![](\"files/images/bayesnet.png\")
\n",
"\n",
"The alarm node can be made as follows: "
]
},
{
"cell_type": "code",
"execution_count": 17,
"metadata": {},
"outputs": [],
"source": [
"alarm_node = BayesNode('Alarm', ['Burglary', 'Earthquake'], \n",
" {(True, True): 0.95,(True, False): 0.94, (False, True): 0.29, (False, False): 0.001})"
]
},
{
"cell_type": "code",
"execution_count": 18,
"metadata": {},
"outputs": [],
"source": [
"john_node = BayesNode('JohnCalls', ['Alarm'], {True: 0.90, False: 0.05})\n",
"mary_node = BayesNode('MaryCalls', 'Alarm', {(True, ): 0.70, (False, ): 0.01}) # Using string for parents.\n",
"# Equivalant to john_node definition."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"The general format used for the alarm node always holds. For nodes with no parents we can also use. "
]
},
{
"cell_type": "code",
"execution_count": 19,
"metadata": {},
"outputs": [],
"source": [
"burglary_node = BayesNode('Burglary', '', 0.001)\n",
"earthquake_node = BayesNode('Earthquake', '', 0.002)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"It is possible to use the node for lookup function using the **p** method. The method takes in two arguments **value** and **event**. Event must be a dict of the type {variable:values, ..} The value corresponds to the value of the variable we are interested in (False or True).The method returns the conditional probability **P(X=value | parents=parent_values)**, where parent_values are the values of parents in event. (event must assign each parent a value.)"
]
},
{
"cell_type": "code",
"execution_count": 20,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"0.09999999999999998"
]
},
"execution_count": 20,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"john_node.p(False, {'Alarm': True, 'Burglary': True}) # P(JohnCalls=False | Alarm=True)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"With all the information about nodes present it is possible to construct a Bayes Network using **BayesNet**. The **BayesNet** class does not take in nodes as input but instead takes a list of **node_specs**. An entry in **node_specs** is a tuple of the parameters we use to construct a **BayesNode** namely **(X, parents, cpt)**. **node_specs** must be ordered with parents before children."
]
},
{
"cell_type": "code",
"execution_count": 21,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"\n",
"\n",
"\n",
"\n",
" class BayesNet:\n",
" """Bayesian network containing only boolean-variable nodes."""\n",
"\n",
" def __init__(self, node_specs=None):\n",
" """Nodes must be ordered with parents before children."""\n",
" self.nodes = []\n",
" self.variables = []\n",
" node_specs = node_specs or []\n",
" for node_spec in node_specs:\n",
" self.add(node_spec)\n",
"\n",
" def add(self, node_spec):\n",
" """Add a node to the net. Its parents must already be in the\n",
" net, and its variable must not."""\n",
" node = BayesNode(*node_spec)\n",
" assert node.variable not in self.variables\n",
" assert all((parent in self.variables) for parent in node.parents)\n",
" self.nodes.append(node)\n",
" self.variables.append(node.variable)\n",
" for parent in node.parents:\n",
" self.variable_node(parent).children.append(node)\n",
"\n",
" def variable_node(self, var):\n",
" """Return the node for the variable named var.\n",
" >>> burglary.variable_node('Burglary').variable\n",
" 'Burglary'"""\n",
" for n in self.nodes:\n",
" if n.variable == var:\n",
" return n\n",
" raise Exception("No such variable: {}".format(var))\n",
"\n",
" def variable_values(self, var):\n",
" """Return the domain of var."""\n",
" return [True, False]\n",
"\n",
" def __repr__(self):\n",
" return 'BayesNet({0!r})'.format(self.nodes)\n",
"def enumerate_all(variables, e, bn):\n",
" """Return the sum of those entries in P(variables | e{others})\n",
" consistent with e, where P is the joint distribution represented\n",
" by bn, and e{others} means e restricted to bn's other variables\n",
" (the ones other than variables). Parents must precede children in variables."""\n",
" if not variables:\n",
" return 1.0\n",
