Update with removing lawrennd/talks and replacing with lawrennd/snippets for posts

Author: lawrennd

_posts/2012-09-06-life-the-universe-and-machine-learning.html

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 <h2 id="discovery-of-ceres">Discovery of Ceres</h2>
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 <p>On New Year’s Eve in 1800, Giuseppe Piazzi, an Italian Priest, born in Lombardy, but installed in a new Observatory at the University viewed a faint smudge through his telescope.</p>
 <p>Piazzi was building a star catalogue.</p>
 <p>Unbeknownst to him, Piazzi was also participating in an international search. One that he’d been volunteered for by the Hungarian astronomer Franz von Zach. But without even knowing that he’d joined the search party, Piazzi had discovered their target a new planet.</p>
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 <h2 id="overdetermined-system">Overdetermined System</h2>
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 <p>The challenge with a linear model is that it has two unknowns, <span class="math inline"><em>m</em></span>, and <span class="math inline"><em>c</em></span>. Observing data allows us to write down a system of simultaneous linear equations. So, for example if we observe two data points, the first with the input value, <span class="math inline">$\inputScalar_1 = 1$</span> and the output value, <span class="math inline">$\dataScalar_1 =3$</span> and a second data point, <span class="math inline">$\inputScalar = 3$</span>, <span class="math inline">$\dataScalar=1$</span>, then we can write two simultaneous linear equations of the form.</p>
 <p>point 1: <span class="math inline">$\inputScalar = 1$</span>, <span class="math inline">$\dataScalar=3$</span> <br /><span class="math display">3 = <em>m</em> + <em>c</em></span><br /> point 2: <span class="math inline">$\inputScalar = 3$</span>, <span class="math inline">$\dataScalar=1$</span> <br /><span class="math display">1 = 3<em>m</em> + <em>c</em></span><br /></p>
 <p>The solution to these two simultaneous equations can be represented graphically as</p>
@@ -310,7 +310,7 @@ <h2 id="overdetermined-system">Overdetermined System</h2>
 <p>This is known as an <em>overdetermined</em> system because there are more data than we need to determine our parameters. The problem arises because the model is a simplification of the real world, and the data we observe is therefore inconsistent with our model.</p>
 <p>The solution was proposed by Pierre-Simon Laplace. His idea was to accept that the model was an incomplete representation of the real world, and the manner in which it was incomplete is <em>unknown</em>. His idea was that such unknowns could be dealt with through probability.</p>
 <h3 id="pierre-simon-laplace">Pierre-Simon Laplace</h3>
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 <p>Laplace’s trick has converted the <em>overdetermined</em> system into an <em>underdetermined</em> system. He has now added three variables, <span class="math inline">$\{\noiseScalar_i\}_{i=1}^3$</span>, which represent the unknown corruptions of the real world. Laplace’s idea is that we should represent that unknown corruption with a <em>probability distribution</em>.</p>
 <h2 id="a-probabilistic-process">A Probabilistic Process</h2>
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 <p>However, it was left to an admirer of Gauss to develop a practical probability density for that purpose. It was Carl Friederich Gauss who suggested that the <em>Gaussian</em> density (which at the time was unnamed!) should be used to represent this error.</p>
 <p>The result is a <em>noisy</em> function, a function which has a deterministic part, and a stochastic part. This type of function is sometimes known as a probabilistic or stochastic process, to distinguish it from a deterministic process.</p>
 <h2 id="hydrodynamica">Hydrodynamica</h2>
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 <p>When Laplace spoke of the curve of a simple molecule of air, he may well have been thinking of Daniel Bernoulli (1700-1782). Daniel Bernoulli was one name in a prodigious family. His father and brother were both mathematicians. Daniel’s main work was known as “Hydrodynamica”.</p>
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 <h2 id="data-fit-term">Data Fit Term</h2>
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