# Plots of prior and posterior distributions for a model

I have an exercise that ask to plot the prior and posterior distribution in the Poisson-Gamma model. I did it (I think it's correct) inspired in the answer of this question https://stats.stackexchange.com/questions/70661/how-does-the-beta-prior-affect-the-posterior-under-a-binomial-likelihood

Is there anything that can improve the code?

colors = c("red","blue","green","orange","purple")

n = 10

lambda = .2

x = rpois(n,lambda)
grid = seq(0,7,.01)

alpha = c(.5,5,1,2,2)
beta = c(.5,1,3,2,5)

plot(grid,grid,type="n",xlim=c(0,5),ylim=c(0,4),xlab="",ylab="Prior Density",
main="Prior Distributions", las=1)
for(i in 1:length(alpha)){
prior = dgamma(grid,shape=alpha[i],rate=1/beta[i])
lines(grid,prior,col=colors[i],lwd=2)
}

legend("topleft", legend=c("Gamma(0.5,0.5)", "Gamma(5,1)", "Gamma(1,3)", "Gamma(2,2)", "Gamma(2,5)"),
lwd=rep(2,5), col=colors, bty="n", ncol=3)

for(i in 1:length(alpha)){
dev.new()
plot(grid,grid,type="n",xlim=c(0,5),ylim=c(0,10),xlab="",ylab="Density",xaxs="i",yaxs="i",
main="Prior and Posterior Distribution")

alpha.star = alpha[i] + sum(x)
beta.star = beta[i] + n
prior = dgamma(grid,shape=alpha[i],rate=1/beta[i])
post = dgamma(grid,shape=alpha.star,rate=beta.star)

lines(grid,post,lwd=2)
lines(grid,prior,col=colors[i],lwd=2)
legend("topright",c("Prior","Posterior"),col=c(colors[i],"black"),lwd=2)

}

• Could I ask why you use 1/beta[i] in the first set of priors and beta[i] in the second set of priors.The first thing I'd do if refactoring your code would be to write a function that does the code that you've written inside the for-loops; then I'd split the data-generation from the data-plotting or plot-appending logic into different functions. If I was rethinking how you present your plots, I'd consider plotting several prior/posterior figures side-by-side, eg, using par(mfrow = c(1, 5)) and add a meaningful x-label – Russ Hyde Jan 25 '19 at 10:26
• I'd also expand the range of the x-axis in your first plot: < 20% of the density is covered for all but the Gamma(0.5, 0.5) distibution – Russ Hyde Jan 25 '19 at 10:32
• I have another query: your choices of alpha and beta jump around quite a bit. The mean of the gamma-dist defined by your alpha/beta pairs varies between 0.3 and 5, and the variance from 0.08 to 5. It might be more illustrative to pick three prior means and two prior variances (say); and then choose alpha/beta pairs that are consistent with the 6 possible combinations of mean/variance – Russ Hyde Jan 25 '19 at 11:10
• @RussHyde ooh, it's a typo, should be 1/beta[i]. With this prior/posterior figures side-by-side you mean one prior in one plane and the posterior in a different plane but next to the prior? – I likeThatMeow Jan 25 '19 at 19:21
• @RussHyde I've change to xlim=c(0,2) now looks complete – I likeThatMeow Jan 25 '19 at 19:22

There was a couple of errors in the code as originally posted. The code was OK as far as base R goes.

First thing I did was run styler and then lintr on your code; these two things help clean up the coding style in your scripts.

That does things like this:

colors = c("red","blue","green","orange","purple")

# changed to (spaces / idiomatic assignment):
colors <- c("red", "blue", "green", "orange", "purple")



Then I changed your 1:length(alphas) to seq_along(alphas). The latter is a bit safer since the former can fail with empty input.