" Y, rest = variables[0], variables[1:]\n",
" Ynode = bn.variable_node(Y)\n",
" if Y in e:\n",
" return Ynode.p(e[Y], e) * enumerate_all(rest, e, bn)\n",
" else:\n",
" return sum(Ynode.p(y, e) * enumerate_all(rest, extend(e, Y, y), bn)\n",
" for y in bn.variable_values(Y))\n",
"def enumeration_ask(X, e, bn):\n",
" """Return the conditional probability distribution of variable X\n",
" given evidence e, from BayesNet bn. [Figure 14.9]\n",
" >>> enumeration_ask('Burglary', dict(JohnCalls=T, MaryCalls=T), burglary\n",
" ... ).show_approx()\n",
" 'False: 0.716, True: 0.284'"""\n",
" assert X not in e, "Query variable must be distinct from evidence"\n",
" Q = ProbDist(X)\n",
" for xi in bn.variable_values(X):\n",
" Q[xi] = enumerate_all(bn.variables, extend(e, X, xi), bn)\n",
" return Q.normalize()\n",
"def make_factor(var, e, bn):\n",
" """Return the factor for var in bn's joint distribution given e.\n",
" That is, bn's full joint distribution, projected to accord with e,\n",
" is the pointwise product of these factors for bn's variables."""\n",
" node = bn.variable_node(var)\n",
" variables = [X for X in [var] + node.parents if X not in e]\n",
" cpt = {event_values(e1, variables): node.p(e1[var], e1)\n",
" for e1 in all_events(variables, bn, e)}\n",
" return Factor(variables, cpt)\n",
"def all_events(variables, bn, e):\n",
" """Yield every way of extending e with values for all variables."""\n",
" if not variables:\n",
" yield e\n",
" else:\n",
" X, rest = variables[0], variables[1:]\n",
" for e1 in all_events(rest, bn, e):\n",
" for x in bn.variable_values(X):\n",
" yield extend(e1, X, x)\n",
" def pointwise_product(self, other, bn):\n",
" """Multiply two factors, combining their variables."""\n",
" variables = list(set(self.variables) | set(other.variables))\n",
" cpt = {event_values(e, variables): self.p(e) * other.p(e)\n",
" for e in all_events(variables, bn, {})}\n",
" return Factor(variables, cpt)\n",
"def pointwise_product(factors, bn):\n",
" return reduce(lambda f, g: f.pointwise_product(g, bn), factors)\n",
" def sum_out(self, var, bn):\n",
" """Make a factor eliminating var by summing over its values."""\n",
" variables = [X for X in self.variables if X != var]\n",
" cpt = {event_values(e, variables): sum(self.p(extend(e, var, val))\n",
" for val in bn.variable_values(var))\n",
" for e in all_events(variables, bn, {})}\n",
" return Factor(variables, cpt)\n",
"def sum_out(var, factors, bn):\n",
" """Eliminate var from all factors by summing over its values."""\n",
" result, var_factors = [], []\n",
" for f in factors:\n",
" (var_factors if var in f.variables else result).append(f)\n",
" result.append(pointwise_product(var_factors, bn).sum_out(var, bn))\n",
" return result\n",
"def elimination_ask(X, e, bn):\n",
" """Compute bn's P(X|e) by variable elimination. [Figure 14.11]\n",
" >>> elimination_ask('Burglary', dict(JohnCalls=T, MaryCalls=T), burglary\n",
" ... ).show_approx()\n",
" 'False: 0.716, True: 0.284'"""\n",
" assert X not in e, "Query variable must be distinct from evidence"\n",
" factors = []\n",
" for var in reversed(bn.variables):\n",
" factors.append(make_factor(var, e, bn))\n",
" if is_hidden(var, X, e):\n",
" factors = sum_out(var, factors, bn)\n",
" return pointwise_product(factors, bn).normalize()\n",
"\n", "This happened because the bayesian network in question is pretty small, with just 5 nodes, some of which aren't even required in the inference process.\n", "For more complicated networks, variable elimination will be significantly faster and runtime will reduce not just by a constant factor, but by a polynomial factor proportional to the number of nodes, due to the reduction in repeated calculations." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Approximate Inference in Bayesian Networks\n", "\n", "Exact inference fails to scale for very large and complex Bayesian Networks. This section covers implementation of randomized sampling algorithms, also called Monte Carlo algorithms." ] }, { "cell_type": "code", "execution_count": 44, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\n", "\n", "\n", "\n", "
