![\"Black](\"https://github.com/JamesSample/enviro_mod_notes/blob/master/images/Black_Box_Model.png?raw=true\")
\n",
"\n",
"As before, this model could be anything (e.g. a complex hydrological model), but for this example we're going to assume it simply implements the calculation $y = \\alpha x + \\beta$. In other words, the user chooses values for $\\alpha$ and $\\beta$ by turning the dials and then supplies as input a vector of data, $x = \\{x_1, .., x_n\\}$. After pressing the \"run\" button, the model returns a vector of output values, $y = \\{y_1, .., y_n\\}$, calculated as $y_i = \\alpha x_i + \\beta$.\n",
"\n",
"We want to find estimates for the parameters $\\alpha$ and $\\beta$ that give the best fit to the test data. Just as importantly, we want to understand the **uncertainty** in these estimates, i.e. how much **confidence** should we have that the fitted values are the right ones?\n",
"\n",
"We'll start by defining a simple function to represent our deterministic model."
]
},
{
"cell_type": "code",
"execution_count": 22,
"metadata": {
"collapsed": false
},
"outputs": [],
"source": [
"def deterministic_model(x, params):\n",
" \"\"\" x is a 1D vector of x values\n",
" params is the vector of parameters [alpha, beta]\n",
" \n",
" Returns alpha*x + beta\n",
" \"\"\"\n",
" # Extract parameters\n",
" alpha, beta = params\n",
" \n",
" # Linear model\n",
" return alpha*x + beta"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### 7.3. Define likelihood\n",
"\n",
"Next we need to define our likelihood function. To do this we must think about the error structure we expect between our model and the observations. In this example we will assume independent and identically distributed Gaussian errors with mean 0 and standard deviation, $\\sigma_\\epsilon$. We know that this is the correct error structure in this case because we generated the test data above using exactly these assumptions. In the real-world, however, deciding on the error structure and devising the likelihood function are often the most difficult parts of the calibration exercise.\n",
"\n",
"As described in notebook 2, in order to avoid numerical errors in our code it is best to consider the **log likelihood** and the **log prior**. Also recall that we only need to think about quantities that are **proportional to** the distributions of interest i.e. we only need to evaluate them to within a multiplicative constant. \n",
"\n",
"The log likelihood function in this case is exactly the same as the one described in detail in section 2.3 of notebook 2."
]
},
{
"cell_type": "code",
"execution_count": 23,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"def log_likelihood(x, params, obs):\n",
" \"\"\" x is a 1D vector of x values\n",
" params is a vector of parameter estimates [alpha, beta, sigma_e]\n",
" obs is the observed dataset used for calibration\n",
" \n",
" Returns the log likelihood assuming iid Gaussian errors\n",
" \"\"\"\n",
" # Get number of value pairs\n",
" n = len(obs)\n",
" \n",
" # Extract parameter values\n",
" alpha, beta, sigma_e = params\n",
" \n",
" # Calculate deterministic model results with these parameters\n",
" sim = deterministic_model(x, [alpha, beta])\n",
" \n",
" # Calculate log likelihood (see Section 2 of notebook 2 for explanation)\n",
" ll = -n*np.log(2*np.pi*sigma_e**2)/2 - np.sum(((obs - sim)**2)/(2*sigma_e**2))\n",
" \n",
" return ll"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### 7.4. Define priors\n",
"\n",
"To take a Bayesian approach, we next need to define our **prior beliefs** about the parameters $\\alpha$ and $\\beta$, as well as about the standard deviation of the errors, $\\sigma_\\epsilon$. We'll assume, as in the illustration of the model above, that we have reason to believe that $\\alpha$ and $\\beta$ are both continuous variables lying between 0 and 12. Because we have no reason to believe that any particular values within this range are more likely than others, we will define our **priors** as **uniform distributions** over this interval\n",
"\n",
"$$\\alpha \\sim \\mathcal{U}(0, 12) \\qquad \\beta \\sim \\mathcal{U}(0, 12)$$\n",
"\n",
"We won't make any assumptions about $\\sigma_\\epsilon$, other than it **must be positive** (negative values for the standard deviation don't make any sense). All positive values for $\\sigma_\\epsilon$ will be considered equally likely.\n",
"\n",
"Because we are only interested in values that are proportional to the probability density of interest, for $\\alpha$ and $\\beta$ we can simply associate any values between 0 and 12 with a constant (e.g. zero). Values outside this range have zero probability, which on a log scale becomes $\\log(0) = -\\infty$"
]
},
{
"cell_type": "code",
"execution_count": 24,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"def log_prior(params):\n",
" \"\"\" params is the vector of parameters [alpha, beta]\n",
" \n",
" Return the log prior probability of p\n",
" \"\"\"\n",
" alpha, beta, sigma_e = params\n",
" \n",
" # If all parameters are within allowed ranges, return a constant \n",
