Probability

This IPy notebook acts as supporting material for topics covered in Chapter 13 Quantifying Uncertainty, Chapter 14 Probabilistic Reasoning, Chapter 15 Probabilistic Reasoning over Time, Chapter 16 Making Simple Decisions and parts of Chapter 25 Robotics of the book Artificial Intelligence: A Modern Approach. This notebook makes use of the implementations in probability.py module. Let us import everything from the probability module. It might be helpful to view the source of some of our implementations. Please refer to the Introductory IPy file for more details on how to do so.

CONTENTS

• Probability Distribution
  • Joint probability distribution
  • Inference using full joint distributions
    
• Bayesian Networks
  • BayesNode
  • BayesNet
  • Exact Inference in Bayesian Networks
    • Enumeration
    • Variable elimination
  • Approximate Inference in Bayesian Networks
    • Prior sample
    • Rejection sampling
    • Likelihood weighting
    • Gibbs sampling
      
• Hidden Markov Models
  • Inference in Hidden Markov Models
    • Forward-backward
    • Fixed lag smoothing
    • Particle filtering
      
      
• Monte Carlo Localization
• Decision Theoretic Agent
• Information Gathering Agent

PROBABILITY DISTRIBUTION

Let us begin by specifying discrete probability distributions. The class ProbDist defines a discrete probability distribution. We name our random variable and then assign probabilities to the different values of the random variable. Assigning probabilities to the values works similar to that of using a dictionary with keys being the Value and we assign to it the probability. This is possible because of the magic methods _ getitem _ and _ setitem _ which store the probabilities in the prob dict of the object. You can keep the source window open alongside while playing with the rest of the code to get a better understanding.

The first parameter of the constructor varname has a default value of '?'. So if the name is not passed it defaults to ?. The keyword argument freqs can be a dictionary of values of random variable: probability. These are then normalized such that the probability values sum upto 1 using the normalize method.

Besides the prob and varname the object also separately keeps track of all the values of the distribution in a list called values. Every time a new value is assigned a probability it is appended to this list, This is done inside the _ setitem _ method.

The distribution by default is not normalized if values are added incrementally. We can still force normalization by invoking the normalize method.

It is also possible to display the approximate values upto decimals using the show_approx method.

Joint Probability Distribution

The helper function event_values returns a tuple of the values of variables in event. An event is specified by a dict where the keys are the names of variables and the corresponding values are the value of the variable. Variables are specified with a list. The ordering of the returned tuple is same as those of the variables.

Alternatively if the event is specified by a list or tuple of equal length of the variables. Then the events tuple is returned as it is.

A probability model is completely determined by the joint distribution for all of the random variables. (Section 13.3) The probability module implements these as the class JointProbDist which inherits from the ProbDist class. This class specifies a discrete probability distribute over a set of variables.

Values for a Joint Distribution is a an ordered tuple in which each item corresponds to the value associate with a particular variable. For Joint Distribution of X, Y where X, Y take integer values this can be something like (18, 19).

To specify a Joint distribution we first need an ordered list of variables.

Like the ProbDist class JointProbDist also employes magic methods to assign probability to different values. The probability can be assigned in either of the two formats for all possible values of the distribution. The event_values call inside _ getitem _ and _ setitem _ does the required processing to make this work.

It is also possible to list all the values for a particular variable using the values method.

Inference Using Full Joint Distributions

In this section we use Full Joint Distributions to calculate the posterior distribution given some evidence. We represent evidence by using a python dictionary with variables as dict keys and dict values representing the values.

This is illustrated in Section 13.3 of the book. The functions enumerate_joint and enumerate_joint_ask implement this functionality. Under the hood they implement Equation 13.9 from the book.

$$\textbf{P}(X | \textbf{e}) = \alpha \textbf{P}(X, \textbf{e}) = \alpha \sum_{y} \textbf{P}(X, \textbf{e}, \textbf{y})$$

Here α is the normalizing factor. X is our query variable and e is the evidence. According to the equation we enumerate on the remaining variables y (not in evidence or query variable) i.e. all possible combinations of y

We will be using the same example as the book. Let us create the full joint distribution from Figure 13.3.

Let us now look at the enumerate_joint function returns the sum of those entries in P consistent with e,provided variables is P's remaining variables (the ones not in e). Here, P refers to the full joint distribution. The function uses a recursive call in its implementation. The first parameter variables refers to remaining variables. The function in each recursive call keeps on variable constant while varying others.

Let us assume we want to find P(Toothache=True). This can be obtained by marginalization (Equation 13.6). We can use enumerate_joint to solve for this by taking Toothache=True as our evidence. enumerate_joint will return the sum of probabilities consistent with evidence i.e. Marginal Probability.

