t-df: learning about the degrees of freedom of
a t distribution
This example uses simulated data: we generate n = 1000 observations from a (standard) t distribution with
d
degrees of freedom:
model {
d <- 4
for (i in 1 : 1000) {
y[i] ~ dt(0, 1, d)
}
}
We let the sampler run for a while (just to check that we're simulating from the correct distribution) and then we save a set of simulated data using the 'save state' facility (select
Save State
from the
Model
menu).
Check: should have mean 0, variance 2, etc.
node mean sd MC error 2.5% median 97.5% start sample
y[1] -0.03787 1.452 0.04852 -3.141 -0.003657 2.847 1 1000
To analyse the simulated data: first we try a model for learning about the degrees of freedom as a continuous quantity:
model {
for (i in 1:1000) {
y[i] ~ dt(0, 1, d)
}
d ~ dunif(2, 100) # degrees of freedom must be at least two
}
![[t-df3]](t-df3.bmp)
node mean sd MC error 2.5% median 97.5% start sample
d 3.929 0.3753 0.03154 3.231 3.913 4.67 1001 1000
Now we attempt to model the degrees of freedom parameter with a discrete uniform prior on {2, 3, 4, ..., 50}. The sampler soon converges to
d
= 4 but refuses to move away from there, presumably because the data provide overwhelming evidence for
d
= 4 compared to the alternatives, e.g.
d
= 3 or 5.
model {
for (j in 1:49) {
p[j] <- 1 / 49
d.value[j] <- j + 1
}
for (i in 1:1000) {
y[i] ~ dt(0, 1, d)
}
K ~ dcat(p[])
d <- d.value[K]
}
Initial values:
list(K = 6)
We should get better mixing if we specify the prior for
d
on a finer grid, e.g. {2.0, 2.1, 2.2, ..., 6.0}:
Model:
model {
for (j in 1:41) {
p[j] <- 1 / 41
d.value[j] <- 2 + (j - 1) * 0.1
}
for (i in 1:1000) {
y[i] ~ dt(0, 1, d)
}
K ~ dcat(p[])
d <- d.value[K]
}
Initial values:
list(K = 30)
![[t-df4]](t-df4.bmp)
node mean sd MC error 2.5% median 97.5% start sample
d 3.98 0.385 0.03401 3.3 4.0 4.8 101 1000
Simulated data:
list(
y = c(
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1.075137749844306,-1.86057298660303,0.6696277496598672,-3.751135618205416,0.2954899608859383,
0.2455600714107795,1.145697069057127,0.5001028365348743,0.1970020283749174,0.9000658663833929,
0.5723779714915089,2.036567915247737,-0.9372064194922384,0.1973300460117617,0.1715791912197212,
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