" # (anything will do - I've used 0 here)\n",
" if ((0 <= alpha < 12) and (0 <= beta < 12) and (0 <= sigma_e)):\n",
" return 0\n",
" # Else the prior probability = 0, and the log prob = -inf\n",
" else:\n",
" return -np.inf"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### 7.5. Define posterior\n",
"\n",
"The **posterior distribution** (to within a multiplicative constant) is just the prior multipled by the likelihood, as described in notebook 3. The **log posterior** is therefore the **sum** of the log prior and the log likelihood. \n",
"\n",
"Note that we can save ourselves a bit of computational time by *only* evaluating the likelihood when the prior probability of the parameter set is not zero (i.e. the log prior is finite). If the prior probability is zero then that parameter set is deemed to be impossible, regardless of how well the model results fit the data: the log prior will be $-\\infty$, so no matter how big the likelihood is the log posterior will still be $-\\infty$. In this example, evaluating the likelihood (i.e. running the model) is very fast, so we don't need to worry about performance too much. However, when our models become more complex and computationally expensive it is useful to run them only when absolutely necessary.\n",
"\n",
"As an aside, note also that for the reasons stated above you should be careful when assigning zero prior probability to particular parameter combinations. These parameter sets will be considered invalid by the algorithm, so you need to be certain that they *really are impossible*, rather than just very unlikely. If in doubt, it's better to assign very small but non-zero probabilities to these combinations, for example by using broad Gaussians for your priors rather than uniform distributions."
]
},
{
"cell_type": "code",
"execution_count": 25,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"def log_posterior(x, params, obs):\n",
" \"\"\" x is a 1D vector of x values\n",
" params is a vector of parameter estimates [alpha, beta, sigma_e]\n",
" obs is the observed dataset used for calibration\n",
" \n",
" Returns the log posterior\n",
" \"\"\"\n",
" # Extract parameter values\n",
" alpha, beta, sigma_e = params\n",
" \n",
" # Get log prior prob\n",
" log_pri = log_prior(params)\n",
"\n",
" # Evaluate log likelihood if necessary\n",
" if np.isfinite(log_pri):\n",
" log_like = log_likelihood(x, params, obs)\n",
" \n",
" # Calculate log posterior\n",
" return log_pri + log_like\n",
" \n",
" else:\n",
" # Log prior is -inf, so log posterior is -inf too\n",
" return -np.inf"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### 7.6. Metropolis algorithm\n",
"\n",
"Next, we need a version of the metropolis algorithm capable of working with this multi-dimensional problem. The model we're fitting has three parameters: $\\alpha$, $\\beta$ and $\\sigma_\\epsilon$ (the latter is necessary because it determines the spread of the errors, so although we're only really interested in $\\alpha$ and $\\beta$, we need to estimate $\\sigma_\\epsilon$ as well).\n",
"\n",
"Our Metropolis algorithm must therefore make jumps within this three dimensional parameter space, rather than the 1D space in the examples above. To do this, we'll define a three-dimensional **Gaussian jump distribution**, which involves specifying three additional constants for the jump distribution variances. We will call these variances ${\\sigma_\\alpha}^2$, ${\\sigma_\\beta}^2$ and ${\\sigma_\\sigma}^2$, but it is important not to get confused between these values and the parameters we're trying to estimate ($\\alpha$, $\\beta$ and $\\sigma_\\epsilon$). \n",
"\n",
"$\\sigma_\\alpha$, $\\sigma_\\beta$ and $\\sigma_\\sigma$ are **properties of the Metropolis algorithm**. In theory, the algorithm will converge regardless of the values choosen but, as described above, for efficiency reasons we should aim to **tune** them to achieve acceptance rates of around 20 to 50%.\n",
"\n",
"We will further assume that $\\sigma_\\alpha$, $\\sigma_\\beta$ and $\\sigma_\\sigma$ are independent of one another i.e. the covariance matrix looks something like this (see section 1.3 of notebook 1 for a reminder about covariance matrices)\n",
"\n",
"| |$\\alpha$|$\\beta$|$\\sigma_\\epsilon$|\n",
"|:-:|:-:|:-:|:-:|\n",
"|$\\alpha$|${\\sigma_\\alpha}^2$|0|0|\n",
"|$\\beta$|0|${\\sigma_\\beta}^2$|0|\n",
"|$\\sigma_\\epsilon$|0|0|${\\sigma_\\sigma}^2$|\n",
"\n",
"Finally, it's worth noting that because we're working with log probabilities we need to re-write the code for calculating **acceptance**. As described above in section 3, to determine the acceptance probability we calculate the ratio of the posterior probability of the proposed point to the posterior probability of the current location\n",
"\n",
"$$P_{accept} = \\frac{P(x_{i+1})}{P(x_i)}$$\n",
"\n",
"However, we are currently working with $\\log[P(x_{i+1})]$ and $\\log[P(x_i)]$. From a simple rearrangement of the logarithms, you should be able to see that \n",
"\n",
"$$P_{accept} = exp(\\log[P(x_{i+1})] - \\log[P(x_i)])$$\n",
"\n",
"as used in the code below."
