![](\"./figures/ml-and-scientific-loop-2.png\")
"
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"### Now Solve the Example Problem: Predicting a Coin Toss\n",
"\n",
"- We observe a the following sequence of heads (h) and tails (t) when tossing the same coin repeatedly $$D=\\{hthhtth\\}\\,.$$\n",
"\n",
"- What is the probability that heads (h) comes up next? We solve this in the next slides ..."
]
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"source": [
"### Coin toss example (1): Model Specification\n",
"\n",
"We observe a sequence of $N$ coin tosses $D=\\{x_1,\\ldots,x_N\\}$ with $n$ heads. "
]
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"##### Likelihood\n",
"- Assume a Bernoulli distributed variable $p(x_k=h|\\mu)=\\mu$, which leads to a **binomial** distribution for the likelihood (assume $n$ times heads were thrown):\n",
"$$ \n",
"p(D|\\mu) = \\prod_{k=1}^N p(x_k|\\mu) = \\mu^n (1-\\mu)^{N-n}\n",
"$$"
]
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"source": [
"##### Prior\n",
"- Assume the prior belief is governed by a **beta distribution**\n",
"$$\n",
"p(\\mu) = \\mathcal{B}(\\mu|\\alpha,\\beta) = \\frac{\\Gamma(\\alpha+\\beta)}{\\Gamma(\\alpha)\\Gamma(\\beta)} \\mu^{\\alpha-1}(1-\\mu)^{\\beta-1}\n",
"$$\n",
" "
]
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"- The Beta distribution is a **conjugate prior** for the Binomial distribution, which means that \n",
"$$\n",
"\\text{beta} \\propto \\text{binomial} \\times \\text{beta}\n",
"$$"
]
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"source": [
"- $\\alpha$ and $\\beta$ are called **hyperparameters**, since they parameterize the distribution for another parameter ($\\mu$). E.g., $\\alpha=\\beta=1$ (uniform).\n",
" "
]
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"source": [
"### Coin toss example (2): Parameter estimation\n",
"\n",
"- Infer posterior PDF over $\\mu$ through Bayes rule\n",
"\n",
"$$\\begin{align*}\n",
"p(\\mu|D) &\\propto p(D|\\mu)\\,p(\\mu|\\alpha,\\beta) \\\\ \n",
" &= \\mu^n (1-\\mu)^{N-n} \\times \\mu^{\\alpha-1} (1-\\mu)^{\\beta-1} \\\\\n",
" &= \\mu^{n+\\alpha-1} (1-\\mu)^{N-n+\\beta-1} \n",
"\\end{align*}$$\n",
"\n",
"hence the posterior is also beta distributed as\n",
"\n",
"$$\n",
"p(\\mu|D) = \\mathcal{B}(\\mu|\\,n+\\alpha, N-n+\\beta)\n",
"$$"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "fragment"
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"source": [
"- Essentially, **here ends the machine learning activity**"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "slide"
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"source": [
"### Coin Toss Example (3): Prediction\n",
"\n",
"- Now, we want to **use** the trained model. Let's use it to predict future observations. "
]
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"cell_type": "markdown",
"metadata": {
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"source": [
"- Marginalize over the parameter posterior to get the predictive PDF for a new coin toss $x_\\bullet$, given the data $D$,\n",
"\n",
"$$\\begin{align*}\n",
"p(x_\\bullet=h|D) &= \\int_0^1 p(x_\\bullet=h|\\mu)\\,p(\\mu|D) \\,\\mathrm{d}\\mu \\\\\n",
" &= \\int_0^1 \\mu \\times \\mathcal{B}(\\mu|\\,n+\\alpha, N-n+\\beta) \\,\\mathrm{d}\\mu \\\\\n",
" &= \\frac{n+\\alpha}{N+\\alpha+\\beta} \\qquad \\mbox{(a.k.a. Laplace rule)}\\hfill\n",
"\\end{align*}$$"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "fragment"
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"source": [
"- Finally, we're ready to solve our example problem: for $D=\\{hthhtth\\}$ and uniform prior ($\\alpha=\\beta=1$), we get\n",
"\n",
"$$ p(x_\\bullet=h|D)=\\frac{n+1}{N+2} = \\frac{4+1}{7+2} = \\frac{5}{9}$$\n",
" "
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "slide"
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},
"source": [
"### Coin Toss Example: What did we learn?\n",
"\n",
"- What did we learn from the data? Before seeing any data, we think that $$p(x_\\bullet=h)=\\left. p(x_\\bullet=h|D) \\right|_{n=N=0} = \\frac{\\alpha}{\\alpha + \\beta}\\,.$$ "
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "fragment"
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"source": [
"- After the $N$ coin tosses, we think that $p(x_\\bullet=h|D) = \\frac{n+\\alpha}{N+\\alpha+\\beta}$."
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "fragment"
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"source": [
"- Note the following decomposition\n",
"\n",
"$$\\begin{align*}\n",
" p(x_\\bullet=h|\\,D) &= \\frac{n+\\alpha}{N+\\alpha+\\beta} = \\frac{n}{N+\\alpha+\\beta} + \\frac{\\alpha}{N+\\alpha+\\beta} \\\\\n",
" &= \\frac{N}{N+\\alpha+\\beta}\\cdot \\frac{n}{N} + \\frac{\\alpha+\\beta}{N+\\alpha+\\beta} \\cdot \\frac{\\alpha}{\\alpha+\\beta} \\\\\n",
" &= \\underbrace{\\frac{\\alpha}{\\alpha+\\beta}}_{prior} + \\underbrace{\\frac{N}{N+\\alpha+\\beta}}_{gain}\\cdot \\big( \\underbrace{\\frac{n}{N}}_{MLE} - \\underbrace{\\frac{\\alpha}{\\alpha+\\beta}}_{prior} \\big)\n",
"\\end{align*}$$"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "fragment"
}
},
"source": [
"- Note that, since $0\\leq\\text{gain}\\lt 1$, the Bayesian estimate lies between prior and maximum likelihood estimate."
]
},
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"cell_type": "markdown",
"metadata": {
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"slide_type": "fragment"
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"source": [
"- For large $N$, the gain goes to $1$ and $p(x_\\bullet=h|D)$ goes to the maximum likelihood estimate (the relative frequency) $n/N$."
]
},
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"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "slide"
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"source": [
"#### CODE EXAMPLE\n",
"\n",
"**Bayesian evolution of $p(\\mu|D)$ for the coin toss**\n",
"\n",
"Let's see how $p(\\mu|D)$ evolves as we increase the number of coin tosses $N$. We'll use two different priors to demonstrate the effect of the prior on the posterior (set $N=0$ to inspect the prior)."
]
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"| Bayesian | Maximum Likelihood | |
| 1. Model Specification | Choose a model $m$ with data generating distribution $p(x|\\theta,m)$ and parameter prior $p(\\theta|m)$ | Choose a model $m$ with same data generating distribution $p(x|\\theta,m)$. No need for priors. |
| 2. Learning | use Bayes rule to find the parameter posterior,\n", "$$\n", "p(\\theta|D) = \\propto p(D|\\theta) p(\\theta)\n", "$$ | By Maximum Likelihood (ML) optimization,\n", "$$ \n", " \\hat \\theta = \\arg \\max_{\\theta} p(D |\\theta)\n", "$$ |
| 3. Prediction | $$\n", "p(x|D) = \\int p(x|\\theta) p(\\theta|D) \\,\\mathrm{d}\\theta\n", "$$ | \n", "$$ \n", " p(x|D) = p(x|\\hat\\theta)\n", "$$ |