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357 | 357 | "cell_type": "markdown", |
358 | 358 | "metadata": {}, |
359 | 359 | "source": [ |
360 | | - "In [Bayesian statistics](https://en.wikipedia.org/wiki/Bayesian_probability) this is called a [*prior*](https://en.wikipedia.org/wiki/Prior_probability). It is the probability prior to incorporating measurements or other information. More completely, this is called the *prior probability distribution*. A [*probability distribution*](https://en.wikipedia.org/wiki/Probability_distribution) is a collection of all possible probabilities for an event. Probability distributions always to sum to 1 because something had to happen; the distribution lists all possible events and the probability of each.\n", |
| 360 | + "In [Bayesian statistics](https://en.wikipedia.org/wiki/Bayesian_probability) this is called a [*prior*](https://en.wikipedia.org/wiki/Prior_probability). It is the probability prior to incorporating measurements or other information. More completely, this is called the *prior probability distribution*. A [*probability distribution*](https://en.wikipedia.org/wiki/Probability_distribution) is a collection of all possible probabilities for an event. Probability distributions always sum to 1 because something had to happen; the distribution lists all possible events and the probability of each.\n", |
361 | 361 | "\n", |
362 | 362 | "I'm sure you've used probabilities before - as in \"the probability of rain today is 30%\". The last paragraph sounds like more of that. But Bayesian statistics was a revolution in probability because it treats probability as a belief about a single event. Let's take an example. I know that if I flip a fair coin infinitely many times I will get 50% heads and 50% tails. This is called [*frequentist statistics*](https://en.wikipedia.org/wiki/Frequentist_inference) to distinguish it from Bayesian statistics. Computations are based on the frequency in which events occur.\n", |
363 | 363 | "\n", |
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5504 | 5504 | "\n", |
5505 | 5505 | "The second problem is that the filter is discrete, but we live in a continuous world. The histogram requires that you model the output of your filter as a set of discrete points. A 100 meter hallway requires 10,000 positions to model the hallway to 1cm accuracy. So each update and predict operation would entail performing calculations for 10,000 different probabilities. It gets exponentially worse as we add dimensions. A 100x100 m$^2$ courtyard requires 100,000,000 bins to get 1cm accuracy.\n", |
5506 | 5506 | "\n", |
5507 | | - "A third problem is that the filter is multimodal. In the least example we ended up with strong beliefs that the dog was in position 4 or 9. This is not always a problem. Particle filters, which we will study later, are multimodal and are often used because of this property. But imagine if the GPS in your car reported to you that it is 40% sure that you are on D street, and 30% sure you are on Willow Avenue. \n", |
| 5507 | + "A third problem is that the filter is multimodal. In the last example we ended up with strong beliefs that the dog was in position 4 or 9. This is not always a problem. Particle filters, which we will study later, are multimodal and are often used because of this property. But imagine if the GPS in your car reported to you that it is 40% sure that you are on D street, and 30% sure you are on Willow Avenue. \n", |
5508 | 5508 | "\n", |
5509 | 5509 | "A forth problem is that it requires a measurement of the change in state. We need a motion sensor to detect how much the dog moves. There are ways to work around this problem, but it would complicate the exposition of this chapter, so, given the aforementioned problems, I will not discuss it further.\n", |
5510 | 5510 | "\n", |
|
5824 | 5824 | "\n", |
5825 | 5825 | "$$P(A \\mid B) = \\frac{P(B \\mid A)\\, P(A)}{\\int P(B \\mid A_j) P(A_j) \\mathtt{d}A_j}\\cdot$$\n", |
5826 | 5826 | "\n", |
5827 | | - "In practice the denominator can be fiendishly difficult to solve analytically (a recent opinion piece for the Royal Statistical Society [called it](http://www.statslife.org.uk/opinion/2405-we-need-to-rethink-how-we-teach-statistics-from-the-ground-up) a \"dog's breakfast\" [8]. Filtering textbooks are filled with integral laden equations which you cannot be expected to solve. We will learn more techniques to handle this in the **Particle Filters** chapter. Until then, recognize that in practice it is just a normalization term over which we can sum. What I'm trying to say is that when you are faced with a page of integrals, just think of them of sums, and relate them back to this chapter, and often the difficulties will fade. Ask yourself \"why are we summing these values\", and \"why am I dividing by this term\". Surprisingly often the answer is readily apparent." |
| 5827 | + "In practice the denominator can be fiendishly difficult to solve analytically (a recent opinion piece for the Royal Statistical Society [called it](http://www.statslife.org.uk/opinion/2405-we-need-to-rethink-how-we-teach-statistics-from-the-ground-up) a \"dog's breakfast\" [8]. Filtering textbooks are filled with integral laden equations which you cannot be expected to solve. We will learn more techniques to handle this in the **Particle Filters** chapter. Until then, recognize that in practice it is just a normalization term over which we can sum. What I'm trying to say is that when you are faced with a page of integrals, just think of them as sums, and relate them back to this chapter, and often the difficulties will fade. Ask yourself \"why are we summing these values\", and \"why am I dividing by this term\". Surprisingly often the answer is readily apparent." |
5828 | 5828 | ] |
5829 | 5829 | }, |
5830 | 5830 | { |
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