]
},
{
"cell_type": "code",
"execution_count": 26,
"metadata": {
"collapsed": false
},
"outputs": [],
"source": [
"def metropolis_3d(start, jump_sigma, n):\n",
" \"\"\" start is the starting location for the chain as [alpha, beta, sigma_e]\n",
" jump_sigma gives the std. devs. for the jump distribution [sigma_alpha, sigma_beta, sigma_sigma]\n",
" n is the number of steps to take\n",
" \n",
" Returns the chain and acceptance rate\n",
" \"\"\"\n",
" # Extract jump covariances\n",
" sigma_alpha, sigma_beta, sigma_sigma = jump_sigma\n",
" \n",
" # Build covariance matrix\n",
" cov = [[sigma_alpha**2, 0, 0], \n",
" [0, sigma_beta**2, 0], \n",
" [0, 0, sigma_sigma**2]]\n",
" \n",
" # Counter for number of accepted jumps\n",
" n_accept = 0.\n",
"\n",
" # Create a chain and add start position to it\n",
" chain = np.array([start,])\n",
"\n",
" # Draw n random samples from Gaussian proposal distribution to use for \n",
" # determining step size\n",
" jumps = np.random.multivariate_normal(mean=[0,0,0],\n",
" cov=cov, \n",
" size=n)\n",
"\n",
" # Draw n random samples from U(0, 1) for determining acceptance\n",
" y_i = np.random.uniform(low=0, high=1, size=n)\n",
" \n",
" # Random walk\n",
" for step in xrange(1, n):\n",
" # Get current position of chain\n",
" cur = chain[-1]\n",
" \n",
" # Get a candidate for x_i+1 based on x_i and jump\n",
" cand = cur + jumps[step]\n",
" \n",
" # Calculate acccept prob\n",
" acc_prob = np.exp(log_posterior(x, cand, obs_y) - log_posterior(x, cur, obs_y))\n",
" \n",
" # Accept candidate if y_i <= alpha\n",
" if y_i[step] <= acc_prob:\n",
" cur = cand\n",
" n_accept += 1\n",
" chain = np.append(chain, [cur,], axis=0)\n",
"\n",
" # Acceptance rate\n",
" acc_rate = (100*n_accept/n)\n",
"\n",
" return [chain, acc_rate]"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### 7.7. Choose a starting location\n",
"\n",
"We're now all set to run the calibration algorithm, but first we need to make some choices regarding where to start our Markov chain. As we saw in the example above, if we start from a region of low probability density the chain will quickly move \"uphill\", so for a simple problem like this one we don't need to worry too much: we can choose almost any starting location and the algorithm should still converge quickly. However, for more complex problems in high-dimensional parameter spaces, it's a good idea to find a starting location with high probability density as this will reduce the burn-in time required for the algorithm to converge. One simple way to do this is to use an optimiser to search for the **[Maximum a Posteriori (MAP)](https://en.wikipedia.org/wiki/Maximum_a_posteriori_estimation)** estimate for the parameters and start the chain from there.\n",
"\n",
"The MAP is very similar to the **[Maximum Likelihood Estimate (MLE)](https://en.wikipedia.org/wiki/Maximum_likelihood)** that was introduced in section 2.3 of notebook 2. The MLE is simply the maximum of the likelihood function, whereas the MAP is the maximum of the *posterior distribution* (i.e. the likelihood multiplied by the prior). Although conceptually similar, the MLE and MAP are not necessarily in the same place.\n",
"\n",
"In notebook 2 we wrote down a function for the negative likelihood and then used `scipy.optimize` to minimise this, which is the same as maximising the likelihood. In the sections above we've already written down a function for the log posterior, so if we now write down the negative log posterior we can use exactly the same approach to find the MAP."
