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Remove find_MAP demonstration in the getting started tutorial #145

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Delete getting_started.ipynb
Aut0R3V Apr 21, 2021
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Aut0R3V Apr 21, 2021
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167 changes: 3 additions & 164 deletions examples/getting_started.ipynb
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Expand Up @@ -283,167 +283,6 @@
"Notice that, unlike for the priors of the model, the parameters for the normal distribution of `Y_obs` are not fixed values, but rather are the deterministic object `mu` and the stochastic `sigma`. This creates parent-child relationships between the likelihood and these two variables."
]
},
{
"cell_type": "markdown",
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"### Model fitting\n",
"\n",
"Having completely specified our model, the next step is to obtain posterior estimates for the unknown variables in the model. Ideally, we could calculate the posterior estimates analytically, but for most non-trivial models, this is not feasible. We will consider two approaches, whose appropriateness depends on the structure of the model and the goals of the analysis: finding the *maximum a posteriori* (MAP) point using optimization methods, and computing summaries based on samples drawn from the posterior distribution using Markov Chain Monte Carlo (MCMC) sampling methods.\n",
"\n",
"#### Maximum a posteriori methods\n",
"\n",
"The **maximum a posteriori (MAP)** estimate for a model, is the mode of the posterior distribution and is generally found using numerical optimization methods. This is often fast and easy to do, but only gives a point estimate for the parameters and can be biased if the mode isn't representative of the distribution. PyMC3 provides this functionality with the `find_MAP` function.\n",
"\n",
"Below we find the MAP for our original model. The MAP is returned as a parameter **point**, which is always represented by a Python dictionary of variable names to NumPy arrays of parameter values. "
]
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"\n",
" <div>\n",
" <style>\n",
" /* Turns off some styling */\n",
" progress {\n",
" /* gets rid of default border in Firefox and Opera. */\n",
" border: none;\n",
" /* Needs to be in here for Safari polyfill so background images work as expected. */\n",
" background-size: auto;\n",
" }\n",
" .progress-bar-interrupted, .progress-bar-interrupted::-webkit-progress-bar {\n",
" background: #F44336;\n",
" }\n",
" </style>\n",
" <progress value='18' class='' max='18' style='width:300px; height:20px; vertical-align: middle;'></progress>\n",
" 100.00% [18/18 00:00<00:00 logp = -157.69, ||grad|| = 14.913]\n",
" </div>\n",
" "
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"<IPython.core.display.HTML object>"
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"\n"
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"{'alpha': array(0.95724679),\n",
" 'beta': array([1.10071814, 2.9511438 ]),\n",
" 'sigma_log__': array(0.03540151),\n",
" 'sigma': array(1.0360356)}"
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"execution_count": 7,
"metadata": {},
"output_type": "execute_result"
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"source": [
"map_estimate = pm.find_MAP(model=basic_model)\n",
"map_estimate"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"By default, `find_MAP` uses the Broyden–Fletcher–Goldfarb–Shanno (BFGS) optimization algorithm to find the maximum of the log-posterior but also allows selection of other optimization algorithms from the `scipy.optimize` module. For example, below we use Powell's method to find the MAP."
]
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{
"cell_type": "code",
"execution_count": 8,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"\n",
" <div>\n",
" <style>\n",
" /* Turns off some styling */\n",
" progress {\n",
" /* gets rid of default border in Firefox and Opera. */\n",
" border: none;\n",
" /* Needs to be in here for Safari polyfill so background images work as expected. */\n",
" background-size: auto;\n",
" }\n",
" .progress-bar-interrupted, .progress-bar-interrupted::-webkit-progress-bar {\n",
" background: #F44336;\n",
" }\n",
" </style>\n",
" <progress value='214' class='' max='214' style='width:300px; height:20px; vertical-align: middle;'></progress>\n",
" 100.00% [214/214 00:01<00:00 logp = -156.45, ||grad|| = 0.24264]\n",
" </div>\n",
" "
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"text/plain": [
"<IPython.core.display.HTML object>"
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"text": [
"/Users/CloudChaoszero/opt/anaconda3/envs/pymc3-dev-py38/lib/python3.8/site-packages/scipy/optimize/_minimize.py:519: RuntimeWarning: Method powell does not use gradient information (jac).\n",
" warn('Method %s does not use gradient information (jac).' % method,\n"
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{
"name": "stdout",
"output_type": "stream",
"text": [
"\n"
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"text/plain": [
"{'alpha': array(0.95835519),\n",
" 'beta': array([1.1017629 , 2.95394057]),\n",
" 'sigma_log__': array(0.03638187),\n",
" 'sigma': array(1.03705179)}"
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"execution_count": 8,
"metadata": {},
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"source": [
"map_estimate = pm.find_MAP(model=basic_model, method=\"powell\")\n",
"map_estimate"
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"cell_type": "markdown",
"metadata": {},
"source": [
"It is important to note that the MAP estimate is not always reasonable, especially if the mode is at an extreme. This can be a subtle issue; with high dimensional posteriors, one can have areas of extremely high density but low total probability because the volume is very small. This will often occur in hierarchical models with the variance parameter for the random effect. If the individual group means are all the same, the posterior will have near infinite density if the scale parameter for the group means is almost zero, even though the probability of such a small scale parameter will be small since the group means must be extremely close together. \n",
"\n",
"Most techniques for finding the MAP estimate also only find a *local* optimum (which is often good enough), but can fail badly for multimodal posteriors if the different modes are meaningfully different.\n",
"\n",
"In summary, while PyMC3 provides the function `find_MAP()`, at this point mostly for historical reasons, this function is of little use in most scenarios. If you want a point estimate you should get it from the posterior. In the next section we will see how to get a posterior using sampling methods."
]
},
{
"cell_type": "markdown",
"metadata": {},
Expand Down Expand Up @@ -3811,9 +3650,9 @@
"metadata": {
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"kernelspec": {
"display_name": "Python PyMC3 (Dev)",
"display_name": "Python 3",
"language": "python",
"name": "pymc3-dev-py38"
"name": "python3"
},
"language_info": {
"codemirror_mode": {
Expand All @@ -3825,7 +3664,7 @@
"name": "python",
"nbconvert_exporter": "python",
"pygments_lexer": "ipython3",
"version": "3.8.6"
"version": "3.7.6"
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"nbformat": 4,
Expand Down