0.1 About

In this topic we discuss how to do multivariate predictions! I recommend reading btopic102 first, as this explains the data and the model. I recommend reading btopic112 first, as this explains the univariate predictions.

1 Initialise R session

We load any packages and set global options. You may need to install these libraries (Installation and general troubleshooting).

library(INLA)
rm(list=ls())
options(width=70, digits=2)
set.seed(2017)

2 Load data, rename, rescale

data(Seeds)
df = data.frame(y = Seeds$r, Ntrials = Seeds$n, Seeds[, 3:5])

2.1 Remove out hold out set

If you want to predict to new locations, you do not have to do this step. Instead you define rows for the “locations” (covariates) where you want to hold out points.

holdout = c(7, 12)
# - alternatively: sort(sample(1:nrow(df), 2))
df.holdout = df
df.holdout$y[holdout] = NA

This entire example code is what you need to do in a for-loop for a complete L2OCV-LPD (Leave-2-Out Log Predictive Density). L2OCV-LPD is useful for model comparison.

3 Model

3.1 Observation Likelihood

family1 = "binomial"
control.family1 = list(control.link=list(model="logit"))
# number of trials is df$Ntrials

3.2 Formula

hyper1 = list(theta = list(prior="pc.prec", param=c(1,0.01)))
formula1 = y ~ x1 + x2 + f(plate, model="iid", hyper=hyper1)

4 Call INLA

res1 = inla(formula=formula1, data=df.holdout, 
            family=family1, Ntrials=Ntrials, 
            control.family=control.family1,
            control.predictor=list(compute=T),
            control.compute=list(config=T))

4.1 Look at results

summary(res1)
## 
## Call:
##    c("inla.core(formula = formula, family = family, contrasts = 
##    contrasts, ", " data = data, quantiles = quantiles, E = E, 
##    offset = offset, ", " scale = scale, weights = weights, 
##    Ntrials = Ntrials, strata = strata, ", " lp.scale = lp.scale, 
##    link.covariates = link.covariates, verbose = verbose, ", " 
##    lincomb = lincomb, selection = selection, control.compute = 
##    control.compute, ", " control.predictor = control.predictor, 
##    control.family = control.family, ", " control.inla = 
##    control.inla, control.fixed = control.fixed, ", " control.mode 
##    = control.mode, control.expert = control.expert, ", " 
##    control.hazard = control.hazard, control.lincomb = 
##    control.lincomb, ", " control.update = control.update, 
##    control.lp.scale = control.lp.scale, ", " control.pardiso = 
##    control.pardiso, only.hyperparam = only.hyperparam, ", " 
##    inla.call = inla.call, inla.arg = inla.arg, num.threads = 
##    num.threads, ", " blas.num.threads = blas.num.threads, keep = 
##    keep, working.directory = working.directory, ", " silent = 
##    silent, inla.mode = inla.mode, safe = FALSE, debug = debug, ", 
##    " .parent.frame = .parent.frame)") 
## Time used:
##     Pre = 2.95, Running = 0.476, Post = 0.052, Total = 3.48 
## Fixed effects:
##              mean   sd 0.025quant 0.5quant 0.97quant  mode kld
## (Intercept) -0.43 0.19      -0.80    -0.43    -0.061 -0.43   0
## x1          -0.40 0.25      -0.93    -0.39     0.055 -0.37   0
## x2           1.05 0.24       0.57     1.04     1.511  1.04   0
## 
## Random effects:
##   Name     Model
##     plate IID model
## 
## Model hyperparameters:
##                      mean    sd 0.025quant 0.5quant 0.97quant mode
## Precision for plate 20.63 41.43       2.70     9.96     86.66 5.59
## 
## Marginal log-Likelihood:  -62.59 
##  is computed 
## Posterior summaries for the linear predictor and the fitted values are computed
## (Posterior marginals needs also 'control.compute=list(return.marginals.predictor=TRUE)')

5 Draw posterior samples

n.samples = 1000
samples = inla.posterior.sample(n.samples, result = res1)

This function draws posterior samples from the linear predictor and all its components. The samples from the hyper-parameters are only samples from the integration grid (which represents the only hyper-parameter values to compute the posterior of all the latent variables).

6 Prediction

We want to compute a prediction where we have several datapoints.

df[holdout, ]
##     y Ntrials x1 x2 plate
## 7  53      74  0  1     7
## 12  8      16  1  0    12

6.1 Using the linear predictor

## See one example value
samples[[1]]$latent[holdout, , drop=F]
##               [,1]
## Predictor:7   0.17
## Predictor:12 -2.13
## Draw this part of every sample
samp = lapply(samples, function(x) x$latent[holdout])
samp = matrix(unlist(samp), ncol = 2, byrow = T)

Plot a density of this sample. You could also do a histogram with truehist.

plot(density(samp[, 1]))
lines(density(samp[, 2]))

6.2 Transforming the predictor samples to data-value samples

We now need to transform through the link function and the data sampling. We know we use the logit transform, but we can doublecheck this:

res1$.args$control.family$control.link$model
## NULL
samp.link = inla.link.invlogit(samp)

You can set up the logit function yourself, or use inla.link.invlogit(x). We transform the samples through the link.

plot(density(samp.link[, 1]))
## Add the naive estimate of binomial probability y/N:
abline(v = df$y[holdout[1]]/df$Ntrials[holdout[1]], col="blue")

plot(density(samp.link[, 2]))
## Add the naive estimate of binomial probability y/N:
abline(v = df$y[holdout[2]]/df$Ntrials[holdout[2]], col="blue")

6.3 Adding observation likelihood randomness

The following code is specific to discrete likelihoods, as we can compute all the densities exactly. What we do here is discrete integration, computing the probabilities exactly (on the samples that we have).

## What range of observations do we want to compute
true.values = df$y[holdout]
if (any(is.na(true.values))) stop("Put the true values here!")

probs = rep(0, length(holdout))
for (i in 1:nrow(samp.link)) {
  probs = probs + dbinom(true.values, size=df$Ntrials[holdout], prob = samp.link[i, ])
}

## Numerical integration needs to divide by the number of samples
probs = probs / length(samples)
names(probs) = paste("holdout", holdout)

## Because of conditional independence, with the same sample, we can sum up the log probabilities
probs = c(probs, exp(sum(log(probs))))
names(probs)[length(probs)] = "Joint probability"

## The result for this hold-out combo
print(probs)
##         holdout 7        holdout 12 Joint probability 
##            0.0422            0.0696            0.0029

7 Comments

One proper scoring rule is the log probability. It is common to multiply the log probability by -2.