Then I replaced your 5-separate prior/posterior plots with a single plot that contains 5 panels. This makes it easier to compare the appropriateness of the different priors. To do this, I removed your dev.new()s, added a call to par(mfrow = c(number_of_rows, number_of_columns)) and obviously, tidied this up afterwards (returning to a 1x1 grid)

par(mfrow = c(2, ceiling(length(alpha) / 2)))

for (i in seq_along(alpha)) {
# removed dev.new()
... plotting code ...
)
}

par(mfrow = c(1, 1))


Then I cleaned up your experimental data and your prior-parameters / plotting parameters; they were all winding around each other. I also renamed your alpha / beta vectors - in R, these correspond to the shape and rate parameters that are passed into dgamma:

# ---- experimental data

num_observations <- 10

lambda <- .2

x <- rpois(num_observations, lambda)

# ---- prior parameters

# assumed 'beta' was a rate parameter
# - this, since there was confusion in the parameterisation of dgamma():
#   - early section used rate = 1 / beta[i];
#   - later section used rate = beta[i]; and
#   - definition of beta_star = beta[i] + n; implied beta was definitely a rate

shape <- c(.5, 5, 1, 2, 2)
rate <- c(.5, 1, 3, 2, 5)

# ---- plotting parameters

colors <- c("red", "blue", "green", "orange", "purple")

# ---- search parameters

grid <- seq(0, 2, .01)



Then I made a function to do your prior-comparison stuff (the first set of plots). Any parameters that were to be passed through to plot were passed in using the ... argument.

# ---- comparison of the prior distributions

plot_priors <- function(grid, shapes, rates, colors,
legend_text, lwd = 2, ...) {
plot(grid, grid, type = "n", ...)

for (i in seq_along(shape)) {
prior <- dgamma(grid, shape = shape[i], rate = rate[i])
lines(grid, prior, col = colors[i], lwd = lwd)
}

legend(
"topleft",
legend = legend_text, lwd = lwd, col = colors, bty = "n", ncol = 2
)
}



This can be called like:

plot_priors(
grid, shape, rate, colors,
legend_text = paste0("Gamma(", c("0.5,0.5", "5,1", "1,3", "2,2", "2,5"), ")"),
xlim = c(0, 1), ylim = c(0, 4), xlab = "", ylab = "Prior Density",
main = "Prior Distributions", las = 1
)


It's useful to split your computations away from your plotting code - so I extracted the code you used to compute the posterior params:

compute_posterior_parameters <- function(observations,
prior_shape,
prior_rate) {
list(
shape = prior_shape + sum(observations),
rate = prior_rate + length(observations)
)
}


Then I pulled the plotting code for your prior/posterior comparisons into a function (similarly to the above)

plot_prior_post_comparison <- function(
observations,
grid, shapes, rates, colors,
lwd = 2,
...) {
# make a grid for plotting
par(mfrow = c(2, ceiling(length(shapes) / 2)))

for (i in seq_along(shapes)) {
# details of the prior and post distributions
posterior_params <- compute_posterior_parameters(
observations,
prior_shape = shapes[i], prior_rate = rates[i]
)
prior <- dgamma(
grid,
shape = shapes[i],
rate = rates[i]
)
post <- dgamma(
grid,
shape = posterior_params$shape, rate = posterior_params$rate
)

# plotting code
plot(grid, grid, type = "n", ...)
lines(grid, post, lwd = lwd)
lines(grid, prior, col = colors[i], lwd = lwd)
legend("topright",
c("Prior", "Posterior"),
col = c(colors[i], "black"), lwd = lwd
)
}

# revert the plotting grid back to 1x1
par(mfrow = c(1, 1))
}



note that the par calls, which change the plotting grid are all nested inside the function, so any subsequent plots should be unaffected.

Then I called that function:

# ---- prior/posterior comparison

plot_prior_post_comparison(
observations = x,
grid = grid, shapes = shape, rates = rate, colors = colors,
xlim = c(0, 1), ylim = c(0, 10), xlab = "", ylab = "Density",
xaxs = "i", yaxs = "i",
main = "Prior and Posterior Distribution"
)



Then I put all the functions at the start and all the calls at the end of a script:

# The full code:

# ---- comparison of the prior distributions

plot_priors <- function(grid, shapes, rates, colors,
legend_text, lwd = 2, ...) {
plot(grid, grid, type = "n", ...)

