set.seed(812)
f0 <- rnorm(200, 125, 25) |> round()
f0[1:10] [1] 79 146 111 114 112 147 167 98 143 102QML - Week 5
Intro to statistical modelling
1 Statistical modelling
2 Gaussian models
\[y \sim Gaussian(\mu, \sigma)\]
3 Simulate Gaussian data
rnorm(n, mean, sd)
. . .
Simulate fundamental frequency f0.
4 f0 density
5 Simulation vs estimation
SIMULATION
You have \(\mu, \sigma\) and you generate \(y\).
ESTIMATION
You have \(y\) and you estimate \(\mu\) and \(\sigma\).
When you simulate data, you know the population mean and SD. In research, you don’t. You just have observations.
Statistical modelling allows you to estimate the mean and SD of the population from the sample. This is statistical inference. Bayesian Gaussian models do exactly that: estimate mean and SD.
6 Gaussian model of f0
\[\begin{align} f0 & \sim Gaussian(\mu, \sigma)\\ \mu & = ...\\ \sigma & = ... \end{align}\]
7 Gaussian model of f0
\[\begin{align} f0 & \sim Gaussian(\mu, \sigma)\\ \mu & = P_\mu\\ \sigma & = P_\sigma \end{align}\]
8 Gaussian model of f0: brms
f0_tib <- tibble(f0 = f0)
library(brms)
f0_bm <- brm(
f0 ~ 1,
family = gaussian,
data = f0_tib,
file = "cache/f0_bm"
)9 Gaussian model of height: posteriors
10 Posterior summary
posterior_summary(f0_bm, probs = c(0.1, 0.9))[1:2,] |> round() Estimate Est.Error Q10 Q90
b_Intercept 126 2 124 129
sigma 27 1 25 2811 Sample size and uncertainty
the lower the sample size
the higher the uncertainty
12 A bit harder: N = 5
set.seed(4256)
f0_tib <- tibble(f0 = sample(f0, 5))
f0_bm_small <- brm(
f0 ~ 1,
family = gaussian,
data = f0_tib,
file = "cache/f0_bm_small"
)13 A bit harder: N = 5
14 A bit harder: N = 5
posterior_summary(f0_bm_small, probs = c(0.1, 0.9))[1:2,] |> round() Estimate Est.Error Q10 Q90
b_Intercept 122 7 114 130
sigma 22 7 15 3115 Statistical modelling