def sample(self, event):\n",
" """Sample from the distribution for this variable conditioned\n",
" on event's values for parent_variables. That is, return True/False\n",
" at random according with the conditional probability given the\n",
" parents."""\n",
" return probability(self.p(True, event))\n",
"def prior_sample(bn):\n",
" """Randomly sample from bn's full joint distribution. The result\n",
" is a {variable: value} dict. [Figure 14.13]"""\n",
" event = {}\n",
" for node in bn.nodes:\n",
" event[node.variable] = node.sample(event)\n",
" return event\n",
"![](\"files/images/sprinklernet.jpg\")
\n",
"\n",
"Traversing the graph in topological order is important.\n",
"There are two possible topological orderings for this particular directed acyclic graph.\n",
"\n", "1. `Cloudy -> Sprinkler -> Rain -> Wet Grass`\n", "2. `Cloudy -> Rain -> Sprinkler -> Wet Grass`\n", "
\n", "
\n", "We can follow any of the two orderings to sample from the network.\n", "Any ordering other than these two, however, cannot be used.\n", "
\n", "One way to think about this is that `Cloudy` can be seen as a precondition of both `Rain` and `Sprinkler` and just like we have seen in planning, preconditions need to be satisfied before a certain action can be executed.\n", "
\n", "We store the samples on the observations. Let us find **P(Rain=True)** by taking 1000 random samples from the network." ] }, { "cell_type": "code", "execution_count": 46, "metadata": {}, "outputs": [], "source": [ "N = 1000\n", "all_observations = [prior_sample(sprinkler) for x in range(N)]" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Now we filter to get the observations where Rain = True" ] }, { "cell_type": "code", "execution_count": 47, "metadata": {}, "outputs": [], "source": [ "rain_true = [observation for observation in all_observations if observation['Rain'] == True]" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Finally, we can find **P(Rain=True)**" ] }, { "cell_type": "code", "execution_count": 48, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "0.489\n" ] } ], "source": [ "answer = len(rain_true) / N\n", "print(answer)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Sampling this another time might give different results as we have no control over the distribution of the random samples" ] }, { "cell_type": "code", "execution_count": 49, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "0.512\n" ] } ], "source": [ "N = 1000\n", "all_observations = [prior_sample(sprinkler) for x in range(N)]\n", "rain_true = [observation for observation in all_observations if observation['Rain'] == True]\n", "answer = len(rain_true) / N\n", "print(answer)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "To evaluate a conditional distribution. We can use a two-step filtering process. We first separate out the variables that are consistent with the evidence. Then for each value of query variable, we can find probabilities. For example to find **P(Cloudy=True | Rain=True)**. We have already filtered out the values consistent with our evidence in **rain_true**. Now we apply a second filtering step on **rain_true** to find **P(Rain=True and Cloudy=True)**" ] }, { "cell_type": "code", "execution_count": 50, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "0.7890625\n" ] } ], "source": [ "rain_and_cloudy = [observation for observation in rain_true if observation['Cloudy'] == True]\n", "answer = len(rain_and_cloudy) / len(rain_true)\n", "print(answer)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Rejection Sampling\n", "\n", "Rejection Sampling is based on an idea similar to what we did just now. \n", "First, it generates samples from the prior distribution specified by the network. \n", "Then, it rejects all those that do not match the evidence. \n", "
\n", "Rejection sampling is advantageous only when we know the query beforehand.\n", "While prior sampling generally works for any query, it might fail in some scenarios.\n", "
