Bayesianische Statistik

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# Bayesianische Statistik
## Teil 2<br/>Von Grid Approximation zu Sampling
### Andrew Ellis
### Methodenkurs Neurowissenschaft im Computerlab
### 2021-03-16

---

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<a href="https://kogpsy.github.io/neuroscicomplab" target="_blank"><svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 576 512" class="rfa" style="height:0.75em;fill:#0F4C81;position:relative;"><path d="M280.37 148.26L96 300.11V464a16 16 0 0 0 16 16l112.06-.29a16 16 0 0 0 15.92-16V368a16 16 0 0 1 16-16h64a16 16 0 0 1 16 16v95.64a16 16 0 0 0 16 16.05L464 480a16 16 0 0 0 16-16V300L295.67 148.26a12.19 12.19 0 0 0-15.3 0zM571.6 251.47L488 182.56V44.05a12 12 0 0 0-12-12h-56a12 12 0 0 0-12 12v72.61L318.47 43a48 48 0 0 0-61 0L4.34 251.47a12 12 0 0 0-1.6 16.9l25.5 31A12 12 0 0 0 45.15 301l235.22-193.74a12.19 12.19 0 0 1 15.3 0L530.9 301a12 12 0 0 0 16.9-1.6l25.5-31a12 12 0 0 0-1.7-16.93z"/></svg></a> Methodenkurs Neurowissenschaft im Computerlab
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---

## Letzte Sitzung:

- Parameterschätzung: Wahrscheinlichkeitsparameter einer Binomialverteilung
- Beta Verteilungen
- Grid Approximation

---

## Bayes' Theorem

$$ P(\theta|Data) = \frac{ P(Data|\theta) * P(\theta) } {P(Data)} $$
oder ohne Normalisierungskonstante:

$$ P(\theta|Data) \propto P(Data|\theta) * P(\theta) $$

### Binomial Likelihood

$$ P(x = k) = {n \choose k} \theta^k (1-\theta)^{n-k} $$

---

## Parameterschätzung: Parameter einer Binomialverteilung

```r
wins <- 6
games <- 9
```

### Maximum Likelihood Schätzung

```r
theta <- wins/games
theta
```

```
## [1] 0.6666667
```

---

---

## Posterior durch Grid Approximation

.panelset[
.panel[.panel-name[Grid]

```r
n_points <- 100
theta_grid <- seq(from = 0 , to = 1 , length.out = n_points)
```

]

.panel[.panel-name[Likelihood]

```r
likelihood <- dbinom(wins , size = games , prob = theta_grid)
```
]

.panel[.panel-name[Prior]

```r
prior <- dbeta(x = theta_grid, shape1 = 4,  shape2 = 4)
```
]
]

---

## Posterior durch Grid Approximation

.panelset[
.panel[.panel-name[Posterior]

```r
unstandardized_posterior <- likelihood * prior
posterior <- unstandardized_posterior / sum(unstandardized_posterior)
```

]

.panel[.panel-name[Code für Plot]

```r
d <- tibble(theta_grid, prior, likelihood, posterior)

d %>%
  pivot_longer(-theta_grid, names_to = "distribution", values_to = "density") %>% 
  mutate(distribution = as_factor(distribution)) %>% 
  ggplot(aes(theta_grid, density, color = distribution)) +
  geom_line(size = 1.5) +
  geom_vline(xintercept = 9/10, linetype = "dashed") +
  scale_color_viridis_d(end = 0.8) +
  xlab("Theta Werte") +
  ylab("") +
  facet_wrap(~distribution, scales = "free_y") +
  theme_bw() +
  theme(legend.position = "none")
```
]

.panel[.panel-name[Plot]

<img src="02-bayesian-stats-slides_files/figure-html/unnamed-chunk-10-1.png" width="100%" />
]
]

---

## Posterior zusammenfassen

Samples aus dem Posterior ziehen:

.panelset[
.panel[.panel-name[N. Samples]

```r
n_samples <- 1e4
set.seed(3) # wegen Reproduzierbarkeit
```

```r
d %>%
    paged_table(options = list(rows.print = 6))
```