for (i in seq_along(shape)) {
prior <- dgamma(grid, shape = shape[i], rate = rate[i])
lines(grid, prior, col = colors[i], lwd = lwd)
}

legend(
"topleft",
legend = legend_text, lwd = lwd, col = colors, bty = "n", ncol = 2
)
}

# ---- prior:posterior analysis

compute_posterior_parameters <- function(observations,
prior_shape,
prior_rate) {
list(
shape = prior_shape + sum(observations),
rate = prior_rate + length(observations)
)
}

plot_prior_post_comparison <- function(
observations,
grid, shapes, rates, colors,
lwd = 2,
...) {
# make a grid for plotting
par(mfrow = c(2, ceiling(length(shapes) / 2)))

for (i in seq_along(shapes)) {
# details of the prior and post distributions
posterior_params <- compute_posterior_parameters(
observations,
prior_shape = shapes[i], prior_rate = rates[i]
)
prior <- dgamma(
grid,
shape = shapes[i],
rate = rates[i]
)
post <- dgamma(
grid,
shape = posterior_params$shape, rate = posterior_params$rate
)

# plotting code
plot(grid, grid, type = "n", ...)
lines(grid, post, lwd = lwd)
lines(grid, prior, col = colors[i], lwd = lwd)
legend("topright",
c("Prior", "Posterior"),
col = c(colors[i], "black"), lwd = lwd
)
}

# revert the plotting grid back to 1x1
par(mfrow = c(1, 1))
}

# ----

# ---- experimental data

num_observations <- 10

lambda <- .2

x <- rpois(num_observations, lambda)

# ---- prior parameters

# assumed 'beta' was a rate parameter
# - this, since there was confusion in the parameterisation of dgamma():
#   - early section used rate = 1 / beta[i];
#   - later section used rate = beta[i]; and
#   - definition of beta_star = beta[i] + n; implied beta was definitely a rate

shape <- c(.5, 5, 1, 2, 2)
rate <- c(.5, 1, 3, 2, 5)

# ---- plotting parameters

colors <- c("red", "blue", "green", "orange", "purple")

# ---- search parameters

grid <- seq(0, 2, .01)

# ---- comparison of priors

plot_priors(
grid, shape, rate, colors,
legend_text = paste0("Gamma(", c("0.5,0.5", "5,1", "1,3", "2,2", "2,5"), ")"),
xlim = c(0, 1), ylim = c(0, 4), xlab = "", ylab = "Prior Density",
main = "Prior Distributions", las = 1
)

# ---- prior/posterior comparison

plot_prior_post_comparison(
observations = x,
grid = grid, shapes = shape, rates = rate, colors = colors,
xlim = c(0, 1), ylim = c(0, 10), xlab = "", ylab = "Density",
xaxs = "i", yaxs = "i",
main = "Prior and Posterior Distribution"
)



I don't thnk its perfect - the plotting and calculation steps are still pretty tied together - but it should be more easy to add extra pairs of rate/shape values into your prior distributions, for example.

One place the code coould be further improved is by passing in a data.frame where each row contains the shape/rate/colour values and any annotations for a given prior. The functions contain too many arguments at the moment, and this would fix that (and guarantee that there is a shape for each rate).

• Thank you so much Russ Hyde. I've tested the full code and the output looks really nice :) . I'll read in detail your full answer later to understand the code. – I likeThatMeow Jan 25 '19 at 20:46
• With this: the plotting and calculation steps are still pretty tied together, you meant the code could be improved? Or you meant the output of the code could be improved ? Or both? – I likeThatMeow Jan 26 '19 at 18:42
• More the code than the output. It's important to think about what bits of the code you would need to change if you wanted to do X. X could be any number of things: i) you might want to sample from a different class of prior; ii) you might want to add an extra example plot or change the parameters for your current examples; iii) you might want to plot using ggplot2 instead of base etc. At present, to do any of these things you'd have to change a few different places in the code, so there's still improvements that could be made – Russ Hyde Jan 27 '19 at 11:57
• I see. For example, What new parameters would be a good idea to use? – I likeThatMeow Jan 27 '19 at 19:11
• Haha, that's for you to decide. I was just giving an example of how the code might evolve... – Russ Hyde Jan 28 '19 at 10:19