.smaller} - $X(r,t)$ discrete (usually finite), $r$ on a regular lattice, $t$ discrete - Recursive (option 1) 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) - A fun (?) exercise: code up the spatial SIRS model: + Each cell is in one of the 3 states S, I, R + An S can get infected from a _neighboring_ I, with some probability each time step + An I turns in to an R at some rate + Rs turn back in to Ss at some other rate + Tune the parameters right and you see traveling waves of infection, sometimes [quite pretty](http://psoup.math.wisc.edu/archive/recipe2.html): ![](http://psoup.math.wisc.edu/archive/soup2.gif) ## 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 rarely 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 - Recursive spatio-temporal Markov random fields are, despite the name, actually a lot simpler, and a very powerful class of models ## Backup: Gibbs-Markov Theorem - $X=$ the whole random field, the vector of $X(r)$ for all $r$ - $X$ has a **Gibbs distribution** when \[ \Prob{X=x} = \frac{1}{Z}\exp{\left\{-\left(\sum_{A}{V_A(x)}\right)\right\}} \] + $A =$ subsets of sites + $V_A=$ **potential function**, depending only on the state of sites in A + $Z \equiv \sum_{x}{\exp{\left\{-\left(\sum_{A}{V_A(x)}\right)\right\}}} =$ **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 even from the Gibbs form + E.g., read [this](https://www.quantamagazine.org/the-cartoon-picture-of-magnets-that-has-transformed-science-20200624/) for the saga of calculating $Z$ for the 2D and 3D Ising model - Sampling from Gibbs distributions: + Use Gibbs sampler (since it's Markov) + Use [MCMC](http://www.stat.cmu.edu/~cshalizi/statcomp/13/lectures/16/mcmc.pdf), 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 slights everyone equally ## Backup: A Conditional Likelihood for a Fraction of the Data - On a square lattice, color the sites alternately blue and red - Each red site has only blue neighbors - White sites are independent given the blue sites (by Markov property) - Multiply the local characteristics of the red sites to get a conditional likelihood \[ \Prob{X(\text{red})=x(\text{red})|X(\text{blue})=x(\text{blue})} = \prod_{r \in \text{red}}{\Prob{X(r)=x(r)|X(\Neighbors(r)) = x(\Neighbors(r))}} \] - @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