Bayesian Linear Models

.title[
# Bayesian Linear Models
]
.subtitle[
## 02 - BRM basics
]
.author[
### Stefano Coretta
]
.institute[
### University of Edinburgh
]
.date[
### 2023/07/07
]

---

## Estimate mean and SD

---

---

.f3[
$$
`\begin{aligned}
RT_i & \sim Gaussian(\mu, \sigma) \\
\\
\mu & = 1046 \\
\sigma & = 348
\end{aligned}`
$$
]

- The RT values are distributed according to a Gaussian distribution with mean `$\mu$` and standard deviation `$\sigma$`.

- The mean and SD from the sample do not take into consideration the **uncertainty and variability** in the sample.

---

.f3[
$$
`\begin{aligned}
RT_i & \sim Gaussian(\mu, \sigma) \\
\\
\mu & \sim Gaussian(\mu_1, \sigma_1) \\
\sigma & \sim Gaussian_+(\mu_2, \sigma_2)
\end{aligned}`
$$
]

- We can account for uncertainty and variability by assuming that the mean and SD are themselves values coming from a probability distribution.

- The mean `$\mu$` is a value from a Gaussian distribution with mean `$\mu_1$` and SD `$\sigma_1$`.

- The SD `$\sigma$` is a value from a Gaussian distribution with mean `$\mu_2$` and SD `$\sigma_2$`.

- SD can only be positive so the Gaussian distribution is truncated to positive values only ( `$Gaussian_+$` ).

---

```r
# Attach the brms package
library(brms)

# Run a Bayesian model
rt_bm <- brm(
  # This is the formula of the model.
  RT ~ 1,
  # This is the probability distribution family.
  family = gaussian(),
  # And the data.
  data = mald
)
```

---

```
##  Family: gaussian 
##   Links: mu = identity; sigma = identity 
## Formula: RT ~ 1 
##    Data: mald (Number of observations: 3000) 
##   Draws: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
##          total post-warmup draws = 4000
## 
## Population-Level Effects: 
##           Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## Intercept  1046.44      6.26  1034.22  1058.64 1.00     3527     2803
## 
## Family Specific Parameters: 
##       Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma   347.48      4.38   339.16   356.06 1.00     3292     2558
## 
## Draws were sampled using sample(hmc). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).
```

---

```
##  Family: gaussian 
##   Links: mu = identity; sigma = identity 
## Formula: RT ~ 1 
##    Data: mald (Number of observations: 3000) 
##   Draws: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
##          total post-warmup draws = 4000
## 
## Population-Level Effects: 
##           Estimate Est.Error l-70% CI u-70% CI Rhat Bulk_ESS Tail_ESS
## Intercept  1046.44      6.26  1040.04  1052.96 1.00     3527     2803
## 
## Family Specific Parameters: 
##       Estimate Est.Error l-70% CI u-70% CI Rhat Bulk_ESS Tail_ESS
## sigma   347.48      4.38   342.91   352.13 1.00     3292     2558
## 
## Draws were sampled using sample(hmc). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).
```

---

.f3[
$$
`\begin{aligned}
RT_i & \sim Gaussian(\mu, \sigma) \\
\\
\mu & \sim Gaussian(1046, 6.26) \\
\sigma & \sim Gaussian_+(347, 4.38)
\end{aligned}`
$$
]

- There is a 95% probability that `$\mu$` is between 1034 and 1059 ms. There is a 70% probability that it is between 1040 and 1053.

- There is a 95% probability that `$\sigma$` is between 339 and 356 ms. There is a 70% probability that it is between 343 and 352.

Great! But what about the RT when the word is a real word vs when it is not?

---

## Real vs nonce words

---

```r
# Attach the brms package
library(brms)

# Run a Bayesian model
rt_bm_2 <- brm(
  # This is the formula of the model.
  RT ~ IsWord,
  # This is the probability distribution family.
  family = gaussian(),
  # And the data.
  data = mald
)
```

---

```
##  Family: gaussian 
##   Links: mu = identity; sigma = identity 
## Formula: RT ~ IsWord 
##    Data: mald (Number of observations: 3000) 
##   Draws: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
##          total post-warmup draws = 4000
## 
## Population-Level Effects: 
##             Estimate Est.Error l-89% CI u-89% CI Rhat Bulk_ESS Tail_ESS
## Intercept     981.30      8.63   967.63   995.52 1.00     3804     3037
## IsWordFALSE   132.77     12.38   113.01   152.14 1.00     3822     2809
## 
## Family Specific Parameters: 
##       Estimate Est.Error l-89% CI u-89% CI Rhat Bulk_ESS Tail_ESS
## sigma   341.12      4.42   334.06   348.38 1.00     3932     2800
## 
## Draws were sampled using sample(hmc). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).
```

---

.bg-washed-yellow.b--gold.ba.bw2.br3.shadow-5.ph4.mt2[
- The `Intercept` is the mean RT when [IsWord=TRUE].

- `IsWordFALSE` is the **difference** between the mean RT when [IsWord=FALSE] and the mean RT when [IsWord=TRUE].
]

---

.bg-washed-blue.b--dark-blue.ba.bw2.br3.shadow-5.ph4.mt2[
There is an **89% probability** that the mean RT when [IsWord=FALSE] is **between 113 to 152 ms longer** than the mean RT when [IsWord=TRUE].
]

---

.f3[
$$
`\begin{aligned}
RT_i & \sim Gaussian(\mu, \sigma) \\
\\
\mu & = \beta_0 + \beta_1 \times IsWord_{F} \\
\end{aligned}`
$$
]

.bg-washed-yellow.b--gold.ba.bw2.br3.shadow-5.ph4.mt2[
- `$\beta_0$` is mean RT when [IsWord=TRUE].

- `$\beta_1$` is the **difference** between the mean RT when [IsWord=FALSE] and the mean RT when [IsWord=TRUE].

- `$IsWord_F$` is `0` when [IsWord=TRUE] and `1` when [IsWord=FALSE].
]

---

.f3[
$$
`\begin{aligned}
RT_i & \sim Gaussian(\mu, \sigma) \\
\\
\mu & = \beta_0 + \beta_1 \times IsWord_{F} \\
\end{aligned}`
$$
]

.bg-washed-blue.b--dark-blue.ba.bw2.br3.shadow-5.ph4.mt2[
**Mean RT when [IsWord=TRUE].**

$$
`\begin{aligned}
\mu & = \beta_0 + \beta_1 \times IsWord_{F} = \beta_0 + \beta_1 \times 0 = \beta_0 \\
\end{aligned}`
$$
**Mean RT when [IsWord=FALSE].**

$$
`\begin{aligned}
\mu & = \beta_0 + \beta_1 \times IsWord_{F} = \beta_0 + \beta_1 \times 1 = \beta_0 + \beta_1 \\
\end{aligned}`
$$
]

---

.f3[
$$
`\begin{aligned}
RT_i & \sim Gaussian(\mu, \sigma) \\
\\
\mu & = \beta_0 + \beta_1 \times IsWord_{F} \\
\beta_0 & \sim Gaussian(\mu_0, \sigma_0) \\
\beta_1 & \sim Gaussian(\mu_1, \sigma_1) \\
\\
\sigma & \sim Gaussian_+(\mu_2, \sigma_2)
\end{aligned}`
$$
]