]
},
{
"cell_type": "code",
"execution_count": 27,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Optimization terminated successfully.\n",
" Current function value: 295.100848\n",
" Iterations: 87\n",
" Function evaluations: 157\n",
"\n",
"\n",
"Estimated alpha: 7.20. True value 7.00\n",
"Estimated beta: 2.26. True value 3.00\n",
"Estimated error sigma: 4.63. True value 5.00\n"
]
}
],
"source": [
"from scipy import optimize\n",
"\n",
"def neg_log_posterior(params, x, obs):\n",
" \"\"\" Maximising the log posterior is the same as minimising the negative log\n",
" posterior.\n",
" \"\"\"\n",
" return -log_posterior(x, params, obs)\n",
"\n",
"# Guess some starting values for [alpha, beta, sigma_e]\n",
"param_guess = [6., 6., 1.]\n",
"\n",
"# Run optimiser\n",
"param_est = optimize.fmin(neg_log_posterior, param_guess, args=(x, obs_y))\n",
"\n",
"# Print results\n",
"print '\\n'\n",
"print 'Estimated alpha: %.2f. True value %.2f' % (param_est[0], alpha_true) \n",
"print 'Estimated beta: %.2f. True value %.2f' % (param_est[1], beta_true) \n",
"print 'Estimated error sigma: %.2f. True value %.2f' % (param_est[2], sigma_e_true)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"You can see that the very simple code above does a pretty good job of identifying the true values for $\\alpha$, $\\beta$ and $\\sigma_\\epsilon$, **without requiring any MCMC at all**. However, the optimiser only gives a single point estimate for each parameter - it does not give any indication of **uncertainty**, which is important in real-world applications where we don't have the true values to compare to. For example, in the real world, the optimiser tells us the best estimate for $\\alpha$, but we don't know whether this estimate is well-constrained by the data. Perhaps the model is largely insensitive to $\\alpha$, so the value chosen doesn't matter much? Or perhaps $\\alpha$ is a very important parameter indeed? \n",
"\n",
"To move beyond simple point estimates to something more useful, we need to characterise the *entire posterior distribution*, not just look for the location of the highest peak. This is where MCMC comes in, but because the point estimate from the optimiser is easy to calculate it can provide a good starting point for our Markov chain.\n",
"\n",
"### 7.8. Run the MCMC\n",
"\n",
"We can now run the Metropolis algorithm using the starting values estimated above. For this simple, non-adaptive algorithm, it's likely we will need to do this several times, experimenting with the values of $\\sigma_\\alpha$, $\\sigma_\\beta$ and $\\sigma_\\sigma$ until the accepatnce rate is between about 20% and 50%.\n",
"\n",