\n", "Let's say we have a generic Bayesian network and we have evidence `e`, and we want to know how many times a state `A` is true, given evidence `e` is true.\n", "Normally, prior sampling can answer this question, but let's assume that the probability of evidence `e` being true in our actual probability distribution is very small.\n", "In this situation, it might be possible that sampling never encounters a data-point where `e` is true.\n", "If our sampled data has no instance of `e` being true, `P(e) = 0`, and therefore `P(A | e) / P(e) = 0/0`, which is undefined.\n", "We cannot find the required value using this sample.\n", "
\n", "We can definitely increase the number of sample points, but we can never guarantee that we will encounter the case where `e` is non-zero (assuming our actual probability distribution has atleast one case where `e` is true).\n", "To guarantee this, we would have to consider every single data point, which means we lose the speed advantage that approximation provides us and we essentially have to calculate the exact inference model of the Bayesian network.\n", "
\n", "
\n", "Rejection sampling will be useful in this situation, as we already know the query.\n", "
\n", "While sampling from the network, we will reject any sample which is inconsistent with the evidence variables of the given query (in this example, the only evidence variable is `e`).\n", "We will only consider samples that do not violate **any** of the evidence variables.\n", "In this way, we will have enough data with the required evidence to infer queries involving a subset of that evidence.\n", "
\n", "
\n", "The function **rejection_sampling** implements the algorithm described by **Figure 14.14**" ] }, { "cell_type": "code", "execution_count": 51, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\n", "\n", "\n", "\n", "
def rejection_sampling(X, e, bn, N=10000):\n",
" """Estimate the probability distribution of variable X given\n",
" evidence e in BayesNet bn, using N samples. [Figure 14.14]\n",
" Raises a ZeroDivisionError if all the N samples are rejected,\n",
" i.e., inconsistent with e.\n",
" >>> random.seed(47)\n",
" >>> rejection_sampling('Burglary', dict(JohnCalls=T, MaryCalls=T),\n",
" ... burglary, 10000).show_approx()\n",
" 'False: 0.7, True: 0.3'\n",
" """\n",
" counts = {x: 0 for x in bn.variable_values(X)} # bold N in [Figure 14.14]\n",
" for j in range(N):\n",
" sample = prior_sample(bn) # boldface x in [Figure 14.14]\n",
" if consistent_with(sample, e):\n",
" counts[sample[X]] += 1\n",
" return ProbDist(X, counts)\n",
"def consistent_with(event, evidence):\n",
" """Is event consistent with the given evidence?"""\n",
" return all(evidence.get(k, v) == v\n",
" for k, v in event.items())\n",
"def weighted_sample(bn, e):\n",
" """Sample an event from bn that's consistent with the evidence e;\n",
" return the event and its weight, the likelihood that the event\n",
" accords to the evidence."""\n",
" w = 1\n",
" event = dict(e) # boldface x in [Figure 14.15]\n",
" for node in bn.nodes:\n",
" Xi = node.variable\n",
" if Xi in e:\n",
" w *= node.p(e[Xi], event)\n",
" else:\n",
" event[Xi] = node.sample(event)\n",
" return event, w\n",
"def likelihood_weighting(X, e, bn, N=10000):\n",
" """Estimate the probability distribution of variable X given\n",
" evidence e in BayesNet bn. [Figure 14.15]\n",
" >>> random.seed(1017)\n",
" >>> likelihood_weighting('Burglary', dict(JohnCalls=T, MaryCalls=T),\n",
" ... burglary, 10000).show_approx()\n",
" 'False: 0.702, True: 0.298'\n",
" """\n",
" W = {x: 0 for x in bn.variable_values(X)}\n",
" for j in range(N):\n",
" sample, weight = weighted_sample(bn, e) # boldface x, w in [Figure 14.15]\n",
" W[sample[X]] += weight\n",
" return ProbDist(X, W)\n",
"def gibbs_ask(X, e, bn, N=1000):\n",
" """[Figure 14.16]"""\n",
" assert X not in e, "Query variable must be distinct from evidence"\n",
" counts = {x: 0 for x in bn.variable_values(X)} # bold N in [Figure 14.16]\n",
" Z = [var for var in bn.variables if var not in e]\n",
" state = dict(e) # boldface x in [Figure 14.16]\n",
" for Zi in Z:\n",
" state[Zi] = random.choice(bn.variable_values(Zi))\n",
" for j in range(N):\n",
" for Zi in Z:\n",