]
.panel[.panel-name[Ziehen mit Zurücklegen]

```r
samples <-
  d %>% 
  slice_sample(n = n_samples, weight_by = posterior, replace = TRUE) %>%
  mutate(sample_number = 1:n())
```
]

.panel[.panel-name[Samples]

```r
samples %>%
    paged_table(options = list(rows.print = 6))
```

<div data-pagedtable="false">
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  </script>
</div>
]
]

---

## Posterior zusammenfassen

.panelset[
.panel[.panel-name[Markov Chain]
<img src="02-bayesian-stats-slides_files/figure-html/unnamed-chunk-15-1.png" width="100%" />
]

.panel[.panel-name[Histogram/Density]
<img src="02-bayesian-stats-slides_files/figure-html/unnamed-chunk-16-1.png" width="100%" />
]
]

---

## Intervalle mit definierten Endpunkten

### Posterior Approximation

```r
sum(posterior[theta_grid > 0.5])
```

```
## [1] 0.7728132
```

```r
d %>% 
  filter(theta_grid > .5) %>% 
  summarise(sum = sum(posterior)) 
```

```
## # A tibble: 1 x 1
##     sum
##   <dbl>
## 1 0.773
```

---

## Intervalle mit definierten Endpunkten

### Posterior Samples

```r
*samples %>%
* filter(theta_grid > .5) %>%
* summarise(sum = n() / n_samples)
```

```
## # A tibble: 1 x 1
##     sum
##   <dbl>
## 1 0.773
```

```r
samples %>% 
  count(theta_grid > .5) %>% 
  mutate(probability = n / sum(n))
```

```
## # A tibble: 2 x 3
##   `theta_grid > 0.5`     n probability
##   <lgl>              <int>       <dbl>
## 1 FALSE               2273       0.227
## 2 TRUE                7727       0.773
```

---
## Intervalle mit definierter Masse

### Quantile

```r
quantile(samples$theta_grid, prob = c(.25, .75))
```

```
##       25%       75% 
## 0.5050505 0.6666667
```

```r
library(tidybayes)
median_qi(samples$theta_grid, .width = .5)
```

```
##           y      ymin      ymax .width .point .interval
## 1 0.5959596 0.5050505 0.6666667    0.5 median        qi
```

```r
library(tidybayes)
median_qi(samples$theta_grid, .width = c(.5, .8, .99))
```

```
##           y      ymin      ymax .width .point .interval
## 1 0.5959596 0.5050505 0.6666667   0.50 median        qi
## 2 0.5959596 0.4343434 0.7373737   0.80 median        qi
## 3 0.5959596 0.2929293 0.8484848   0.99 median        qi
```
---

### Highest Posterior Density (HPDI)

```r
mode_hdi(samples$theta_grid, .width = .5)
```

```
##           y      ymin      ymax .width .point .interval
## 1 0.5961325 0.5151515 0.6666667    0.5   mode       hdi
```

```r
hdi(samples$theta_grid, .width = .5)
```

```
##           [,1]      [,2]
## [1,] 0.5151515 0.6666667
```

---

## Quantiles vs Highest Posterior Density

---

## Sampling: Simulation

---

## Sampling: Simulation

---

## Prior Predictive Distribution

---

## Posterior Predictive Distribution

---

## Graphical model

Grafische Darstellung des  <span style="color: var(--text-color)">Generativen Modells</span>:
--

.pull-left[
![an image](images/binomial-graphical-model.png)]

.pull-right[
![an image](images/bernoulli-graphical-model.png)]

---

## Graphical model

Grafische Darstellung eines  <span style="color: var(--text-color)">Generativen Modells</span> mit normalverteilten Daten:
--

.pull-left[
![an image](images/normal-graphical-model.png)]

.pull-right[

```r
n <- 20

mu_sigma <- 5

set.seed(6)
mu <- rnorm(n = 1, mean = 120, 
            sd = mu_sigma)
sigma <- runif(n = 1, 1, 10)
y <- rnorm(n = n, mean = mu, 
           sd = sigma) %>% 
     round(2)
```
]

