# Struggling with Bayesian Truth Serum formula

1. May 19, 2015

### Illmaticus

Hi there,

right now, I am struggling to successfully calculate scores with the Bayesian Truth formula and I hope this is the right place to find someone who can help me along
For everyone, who doesn’t know it, I will summarize it briefly : The Bayesian Truth Serum is an scoring method, that provides truth-telling incentives and elicits “truthful subjective data in situations where the objective truth is unknowable”. The basic idea is, that truthful answers maximize ones individual score.

When using this method, people are asked for dual reports:
• First they are asked to endorse their personal answer to an m multiple-choice question. E.g. Which color do you like most - Red or Blue?
• Second, they are asked to predict what the other individuals in the sample might answer: Form a prediction of the sample distribution of all endorsments: Red __% ; Blue __%

• The respondents are defined as r ∈ {1,2,…}
• xrk indicates whether respondent r has endorsed answer k ∈ {1,…,m} and xrk ∈ {0,1} meaning that xrk=0 for all answers except the one endorsed by r.
• yr = (yr1,…yrm) indicates the prediction of the sample distribution) and yrm>0 and $\sum\limits_{k}^{} y^{r}_{k}$. So in this example it could be (0,6;0,4) for someone who thinks that most people like red.

The formula consists of an information score + prediction score.

• The information score is given as $\sum\limits_{k}^{} x^{r}_k \cdot log \frac{x^{-}_k }{y^{-}_k}$ . $x^{-}_k$ is the fraction endorsing answer k and $y^{-}_k$ is the geometric average of endorsment prediction yr = (yr-,…yrm).
• The prediction score is given as $a\cdot \sum\limits_{k}^{} x^{-}_{k} \cdot log \frac{y^{r}_k }{x^{-}_k }$

Back again to the question about the favorite Color red or blue: I have created a table for a fictional Survey in order to practice the calculation of those BTS-scores.

HTML:
<table border="1" style="background-color:#FFFFCC;border-collapse:collapse;border:1px solid #FFCC00;color:#000000;width:100%" cellpadding="3" cellspacing="3">
<tr>
<td></td>
<td></td>
<td></td>
<td>Predicted fraction</td>
<td></td>
</tr>
<tr>
<td>r</td>
<td>red</td>
<td>blue</td>
<td>l</td>
<td>redl</td>
<td>blue</td>
</tr>
<tr>
<td>1l</td>
<td>1</td>
<td>0</td>
<td></td>
<td>0,8</td>
<td>0,2</td>
</tr>
<tr>
<td>1l</td>
<td>1</td>
<td>0</td>
<td></td>
<td>0,8</td>
<td>0,2</td>
</tr>
<tr>
<td>1l</td>
<td>1</td>
<td>0</td>
<td></td>
<td>0,8</td>
<td>0,2</td>
</tr>
<tr>
<td>1l</td>
<td>1</td>
<td>0</td>
<td></td>
<td>0,8</td>
<td>0,2</td>
</tr>
<tr>
<td>1l</td>
<td>1</td>
<td>0</td>
<td></td>
<td>0,8</td>
<td>0,2</td>
</tr>
<tr>
<td>1l</td>
<td>1</td>
<td>0</td>
<td></td>
<td>0,8</td>
<td>0,2</td>
</tr>
<tr>
<td>1l</td>
<td>1</td>
<td>0</td>
<td></td>
<td>0,8</td>
<td>0,2</td>
</tr>
<tr>
<td>7</td>
<td>0</td>
<td>1</td>
<td></td>
<td>0,6</td>
<td>0,4</td>
</tr>
<tr>
<td>7</td>
<td>0</td>
<td>1</td>
<td></td>
<td>0,6</td>
<td>0,4</td>
</tr>
<tr>
<td>7</td>
<td>0</td>
<td>1</td>
<td></td>
<td>0,6</td>
<td>0,4</td>
</tr>
<tr>
<td>average</td>
<td>0,7</td>
<td>0,3</td>
<td>geometric average</td>
<td>0,73385</td>
<td>0,24623</td>
</tr>
</table>

I also have the Information scores for both answers red and blue given: information score for red should be -0,04722677. The info-score for blue should be 0,24622888.

Now that's where i am struggling. When I calculate it myself, i am getting Information score (red)= -0,0205103 (so for people stating that their favorite color is red) and Information score (blue) = 0,0857823, which is both not correct.
should be -0,04722677. The info-score for blue should be 0,24622888.

I calculated it using the formula $1*log\frac{0,7}{0,73385} + 0*log\frac{0,3}{0,24623}$

What am I doing wrong?

2. May 20, 2015

### Stephen Tashi

Only the html source shows up in my browser. I gather there are 10 respondents. Seven of them pick "red" and all seven estimate the population distribution to be 0.8 for red and 0.2 for blue. Three of the respondents pick "blue" and all three estimate the population distribution to be 0.6 for red and 0.4 for blue.