![\"Open](\"https://colab.research.google.com/assets/colab-badge.svg\"){kind=icon}
"
]
},
{
"cell_type": "code",
"execution_count": 1,
"metadata": {
"id": "w-CIikP0ZZnm",
"slideshow": {
"slide_type": "skip"
},
"tags": [
"remove-cell"
]
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Note: you may need to restart the kernel to use updated packages.\n"
]
}
],
"source": [
"%pip install -q -U gtbook"
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {
"id": "uef6Kglzcrha",
"slideshow": {
"slide_type": "skip"
},
"tags": [
"remove-cell"
]
},
"outputs": [],
"source": [
"import numpy as np\n",
"import gtsam\n",
"\n",
"from gtbook.discrete import Variables\n",
"from gtbook.display import show\n",
"\n",
"from IPython.display import display\n",
"from ipywidgets import interact\n",
"\n",
"import plotly.express as px\n",
"try:\n",
" import google.colab\n",
"except:\n",
" import plotly.io as pio\n",
" pio.renderers.default = \"png\"\n",
"\n",
"import gtbook\n",
"VARIABLES = Variables()\n",
"def pretty(obj): \n",
" return gtbook.display.pretty(obj, VARIABLES)"
]
},
{
"cell_type": "code",
"execution_count": 3,
"metadata": {
"id": "vXWJdNcgTJcQ",
"slideshow": {
"slide_type": "skip"
},
"tags": [
"remove-cell"
]
},
"outputs": [],
"source": [
"# Define variables here\n",
"Conductivity = VARIABLES.binary(\"Conductivity\")\n",
"Detection = VARIABLES.discrete(\"Detection\", [\"bottle\", \"cardboard\", \"paper\"])\n",
"categories = [\"cardboard\", \"paper\", \"can\", \"scrap metal\", \"bottle\"]\n",
"Category = VARIABLES.discrete(\"Category\", categories) # Not an accident that it is defined last."
]
},
{
"attachments": {},
"cell_type": "markdown",
"metadata": {
"id": "5_gRa4NQZsD4",
"slideshow": {
"slide_type": "slide"
}
},
"source": [
"```{index} sensing; sensor models\n",
"```\n",
"\n",
"# Sensors for Sorting Trash\n",
"\n",
"> Probability distributions can be used to model the behavior of sensors.\n",
"\n",
"\n",
"![\"Splash](\"Figures2/S23-Trash_sorting_robot_with_gripper-07.jpg\")
"
]
},
{
"cell_type": "markdown",
"metadata": {
"id": "5_gRa4NQZsD4",
"slideshow": {
"slide_type": "slide"
}
},
"source": [
"How can our trash sorting system know the category of an object that enters the sorting area? Robots use sensors to measure aspects of the world, and then make inferences about the world state based on these measurements. So, sensing can be used to infer the type of the object.\n",
"\n",
"Let us assume our robot cell only has three sensors:\n",
"* a conductivity sensor, outputting the value `True` or `False`\n",
"* a camera with a three detection algorithms: bottle, cardboard, paper\n",
"* a scale, which gives a continuous value in kg.\n",
"\n",
"We will discuss each in turn."
]
},
{
"cell_type": "markdown",
"metadata": {
"id": "fTcEUT_IAu54",
"slideshow": {
"slide_type": "slide"
}
},
"source": [
"```{index} conditional probability\n",
"```\n",
"## Binary Sensors\n",
"\n",
"> A binary sensor can be modeled using conditional probability distributions.\n",
"\n",
"Sadly, sensors are imperfect devices. For example, consider a sensor that measures electrical conductivity. Suppose we have a binary sensor that simply returns `True` or `False` , based on a measurement from an electrical probe, for an object's conductivity. For a metal can, this sensor will return `True` almost every time; however, some cans may be a bit dirty, or the probe may not make good contact, and these can lead to a mis-classification, returning `False`.\n",
"\n",
"Probability theory lets us quantify this. For example, for a metal can, the probability of the sensor returning `True` might be $0.9$, and hence the probability of returning `False` is $1-0.9=0.1$. \n",
"Of course in reality, we cannot know the true values of these probabilities, but we can estimate them using statistics.\n",
"A straightforward approach is to merely perform a set of experiments, and construct\n",
"a histogram of the results.\n",
"\n",
"Because the probability of the outcome depends on the type of the trash item, we represent this with a **conditional probability**, which accords a value \n",
"\n",
"$$P(conductive| trash~category)$$\n",
"\n",
"to every possible outcome (only `True` or `False` for this example), and where the `|` indicates that this depends on the category of the trash item. For example, we already established that\n",
"\n",
"$$P(conductive=True| trash~category=can)=0.9$$\n",
"\n",
"$$P(conductive=False| trash~category=can)=0.1$$\n"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "slide"
}
},
"source": [
"```{index} pair: conditional probability table; CPT\n",
"```\n",
"## Conditional Probabilities\n",
"\n",
"In general, the probability mass function of *any* single variable $X$ can be\n",
"parameterized by a parameter $Y$, whose value we assume as known. \n",
"This corresponds to the notion of a conditional probability,\n",
"which we write as \n",
"\n",
"$$P(X|Y=y).$$\n",
"\n",
"Note that given a particular value of\n",
"$Y$, this is just a probability distribution over $X$, with parameters given by a\n",
"PMF, as before. \n",
"\n",
"There are many ways to specify conditional probabilities, but in this simple case, with a binary outcome and a finite number of discrete categories, the simplest representation is to use a a **conditional probability table** or **CPT**. "
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"An example is shown below for our binary sensor example, taking care to first create the variables for pretty-printing:"
]
},
{
"cell_type": "code",
"execution_count": 4,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"P(Conductivity|Category):
\n", "| Category | false | true |
|---|---|---|
| cardboard | 0.99 | 0.01 |
| paper | 0.99 | 0.01 |
| can | 0.1 | 0.9 |
| scrap metal | 0.15 | 0.85 |
| bottle | 0.95 | 0.05 |
P(Detection|Category):
\n", "| Category | bottle | cardboard | paper |
|---|---|---|---|
| cardboard | 0.02 | 0.88 | 0.1 |
| paper | 0.02 | 0.2 | 0.78 |
| can | 0.33 | 0.33 | 0.34 |
| scrap metal | 0.33 | 0.33 | 0.34 |
| bottle | 0.95 | 0.02 | 0.03 |
P(Category):
\n", "| Category | value |
|---|---|
| cardboard | 0.2 |
| paper | 0.3 |
| can | 0.25 |
| scrap metal | 0.2 |
| bottle | 0.05 |
P(Category|Conductivity,Detection):
\n", "| Conductivity | Detection | cardboard | paper | can | scrap metal | bottle |
|---|---|---|---|---|---|---|
| false | bottle | 0.6 | 0.1 | 0.1 | 0.1 | 0.1 |
| false | cardboard | 0.1 | 0.6 | 0.1 | 0.1 | 0.1 |
| false | paper | 0.1 | 0.1 | 0.6 | 0.1 | 0.1 |
| true | bottle | 0.1 | 0.1 | 0.1 | 0.6 | 0.1 |
| true | cardboard | 0.1 | 0.1 | 0.1 | 0.1 | 0.6 |
| true | paper | 0.3 | 0.1 | 0.2 | 0.1 | 0.3 |
| Variable | value |
|---|
| Variable | Domain |
|---|---|
| Conductivity | false, true |
| Detection | bottle, cardboard, paper |
| Category | cardboard, paper, can, scrap metal, bottle |
| Variable | value |
|---|---|
| Conductivity | true |
| Variable | value |
|---|---|
| Category | scrap metal |