---

---

## Posterior Inference

.pull-left[
```{C++}
// The input data is a vector 'y' of length 'N'.
data {
  int<lower=0> N;
  vector[N] y;
}

// The parameters accepted by the model. Our model
// accepts two parameters 'mu' and 'sigma'.
parameters {
  real mu;
  real<lower=0> sigma;
}

// The model to be estimated. We model the output
// 'y' to be normally distributed with mean 'mu'
// and standard deviation 'sigma'.
model {
  mu ~ normal(120, 5);
  sigma ~ uniform(1, 10);
  y ~ normal(mu, sigma);
}
```
]

.pull-right[

```r
library(brms)
d <- tibble(y)

priors <- set_prior("normal(120, 5)", class = "Intercept") +
    set_prior("uniform(1, 10)", class = "sigma")

fit <- brm(y ~ 1,
           family = gaussian,
           prior = priors,
           data = d,
           cores = parallel::detectCores())
```
]

---

## `brms` Model

```r
library(brms)
d <- tibble(y)
```

```r
get_prior(y ~ 1,
          family = gaussian,
          data = d)
```

```
##                   prior     class coef group resp dpar nlpar bound  source
##  student_t(3, 123, 4.4) Intercept                                  default
##    student_t(3, 0, 4.4)     sigma                                  default
```

---

```r
priors <- set_prior("normal(120, 5)", class = "Intercept") +
  set_prior("cauchy(0, 1)", class = "sigma")
```

```r
m <- brm(y ~ 1,
           family = gaussian,
           prior = priors,
           data = d,
           cores = parallel::detectCores(),
           file = "model_1")
```

---

## `brms` Model

.panelset[
.panel[.panel-name[Summary]

```r
summary(m)
```
]
.panel[.panel-name[Output]

```
##  Family: gaussian 
##   Links: mu = identity; sigma = identity 
## Formula: y ~ 1 
##    Data: d (Number of observations: 20) 
## Samples: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
##          total post-warmup samples = 4000
## 
## Population-Level Effects: 
##           Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## Intercept   121.95      0.81   120.36   123.55 1.00     2605     2128
## 
## Family Specific Parameters: 
##       Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma     3.57      0.60     2.63     4.94 1.00     2482     2468
## 
## Samples were drawn using sampling(NUTS). 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).
```
]
]

.footnote[ Parameter werden durch Mittelwert, SD (Est.Error), 2-seitige 95% Credible
intervals der Posterior-Verteilung zusammengefasst. ]

---

```r
plot(m)
```

---

```r
library(tidybayes)
m %>%
  spread_draws(b_Intercept) %>% 
  median_qi(.width = c(.50, .80, .95)) %>% 
  kableExtra::kbl()
```

<table>
 <thead>
  <tr>
   <th style="text-align:right;"> b_Intercept </th>
   <th style="text-align:right;"> .lower </th>
   <th style="text-align:right;"> .upper </th>
   <th style="text-align:right;"> .width </th>
   <th style="text-align:left;"> .point </th>
   <th style="text-align:left;"> .interval </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:right;"> 121.9468 </td>
   <td style="text-align:right;"> 121.4055 </td>
   <td style="text-align:right;"> 122.5020 </td>
   <td style="text-align:right;"> 0.50 </td>
   <td style="text-align:left;"> median </td>
   <td style="text-align:left;"> qi </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 121.9468 </td>
   <td style="text-align:right;"> 120.9159 </td>
   <td style="text-align:right;"> 123.0010 </td>
   <td style="text-align:right;"> 0.80 </td>
   <td style="text-align:left;"> median </td>
   <td style="text-align:left;"> qi </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 121.9468 </td>
   <td style="text-align:right;"> 120.3588 </td>
   <td style="text-align:right;"> 123.5486 </td>
   <td style="text-align:right;"> 0.95 </td>
   <td style="text-align:left;"> median </td>
   <td style="text-align:left;"> qi </td>
  </tr>
</tbody>
</table>

---

```r
m %>%
  spread_draws(b_Intercept) %>%
  ggplot(aes(x = b_Intercept)) +
  stat_halfeye(.width = c(.50, .80, .95))
```