" state[Zi] = markov_blanket_sample(Zi, state, bn)\n",
" counts[state[X]] += 1\n",
" return ProbDist(X, counts)\n",
"\n", "Likelihood weighting is the fastest out of all as it doesn't involve rejecting samples, but also has a quite high variance." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## HIDDEN MARKOV MODELS\n", "\n", "Often, we need to carry out probabilistic inference on temporal data or a sequence of observations where the order of observations matter.\n", "We require a model similar to a Bayesian Network, but one that grows over time to keep up with the latest evidences.\n", "If you are familiar with the `mdp` module or Markov models in general, you can probably guess that a Markov model might come close to representing our problem accurately.\n", "
\n", "A Markov model is basically a chain-structured Bayesian Network in which there is one state for each time step and each node has an identical probability distribution.\n", "The first node, however, has a different distribution, called the prior distribution which models the initial state of the process.\n", "A state in a Markov model depends only on the previous state and the latest evidence and not on the states before it.\n", "
\n", "A **Hidden Markov Model** or **HMM** is a special case of a Markov model in which the state of the process is described by a single discrete random variable.\n", "The possible values of the variable are the possible states of the world.\n", "
\n", "But what if we want to model a process with two or more state variables?\n", "In that case, we can still fit the process into the HMM framework by redefining our state variables as a single \"megavariable\".\n", "We do this because carrying out inference on HMMs have standard optimized algorithms.\n", "A HMM is very similar to an MDP, but we don't have the option of taking actions like in MDPs, instead, the process carries on as new evidence appears.\n", "
\n", "If a HMM is truncated at a fixed length, it becomes a Bayesian network and general BN inference can be used on it to answer queries.\n", "\n", "Before we start, it will be helpful to understand the structure of a temporal model. We will use the example of the book with the guard and the umbrella. In this example, the state $\\textbf{X}$ is whether it is a rainy day (`X = True`) or not (`X = False`) at Day $\\textbf{t}$. In the sensor or observation model, the observation or evidence $\\textbf{U}$ is whether the professor holds an umbrella (`U = True`) or not (`U = False`) on **Day** $\\textbf{t}$. Based on that, the transition model is \n", "\n", "| $X_{t-1}$ | $X_{t}$ | **P**$(X_{t}| X_{t-1})$| \n", "| ------------- |------------- | ----------------------------------|\n", "| ***${False}$*** | ***${False}$*** | 0.7 |\n", "| ***${False}$*** | ***${True}$*** | 0.3 |\n", "| ***${True}$*** | ***${False}$*** | 0.3 |\n", "| ***${True}$*** | ***${True}$*** | 0.7 |\n", "\n", "And the the sensor model will be,\n", "\n", "| $X_{t}$ | $U_{t}$ | **P**$(U_{t}|X_{t})$| \n", "| :-------------: |:-------------: | :------------------------:|\n", "| ***${False}$*** | ***${True}$*** | 0.2 |\n", "| ***${False}$*** | ***${False}$*** | 0.8 |\n", "| ***${True}$*** | ***${True}$*** | 0.9 |\n", "| ***${True}$*** | ***${False}$*** | 0.1 |\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "HMMs are implemented in the **`HiddenMarkovModel`** class.\n", "Let's have a look." ] }, { "cell_type": "code", "execution_count": 64, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\n", "\n", "\n", "\n", "
class HiddenMarkovModel:\n",
" """A Hidden markov model which takes Transition model and Sensor model as inputs"""\n",
"\n",
" def __init__(self, transition_model, sensor_model, prior=None):\n",
" self.transition_model = transition_model\n",
" self.sensor_model = sensor_model\n",
" self.prior = prior or [0.5, 0.5]\n",
"\n",
" def sensor_dist(self, ev):\n",
" if ev is True:\n",
" return self.sensor_model[0]\n",
" else:\n",
" return self.sensor_model[1]\n",