"Finally, before running the calibration algorithm, it worth downloading and installing **[corner plot](https://github.com/dfm/corner.py)**, which is a very useful package for visualising the output from MCMC analyses. "
]
},
{
"cell_type": "code",
"execution_count": 28,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Acceptance rate: 62.14%\n"
]
},
{
"data": {
"image/png": 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fp/vmv5oaawfzX35pnT9T7GrfqafqejCYWO6VV8bWt/vu2bdxINqfyV8kGMUd\nd1hfX7Iktk0RZ/26rutOZ+qy8/ksye79/ffcy21sNO51ufrT9t8/++/l5ZeH1ncIuv7LL5n14Smn\nWLd3zJj05edCd7euz50bm3bppbFlRwJmWD1vkb8ZMzLr57/+ss737be6fu+96dv79dfWn/+886zr\ns/r74YeB//4L9XfDDYWvw+3O/d4PP8yvbn0IBRkJczeGynzfdEZpGXPcceiTJ+cVJ7T1hRkUP/4I\nxU9Np3fUaFotHMy7GtuizuHNZBqQoPjm2ynPQj2blJtvpnXsNvTuYOxnj5qNnHsuXXIerllfE9hp\nV4rm/xL9wprm/4leXmFZXKZBRgoRNMF39nl0738gtZtvGJPe2BR2/LHbvtRzYsJ9bS/NJDBxr2ib\n2jq6CTR2UHL91ZSmmf2aAxVkS98aYyx/BC2/LSJYVJlTub7rb8Z/+tnUBYPRZ7inu5f2DL4T53dz\nqAkft7X5jYAFKShk4Asrek48KSEYhu/s8yiNz3jLLQSfepqWb/uXe7T2NurmpfYZ3lhaC5GAPdkG\ny6kbHb0XwDNqVcwmYEE0Ijq2lu9+Jri2O6H/Wtu76bUKqmI67jpsEp3O0pi6oqy4Ouy9uvU1E66W\nTst3j7+zG1+Gz3NjzagB//4LReNhx1J/+umFrSSPbV6Na64/pPo6ryAjQoiNgaOA9YG3hBBvCyH2\nsaNhmp7fGCBUU4vv7PMIbLUNnddYr8doFr6Ss9lf2X3IYQmGMrkS8V4V77PY+/D9uL/+itJrr8T1\nQ/8+Zs9L/7OlXrsJjlmT0MqrJM9QWkr7bXcnJFce9c+Yc93hwHv37ZRee2XaOuMds2SD6xdrf/Ga\nr9MyPSM0DfdHH2T9DGtNTdSM3zKmnKFGkcVWzmTfkfOvhcZBXx9lp02hcr/MnFvYRWDPfeiZ2O8o\nI9oeoGb8ltbONjJZOrBhW1moLokYiHtm2u/K3j+CnQS22obgaqvnVUbTj6m3hS19+fWU1+PxT04c\n+BeKnp13peOG/4Cm0bvZ/w1YvemwI8hIavPCwcLlJLTyKrT9N8XaosWLtePeB7Oqpu3ZF6nZxr4v\nNObFnQLXnFlkvxNzAAiFEl5+XQfHCuWkLz7TzNo961tKr0xuZxBTXIYGRdlQctsttN+fNkKtNZpG\n1T6xVsaZGJpFbDsipPMHncnaf650HTYJ76MP2VKW+7NP8D6eY1/mg6bRM3GvpK5OPR99kJCmdSYO\n2OItzfUxlnvpAAAgAElEQVQs3ftaEVpxlPWF8DspVFZOcI0x9Ox3AIHHH83Jh4EdtP33ZarHbZJX\nGXqVlW6hn77Nt8iyxIEb0HZefQOh0SsbJwNozJuOwfVVXkD0TJwyWLwYg6NGZ1VPKM1DmQ0pfYEn\nZM4/ylEy6928CAvfpnl/0PzZtzT+0UTnLbHbUSz30wKOpv4XZKZCG8DzZnYj9kzIxZd6hBKLfcRF\nr7yYNhqW+5OPYs6tAtG4vviM2nVWx/XNV9RM2DrnNqbDd/7FdNjkY9udgXvMgpFm21I8Ed/uzu/n\nUjVhG2rXWZ269cbE5PGfXDiVbt/6hhFU84+/0fq64TCl4+bk0e6yNi7MhQJqfvQcgk4NqCbK3L8F\njviVDcus4M7kB9s7dkuCq6wWPe+46TaC666XXT02emMqeubJzKvNQz0cpQCxlPs2NrZ36eUVhFZb\n3TL4SGCnJFupclweKZR1fP0IaxuCdCTzzOT89ZeU97njgqNUTJlM0ROPx/RL+UnH42hupjQHP85Z\noWl0Tzoa36l5WiaHQpTenOhwyYrA1tvmV5cVOTxTWmcHNduNxT372xjXldEi6+1Z7Vw68x18/z4/\net5x9Q30RPZwu1xRARVaeRVan7d2emPew14wCjg4MHuTyxgbNB4ZYxok+M4+b+DqTcMyK7gzcpno\ndtM2vd/lZ/ehOYSadCUOEHxTz6T97uSO5UM1NZbpziTrrVZ44mZnWaPrMRFzzKFTc6Vt2tME18zA\n/elA/vCGEklegKVXXELJtVdaDgIrTj4Bzxv92+gi67RaR+4BHDIiLPACu+fsTwlIHq0ppiqHg9Yn\n/0vbo5kPXDMluEISlXQKav4vSehem+nbaBP8U8+KngfGT0g64ejdcmtjq9xrb9N238MsnWHYCvdu\ntrllfjsIVVUB0HXM8UnztN+T2dJi07w/aJpr8X7LYVDQs9seBMZmtqSYN6b2RYyHhwLLnODu3m9/\nGv9oShv/NkK+zip0Z+LL1v/vC+hbK7kauvl765lXUdjxTKZo7W1Z5TdTuc/uMb6Euw/JYdASR2DH\nDJySpCIH/8nDiiQv5ZJbbjAC1CS5HuPhLOyUyP156lC4EfrGrJldGyOEBXe8K89s8byS3id32/Rn\nDVfHpQm26XnTN3ac4cI2C8xLNgNKGhVw9yGH0bfxpgT22pe+TQ2B3bNP4WbcveOMpZjuI6zjmLe8\n8zE9+/w9qTbAjF5egV5XR3CluKXILAV386ff0Lfp5rQ999KAGO4l83A32Cxzgrv4uWeymtHl+2LC\n6405NVuxWhEYu6VtazQOC3/ZztmzwCIOrta6NCbWrCcuBGlohSyCOhQIO10ztj6V+xp1wUjzkkoa\nBMOk7s02YlLA7Ps/GyK/oTy1I2WXX5w+U4HXLFuztFoeLJKFDE6JHV03cSKBcVslJOvF4Xdjsu8n\nnN67ZXJbi3gL8KXvfkzLe/2DTj3JYDWZAWAo4tLW5aJnvwOS1msbcW5y7Y7qlitpBbcQ4ojwNq+3\nhRCfCCG6hBAVput7CiE+E0J8JIQ4prDNLQBZGq+ku99KddS7sWGV6Zt6Fh0ZqpYyId47k+O3Xw2D\npS0SZxh1DatSu/G6ScvqOvgw29qVK97HM/d8lm4W1bv9Dvk2x3bKzj875ZprbwHUf3plVfS449qb\nCCXZ+2+m7cHHo0JETxdsxQaCuWoFlhGav5hN6xPPWnqES0sOcQvM6+qtz78CL75I2wszaHn7I7qO\nOra/aJP9jyUZDLji1ex6RSXBtdfpT7B4/3YdfhRNvyYGkmpc3FbQ9XYzwdVWZ+mMN9FrYpcQm+fO\no+nn32ma83NMeu8mia6b9eJiWj75ythOZjP5BhlxAzcCOwHbAccJIbK2NsjWurnl3U9izq18YA8G\nvRtt3L9OaXpBt772Do2L2/D/+3xCK+QWkcmK2rEbU/TsU9FzZyTcaZIAGFGsZm1lZQTiRs6RAYdl\nEXHr9E3z/kiS05qWeH/TgOZPbXVtpvvAQ9LmKZSbQv8JuQUq8Lz1RqL/eRNJ7TJ0HdeXn1Ny47VZ\n1dd9wEExe3B7t9mW5h9/o/nz1NvIArvsZmp04e0RUu77Xw4IrbJqzuunVj73U9H2wGOEKvoHb+aZ\nbXC99em8+ob+tHSGfRkI7pAp6IslpqXGnvBz13XkMdZGvxlqZrr//o/ocfM33yfN13HNjfRu8Dda\n3v8s4VpoxMjockQMXi96RSX6iH4x17iwhdZX307I2jt2S4JrrEn3YZMyanc25BtkZB3gZyllm5Sy\nF/gAw8taVoRq67JSQYTinJ50XHV99GHL2vLPY9+2KrPBUExge0h46DIRPJlQcYJJyZHkh1a93bj+\nk64uih+ONZxrfeK5hDa233IHbc++yNI33rMss+PmuC1eGczkzATXXY/WVHvsTVg+GxlYCy99830a\nF+TunzietmlP03nxFfjOvzj3Qszt7umJ2QLokkleMrpO9W4T0lqS9+yxV+z5LrvF7KENVVSB09m/\nL9WCtgcfj3lpmuOLt74wI3rcfutdBLbaJmV7ktH8yddGPcOQlhR+1AcFi50h/hNPTp7fWxwbvzxo\nMYiPvAdMA/ylLyUuN2RkiJoGs6q8/dEnaVzUSjAcwzxiHBfYetsY9XoyWp9/hd5NN8d3wSW0Pfg4\nXZOOTrpfvuuQw+g+4iha33yfoFibxiVxy4u9qWN2g+FXv+2+h23dWZQpmp7hdgkhxHPALVLKd01p\nWwNTpJQHhc8vARZIKZNaDaym/ZpYoacIvbwcrTlx32qE0MgVcITjcIdWGo3W0oLW5Qc0QisnfxFl\nRE+P0fk5qs0dfy2Evj70khL02rC6q7cXx6K/QNOsX5R9fcZ9NhAavTJac3O4P8JpplmM43fzWrgG\ncYOKSF7HkiXQ0w2eomgIRcDoa18nFBVDTze614teUxeNcZvPd+BYuBBSzRqcTkL1I3As+gu9pDQ6\nKw+tNDpmzT6emM+/eDEEDMM3vboabenSnNoaGrlidKAX26dZlLHCitFY31pzE5rfn+YOw4GF1pq6\nzXp1DXpxccwzpdfUopeU4vjDaGto9MrRl3JM+52uqHOJULzxUCgU7efQqJWigX9CK6+C1tqaNApY\nKkIrjjJ+b11dxv8k6niHw0HIpr2zuX5f8Qw57YCu44ho2gA0Db2yKunzEqofgdbXFw3iE6ofgaOk\nJKaftaZGtK4u4/kxGQzG9KHmIDS6/1mJvAMT6kvSX1rrUrSODvSSUvRkO1r6+tD8fmNSkGSyHWlT\nfFut8qRrlzmfXl6BXlWVkCcVWns7Wpsp5G1RMaHwzHzBAoethhwZDRWEEFVAg1loh2kDzBYV5UBm\nb8WVV4bFiyEQgEAPWtmKkEJwO0wzY4dDgxEjIBQETcORibOVVMQZmGVNWA5qmoYWWYMpKoKaWijy\n4LBal/F4oLYWLPaJZoujuwu6YgWAZZ3mxlrldfb/j7m/rg7qaon8erTI0ahR0NkJZeUp6ktDusc5\nGMRRVASrrmZk/c0Q3I7IC9/sf9jpig4CYtpTUwOLDMtszWIXQKY4nFriGpvXa/TP779b3xRfxqK/\njO+9rBwyENqQmYZQq6hIUJt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n4UVal6dCiGuA8Rhq9XMwVORlUsp7w6rxqzGsyj+UUp5W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\nL1smbEvpVq+2l3fa7UT6b/jw7K/t3z+9HY2mt2fMSKdfvVp9TSpl7z4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+xK7/9VctO24z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\nvX44Vm/E4T21OgDlw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"\n", " \n", " 2. The distribution of errors on the right-hand plot is approximately Gaussian with mean zero.\n", " \n", "## 8. Summary\n", "\n", "1. MCMC techniques offer **better performance** than MC approaches, because the Markov property enables the algorithms to seek out high probability regions, while still exploring low probability areas in proportion to their relative density.
\n", "\n", "2. The simplest MCMC algorithm is the **Metropolis algorithm**, which is guaranteed to converge on the posterior (but not necessarily in a reasonable time frame).
\n", "\n", "3. As a rule of thumb, it is usually a good idea to aim for **acceptance rates** of between about **20% and 50%**.
\n", "\n", "4. MCMC algorithms require, as a minimum, a **starting point** and a **proposal distribution**. The starting point is often best chosen by searching for a maximum in the posterior distribution i.e. the **MAP estimate**. Sampling will be most efficient when the **proposal distribution is similar to the target distribution**.
\n", "\n", "5. More sophisticated algorithms achieve better sampling efficiencies by using **multiple chains** and/or **adapting the proposal distribution** as the chain(s) evolve(s).
\n", "\n", "6. Ready-implemented, state-of-the-art MCMC samplers are available in a variety of languages (e.g. Python, R, Julia etc.). The hardest part of a calibration exercise is therefore often **formulating an appropriate likelihood function**.
\n", "\n", "7. Having estimated parameters for a model using MCMC, it is important to **check that the assumptions of the likelihood function are valid**. This usually involves some kind of analysis of the **model residuals**. If the assumptions are not met, parameter estimates will be **biased** and you may need to repeat the exercise using a different likelihood function." ] } ], "metadata": { "kernelspec": { "display_name": "Python 2", "language": "python", "name": "python2" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 2 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython2", "version": "2.7.10" } }, "nbformat": 4, "nbformat_minor": 0 }