Minto typos.

Courtesy of email from James Kim.
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Roger Labbe 2016-06-22 19:53:33 -07:00
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"cell_type": "markdown",
"metadata": {},
"source": [
"Back to filtering. The results of the filter may strike you as quite silly; of course the data will look good if we assume the conclusion, that our weight gain is around 1 lb/day! Let's see what the filter does if our initial guess is bad. Let's spredict that there is a weight loss of 1 lb a day"
"Back to filtering. The results of the filter may strike you as quite silly; of course the data will look good if we assume the conclusion, that our weight gain is around 1 lb/day! Let's see what the filter does if our initial guess is bad. Let's predict that there is a weight loss of 1 lb a day"
]
},
{

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"\n",
"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",
"\n",
"I flip the coin one more time and let it land. Which way do I believe it landed? Frequentist probability has nothing to say about that; it will merely state that 50% of coin flips land as heads. In some ways it is meaningless to to assign a probability to the current state of the coin. It is either heads or tails, we just don't know which. Bayes treats this as a belief about a single event - the strength of my belief or knowledge that this specific coin flip is heads is 50%. Some object to the term \"belief\"; belief can imply holding something to be true without evidence. In this book it always is a measure of the strength of our knowledge. We'll learn more about this as we go.\n",
"I flip the coin one more time and let it land. Which way do I believe it landed? Frequentist probability has nothing to say about that; it will merely state that 50% of coin flips land as heads. In some ways it is meaningless to assign a probability to the current state of the coin. It is either heads or tails, we just don't know which. Bayes treats this as a belief about a single event - the strength of my belief or knowledge that this specific coin flip is heads is 50%. Some object to the term \"belief\"; belief can imply holding something to be true without evidence. In this book it always is a measure of the strength of our knowledge. We'll learn more about this as we go.\n",
"\n",
"Bayesian statistics takes past information (the prior) into account. We observe that it rains 4 times every 100 days. From this I could state that the chance of rain tomorrow is 1/25. This is not how weather prediction is done. If I know it is raining today and the storm front is stalled, it is likely to rain tomorrow. Weather prediction is Bayesian.\n",
"\n",

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"source": [
"The y-axis depicts the *probability density* — the relative amount of cars that are going the speed at the corresponding x-axis.\n",
"\n",
"You may object that human heights or automobile speeds cannot be less than zero, let alone $-\\infty$ or $-\\infty$. This is true, but this is a common limitation of mathematical modeling. “The map is not the territory” is a common expression, and it is true for Bayesian filtering and statistics. The Gaussian distribution above models the distribution of the measured automobile speeds, but being a model it is necessarily imperfect. The difference between model and reality will come up again and again in these filters. Gaussians are used in many branches of mathematics, not because they perfectly model reality, but because they are easier to use than any other relatively accurate choice. However, even in this book Gaussians will fail to model reality, forcing us to use computationally expensive alternatives. \n",
"You may object that human heights or automobile speeds cannot be less than zero, let alone $-\\infty$ or $\\infty$. This is true, but this is a common limitation of mathematical modeling. “The map is not the territory” is a common expression, and it is true for Bayesian filtering and statistics. The Gaussian distribution above models the distribution of the measured automobile speeds, but being a model it is necessarily imperfect. The difference between model and reality will come up again and again in these filters. Gaussians are used in many branches of mathematics, not because they perfectly model reality, but because they are easier to use than any other relatively accurate choice. However, even in this book Gaussians will fail to model reality, forcing us to use computationally expensive alternatives. \n",
"\n",
"You will see these distributions called *Gaussian distributions* or *normal distributions*. *Gaussian* and *normal* both mean the same thing in this context, and are used interchangeably. I will use both throughout this book as different sources will use either term, and I want you to be used to seeing both. Finally, as in this paragraph, it is typical to shorten the name and talk about a *Gaussian* or *normal* — these are both typical shortcut names for the *Gaussian distribution*. "
]
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"\n",
"In this chapter I used code from [FilterPy](https://github.com/rlabbe/filterpy) to compute and plot Gaussians. I did that to give you a chance to look at the code and see how these functions are implemented. However, Python comes with \"batteries included\" as the saying goes, and it comes with a wide range of statistics functions in the module `scipy.stats`. So let's walk through how to use scipy.stats to compute statistics and probabilities.\n",
"\n",
"The `scipy.stats` module contains a number of objects which you can use to compute attributes of various probability distributions. The full documentation for this module is here: http://http://docs.scipy.org/doc/scipy/reference/stats.html. We will focus on the norm variable, which implements the normal distribution. Let's look at some code that uses `scipy.stats.norm` to compute a Gaussian, and compare its value to the value returned by the `gaussian()` function from FilterPy."
"The `scipy.stats` module contains a number of objects which you can use to compute attributes of various probability distributions. The full documentation for this module is here: http://docs.scipy.org/doc/scipy/reference/stats.html. We will focus on the norm variable, which implements the normal distribution. Let's look at some code that uses `scipy.stats.norm` to compute a Gaussian, and compare its value to the value returned by the `gaussian()` function from FilterPy."
]
},
{

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