Graph of hmc search path

In Michael Betancourt’s youtube video, he showed examples of plotting out the search path when running stan. How was he able to show a 2D plot like that. Was that only for a particular test case where the space is completely known, and kept to 2 dimensions on purpose? Is it meaningful to try to observe my own model where the dimension is high, maybe by focusing on one parameter at a time. But how would I figure out the range of valid parameter values to plot out the boundary?

I don’t think there’s any easy way to get the states of a Hamiltonian trajectory from any of our interfaces in Stan. The algorithm’s very easy to write by hand, though. Michael uses gnuplot, so who knows what he’s doing there—you can do the same thing in ggplot in R using gganimate. Here’s a .gif I made:

ham-sim-stepsize-just-right|480x480

The R code, which includes a complete Hamiltonian simulator, is pretty trivial. It uses a generated Stan model for the implementation of the bivariate normal. I think the animation packages changed and I never could get gganimate installed properly to call the external package, so recently I’ve just been doing it all with ImageMagick from teh command line.

``````# thanks to:
# http://stackoverflow.com/questions/26106086/2d-normal-pdf-with-ggplot-and-r
# http://stackoverflow.com/questions/13487625/overlaying-two-graphs-using-ggplot2-in-r

library(ggplot2);
# library(grid);      # units
# library(reshape2);  # melt
library(rstan);
library(scales);    # mute color

bivariate_normal_code <-
"
data {
real<lower=-1,upper=1> rho;
}
transformed data {
vector[2] mu;
cov_matrix[2] Sigma;
mu[1] <- 0;  mu[2] <- 0;
Sigma[1,1] <- 1;      Sigma[1,2] <- rho;
Sigma[2,1] <- rho;    Sigma[2,2] <- 1;
}
parameters {
vector[2] y;
}
model {
y ~ multi_normal(mu, Sigma);
}
";

fit <- stan(model_code = bivariate_normal_code, data = list(rho=0.95));

log_binorm <- function(y) {
return( log_prob(fit, upars=y) );
}

}

N <- 201;
H <- 5;
L <- -H;
pts <- seq(L, H, length.out=N);
density <- rep(NA, N * N);
y1 <- rep(NA, N * N);
y2 <- rep(NA, N * N);
pos <- 1;
for (m in 1:N) {
for (n in 1:N) {
density[pos] <- sqrt(-log_binorm(c(pts[m], pts[n])));
y1[pos] <- pts[m];
y2[pos] <- pts[n];
pos <- pos + 1;
}
}

lp_df <- data.frame(density,y1,y2);

# use geom_raster to mimic image

q1 <- c();
q2 <- c();
p1 <- c();
p2 <- c();
H <- c();
epsilon = 0.005;
q <- c(-1.5, -2);
p <- c(-2, -1);
#epsilon = 0.01;
#q <- c(-2.5, -2.75);
#p <- c( 0.25, -0.25);
n_leapfrog <- 800;
for (l in 1:n_leapfrog) {
p <- p - (epsilon / 2) * grad;
q <- q + epsilon * p;
p <- p - (epsilon / 2) * grad;

q1[l] <- q[1];
q2[l] <- q[2];
p1[l] <- p[1];
p2[l] <- p[2];
H[l] <- -log_binorm(q) + sum(p^2)
}
T <- 1:n_leapfrog
qp_df <- data.frame(density=c(rep(0,n_leapfrog*2)), # dummy
y1=c(q1,p1), y2=c(q2,p2), time=c(T,T),
H = c(H,H),
params=c(rep("position",n_leapfrog),
rep("momentum",n_leapfrog)));

# don't really need both unless you also want to plot momentum,
# but I never got around to that because it puts it on the
# density background, which is misleading (but faceting makes it
# easy to see)

q_df <- data.frame(density=rep(0,n_leapfrog),
y1=q1, y2=q2, time=T, H=H, params=rep("position",n_leapfrog));
p_df <- data.frame(density=rep(0,n_leapfrog),
y1=p1, y2=p2, time=T, H=H, params=rep("momentum",n_leapfrog));
qp_df <- rbind(q_df, p_df);

# just code to play around with plotting the entire Hamiltonian
gg <- ggplot(lp_df, aes(x=y1, y=y2, fill=density));
gg <- gg + geom_raster();
gg <- gg + coord_equal();
# natural scale (no sqrt(-x))
#gg <- gg + scale_fill_gradient2(low = "lightyellow",  mid="yellow", high="red",
