Sequential Analysis

In a sequential Bayesian analysis, a posterior model is updated incrementally as more data comes in. With the introduction of each new piece of data, the previous posterior model reflecting our understanding prior to observing this data becomes the new prior model.

Let's time travel to the end of 1970

$\pi \sim Beta(14,1)$

bechdel %>% 
  filter(year == 1970)

## # A tibble: 1 x 3
##    year title                          binary
##   <dbl> <chr>                          <chr> 
## 1  1970 Beyond the Valley of the Dolls PASS

The Posterior

summarize_beta_binomial(14, 1, y = 1, n = 1)

##       model alpha beta      mean mode         var         sd
## 1     prior    14    1 0.9333333    1 0.003888889 0.06236096
## 2 posterior    15    1 0.9375000    1 0.003446691 0.05870853

At the end of 1971

$\pi \sim Beta(15,1)$

bechdel %>% 
  filter(year == 1971)

## # A tibble: 5 x 3
##    year title                                   binary
##   <dbl> <chr>                                   <chr> 
## 1  1971 Escape from the Planet of the Apes      FAIL  
## 2  1971 Shaft                                   FAIL  
## 3  1971 Straw Dogs                              FAIL  
## 4  1971 The French Connection                   FAIL  
## 5  1971 Willy Wonka &amp; the Chocolate Factory FAIL

The Posterior

summarize_beta_binomial(15, 1, y = 0, n = 5)

##       model alpha beta      mean      mode         var         sd
## 1     prior    15    1 0.9375000 1.0000000 0.003446691 0.05870853
## 2 posterior    15    6 0.7142857 0.7368421 0.009276438 0.09631427

At the end of 1972

$\pi \sim Beta(15,6)$

bechdel %>% 
  filter(year == 1972)

## # A tibble: 3 x 3
##    year title          binary
##   <dbl> <chr>          <chr> 
## 1  1972 1776           FAIL  
## 2  1972 Pink Flamingos PASS  
## 3  1972 The Godfather  FAIL

The Posterior

summarize_beta_binomial(15, 6, y = 1, n = 3)

##       model alpha beta      mean      mode         var         sd
## 1     prior    15    6 0.7142857 0.7368421 0.009276438 0.09631427
## 2 posterior    16    8 0.6666667 0.6818182 0.008888889 0.09428090

summarize_beta_binomial(14, 1, y = 2, n = 9)

##       model alpha beta      mean      mode         var         sd
## 1     prior    14    1 0.9333333 1.0000000 0.003888889 0.06236096
## 2 posterior    16    8 0.6666667 0.6818182 0.008888889 0.09428090

Let $\theta$ be any parameter of interest with prior pdf $f(\theta)$. Then a sequential analysis in which we first observe a data point $y_1$ and then a second data point $y_2$ will produce the same posterior model of $\theta$ as if we first observe $y_2$ and then $y_1$:

$$f(\theta | y_1,y_2) = f(\theta|y_2,y_1)\;.$$

Similarly, the posterior model is invariant to whether we observe the data all at once or sequentially.