"\n", "The basic inference tasks that must be solved are:\n", "1. **Filtering**: Computing the posterior probability distribution over the most recent state, given all the evidence up to the current time step.\n", "2. **Prediction**: Computing the posterior probability distribution over the future state.\n", "3. **Smoothing**: Computing the posterior probability distribution over a past state. Smoothing provides a better estimation as it incorporates more evidence.\n", "4. **Most likely explanation**: Finding the most likely sequence of states for a given observation\n", "5. **Learning**: The transition and sensor models can be learnt, if not yet known, just like in an information gathering agent\n", "
\n", "
\n", "\n", "There are three primary methods to carry out inference in Hidden Markov Models:\n", "1. The Forward-Backward algorithm\n", "2. Fixed lag smoothing\n", "3. Particle filtering\n", "\n", "Let's have a look at how we can carry out inference and answer queries based on our umbrella HMM using these algorithms." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### FORWARD-BACKWARD\n", "This is a general algorithm that works for all Markov models, not just HMMs.\n", "In the filtering task (inference) we are given evidence **U** in each time **t** and we want to compute the belief $B_{t}(x)= P(X_{t}|U_{1:t})$. \n", "We can think of it as a three step process:\n", "1. In every step we start with the current belief $P(X_{t}|e_{1:t})$\n", "2. We update it for time\n", "3. We update it for evidence\n", "\n", "The forward algorithm performs the step 2 and 3 at once. It updates, or better say reweights, the initial belief using the transition and the sensor model. Let's see the umbrella example. On **Day 0** no observation is available, and for that reason we will assume that we have equal possibilities to rain or not. In the **`HiddenMarkovModel`** class, the prior probabilities for **Day 0** are by default [0.5, 0.5]. " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The observation update is calculated with the **`forward()`** function. Basically, we update our belief using the observation model. The function returns a list with the probabilities of **raining or not** on **Day 1**." ] }, { "cell_type": "code", "execution_count": 67, "metadata": {}, "outputs": [ { "data": { "text/html": [ "\n", "\n", "\n", "\n", "
def forward(HMM, fv, ev):\n",
" prediction = vector_add(scalar_vector_product(fv[0], HMM.transition_model[0]),\n",
" scalar_vector_product(fv[1], HMM.transition_model[1]))\n",
" sensor_dist = HMM.sensor_dist(ev)\n",
"\n",
" return normalize(element_wise_product(sensor_dist, prediction))\n",
"def backward(HMM, b, ev):\n",
" sensor_dist = HMM.sensor_dist(ev)\n",
" prediction = element_wise_product(sensor_dist, b)\n",
"\n",
" return normalize(vector_add(scalar_vector_product(prediction[0], HMM.transition_model[0]),\n",
" scalar_vector_product(prediction[1], HMM.transition_model[1])))\n",
"\n", "Additionally, the forward pass stores $t$ vectors of size $S$ which makes the auxiliary space requirement equivalent to $O(St)$.\n", "
\n", "
\n", "Is there any way we can improve the time or space complexity?\n", "
\n", "Fortunately, the matrix representation of HMM properties allows us to do so.\n", "
\n", "If $f$ and $b$ represent the forward and backward messages respectively, we can modify the smoothing algorithm by first\n", "running the standard forward pass to compute $f_{t:t}$ (forgetting all the intermediate results) and then running\n", "backward pass for both $b$ and $f$ together, using them to compute the smoothed estimate at each step.\n", "This optimization reduces auxlilary space requirement to constant (irrespective of the length of the sequence) provided\n", "the transition matrix is invertible and the sensor model has no zeros (which is sometimes hard to accomplish)\n", "
\n", "
\n", "Let's look at another algorithm, that carries out smoothing in a more optimized way." ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [] } ], "metadata": { "kernelspec": { "display_name": "Python 3", "language": "python", "name": "python3" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.7.6" } }, "nbformat": 4, "nbformat_minor": 2 }