Now we come to the core part of our project: how can we estimate \(S\), without knowing \(\theta, \phi, k_j\)?
Basically, this is the data we have:
import pandas as pd
output_dir = 'data'
df = pd.read_csv(f"{output_dir}/150|200_3.csv").drop(
['k_j', 'S_n', 'affected_or_not'], axis = 1)
df.head()| participant | biomarker | measurement | diseased | |
|---|---|---|---|---|
| 0 | 0 | HIP-FCI | 3.135981 | False |
| 1 | 1 | HIP-FCI | 12.593704 | True |
| 2 | 2 | HIP-FCI | 6.220776 | False |
| 3 | 3 | HIP-FCI | 3.545100 | False |
| 4 | 4 | HIP-FCI | 3.966541 | False |
The main idea is this:
We need to know \(\theta, \phi\) first before we can estimate likelihoods. To estimate \(\theta, \phi\), there are three approaches covered in Chapter 4.
We try many different \(S\) and calculate its associated likelihood. We either accept or reject this \(S\) according to Metropolis–Hastings algorithm.
In the following, We will cover three different approaches to estimate \(S\):