##### March 07, 2014

Bayesian statistics are useful for handling uncertainty of outcomes in many situations. With the growth of small-sats, the latest generation of small and relatively cheap sensors for Earth imaging, the precision and resolution is far less than the best high-end commercial imaging, so there it becomes even more important to understand how to quantify and account for errors when detecting objects in these images. In the course of teaching spatial statistics for GeoINT, I’ve found that it’s useful to have a good grasp of the concept of probability of detection, false positives, and how to bring that information together mathematically to estimate the posterior probabilities for detecting things from space.

## Quick Start Guide

The table below is a calculator for computing the actual probability of a detected object from a sensor, accounting for multiple looks at an object, and for positive and negative results. You can edit the values for the starting, or prior, probability, the sensor detection accuracy, and whether there was a detection or not [True or False]

The probablility updates come from Bayes Theorem, which allows us to update probabilities based on new evidence.

$P[target | evidence] = \frac{P[evidence | target] \times P[target]}{P[evidence]}$

Click the “Calculate” button to generate probabilities updated through Bayesian reasoning after 10 simulated sensor observations. You can change the values of each observation’s Sensor Accuracy and the resulting detection reading from the sensor.

 Starting probability P[H]: 0.0001

### Bayesian Sensor Fusion Calculator

Observation: Sensor Accuracy [0.0 - 1.0] Sensor Reading [True, False] Updated Probability
1 0.9 0
2 0.9 0
3 0.9 0
4 0.9 0
5 0.9 0
6 0.9 0
7 0.9 0
8 0.9 0
9 0.9 0
10 0.9 0

## How to use the calculator

This page calculates the updated probability of an anomaly or target by updating the probability to account for new evidence coming from a sensor. We have provided some example numbers for the Starting Probability and the Probability of Detection of your sensors - you will need to enter your own values for your work. The calculator will then provide updated probabilities for up to 10 observations from any sensor POD you provide. You can also specify when a sensor gives a reading of ‘False’ or ‘Non-detect’. For sensors that provide a measurement or concentration, you should use a threshold and for all sensor readings above your threshold, indicate a reading of ‘True’.