#                                midpoint=-10, space="Lab", limits=c(-40,0), na.value="lightyellow");
gg <- gg + scale_fill_gradient2(low = "red",  mid="yellow", high="gray90",
space="Lab", midpoint=4, limits=c(0,8), na.value="gray90",
guide=guide_colourbar(reverse=TRUE, label=FALSE));
gg <- gg + scale_x_continuous(expand = c(0, 0), breaks=c(-4,-2,0,2,4));
gg <- gg + scale_y_continuous(expand = c(0, 0), breaks=c(-4,-2,0,2,4));
gg <- gg + geom_vline(xintercept=0, colour="gray50", alpha=0.2);
gg <- gg + geom_hline(yintercept=0, colour="gray50", alpha=0.2);
gg <- gg + labs(x="y1", y="y2")
gg <- gg + ggtitle("HAMILTONIAN SIMULATION
bivariate normal (rho = 0.95, sigma=1)
init position: (-1.5, -2)   init momentum: (-2, -1)  stepsize: 0.05
");
gg <- gg + theme(legend.position="none");
# use just q_df for just the position
gg <- gg + geom_point(data=q_df, aes(x=y1, y=y2, colour=time), size=1);
gg <- gg + scale_colour_gradient(low="gray50",high="black", guide=FALSE);
gg <- gg + facet_grid(. ~ params, margins=10);
gg;

# plot Hamiltonian
gg_H <- ggplot(head(q_df, n=n_leapfrog), aes(x=T, y=H));
gg_H <- gg_H + geom_point();
gg_H

draw_curve <- function(cutoff) {
gg <- ggplot(lp_df, aes(x=y1, y=y2, fill=density));
gg <- gg + geom_raster();
gg <- gg + coord_equal();
gg <- gg + scale_fill_gradient2(low = "red",  mid="yellow", high="gray90",
space="Lab", midpoint=4, limits=c(0,8), na.value="gray90",
guide=guide_colourbar(reverse=TRUE, label=FALSE));
gg <- gg + scale_x_continuous(expand = c(0, 0), breaks=c(-4,-2,0,2,4));
gg <- gg + scale_y_continuous(expand = c(0, 0), breaks=c(-4,-2,0,2,4));
gg <- gg + geom_vline(xintercept=0, colour="gray50", alpha=0.2);
gg <- gg + geom_hline(yintercept=0, colour="gray50", alpha=0.2);
gg <- gg + labs(x="y1", y="y2")
gg <- gg + ggtitle("HAMILTONIAN SIMULATION
bivariate normal (rho = 0.95, sigma=1)
init position: (-1.5, -2)   init momentum: (-2, -1)  stepsize: 0.005
");
gg <- gg + theme(legend.position="none");
gg <- gg + geom_point(data=head(q_df, n=cutoff), aes(x=y1, y=y2, colour=time), size=1);
gg <- gg + scale_colour_gradient(low="gray50",high="black", guide=FALSE);
gg <- gg + facet_grid(. ~ params, margins=10);
plot(gg);
}

trace_animate <- function() {
lapply(seq(1,n_leapfrog,20), function(i) {
draw_curve(i)
})
}

library(animation);
saveGIF(trace_animate(), interval = .2, movie.name="ham-sim-stepsize-too-small.gif")
``````
3 Likes

I would love to get this going. This code as it is fails with

Error: Unsupported index type: NULL

I thought it might be the undefined i in

`draw_curve(i)`

I defined i = 1000, but that did not help. tracing it takes me deep into the library with a large number of arguments. Is this something you have seen before?

No idea. But I’d first check to make sure one plot works, then worry about the animation.

got it. both calls to facet_grid created problems.

gg <- gg + facet_grid(. ~ params, margins=10);

1 Like

A couple of things. First the background density plot is not tied to the model output. It is simply a result of the scale_fill_gradient2 call.

Second, I wonder if I can simply have stan dump out sequential values of y1 and y2 and plot them, instead of simulating leapfrog in the plot code. I may miss the rejected samples, but I should be able to get all the accepted data points?

@bhomass: Stan doesn’t output the trajectories, only computes the trajectory, and then samples one location from it. The output of that is basically the traceplot but it doesn’t show you the HMC dynamics, only the MCMC dynamics.

would it be too crazy for me to build from the source and add my own print statements?

I think that’s what @betanalpha did. I think he’s looking at ways to somehow expose some of this, but we can’t export it by default as it’s huge when you have many leapfrog steps per iteration.