You can verify the result from our definition of the full joint distribution. We can use the same function to find more complex probabilities like P(Cavity=True and Toothache=True)

Being able to find sum of probabilities satisfying given evidence allows us to compute conditional probabilities like P(Cavity=True | Toothache=True) as we can rewrite this as $$P(Cavity=True | Toothache = True) = \frac{P(Cavity=True \ and \ Toothache=True)}{P(Toothache=True)}$$

We have already calculated both the numerator and denominator.

We might be interested in the probability distribution of a particular variable conditioned on some evidence. This can involve doing calculations like above for each possible value of the variable. This has been implemented slightly differently using normalization in the function enumerate_joint_ask which returns a probability distribution over the values of the variable X, given the {var:val} observations e, in the JointProbDist P. The implementation of this function calls enumerate_joint for each value of the query variable and passes extended evidence with the new evidence having X = x_i. This is followed by normalization of the obtained distribution.

Let us find P(Cavity | Toothache=True) using enumerate_joint_ask.

You can verify that the first value is the same as we obtained earlier by manual calculation.

BAYESIAN NETWORKS

A Bayesian network is a representation of the joint probability distribution encoding a collection of conditional independence statements.

A Bayes Network is implemented as the class BayesNet. It consisits of a collection of nodes implemented by the class BayesNode. The implementation in the above mentioned classes focuses only on boolean variables. Each node is associated with a variable and it contains a conditional probabilty table (cpt). The cpt represents the probability distribution of the variable conditioned on its parents P(X | parents).

Let us dive into the BayesNode implementation.

The constructor takes in the name of variable, parents and cpt. Here variable is a the name of the variable like 'Earthquake'. parents should a list or space separate string with variable names of parents. The conditional probability table is a dict {(v1, v2, ...): p, ...}, the distribution P(X=true | parent1=v1, parent2=v2, ...) = p. Here the keys are combination of boolean values that the parents take. The length and order of the values in keys should be same as the supplied parent list/string. In all cases the probability of X being false is left implicit, since it follows from P(X=true).

The example below where we implement the network shown in Figure 14.3 of the book will make this more clear.

The alarm node can be made as follows:

It is possible to avoid using a tuple when there is only a single parent. So an alternative format for the cpt is

The general format used for the alarm node always holds. For nodes with no parents we can also use.

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.)

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.

The constructor of BayesNet takes each item in node_specs and adds a BayesNode to its nodes object variable by calling the add method. add in turn adds node to the net. Its parents must already be in the net, and its variable must not. Thus add allows us to grow a BayesNet given its parents are already present.

burglary global is an instance of BayesNet corresponding to the above example.

T, F = True, False

burglary = BayesNet([
    ('Burglary', '', 0.001),
    ('Earthquake', '', 0.002),
    ('Alarm', 'Burglary Earthquake',
     {(T, T): 0.95, (T, F): 0.94, (F, T): 0.29, (F, F): 0.001}),
    ('JohnCalls', 'Alarm', {T: 0.90, F: 0.05}),
    ('MaryCalls', 'Alarm', {T: 0.70, F: 0.01})
])

BayesNet method variable_node allows to reach BayesNode instances inside a Bayes Net. It is possible to modify the cpt of the nodes directly using this method.

Exact Inference in Bayesian Networks

A Bayes Network is a more compact representation of the full joint distribution and like full joint distributions allows us to do inference i.e. answer questions about probability distributions of random variables given some evidence.

Exact algorithms don't scale well for larger networks. Approximate algorithms are explained in the next section.

Inference by Enumeration

We apply techniques similar to those used for enumerate_joint_ask and enumerate_joint to draw inference from Bayesian Networks. enumeration_ask and enumerate_all implement the algorithm described in Figure 14.9 of the book.

enumerate_all recursively evaluates a general form of the Equation 14.4 in the book.

$$\textbf{P}(X | \textbf{e}) = α \textbf{P}(X, \textbf{e}) = α \sum_{y} \textbf{P}(X, \textbf{e}, \textbf{y})$$

such that P(X, e, y) is written in the form of product of conditional probabilities P(variable | parents(variable)) from the Bayesian Network.

enumeration_ask calls enumerate_all on each value of query variable X and finally normalizes them.

Let us solve the problem of finding out P(Burglary=True | JohnCalls=True, MaryCalls=True) using the burglary network. enumeration_ask takes three arguments X = variable name, e = Evidence (in form a dict like previously explained), bn = The Bayes Net to do inference on.

Variable Elimination

The enumeration algorithm can be improved substantially by eliminating repeated calculations. In enumeration we join the joint of all hidden variables. This is of exponential size for the number of hidden variables. Variable elimination employes interleaving join and marginalization.

Before we look into the implementation of Variable Elimination we must first familiarize ourselves with Factors.

In general we call a multidimensional array of type P(Y1 ... Yn | X1 ... Xm) a factor where some of Xs and Ys maybe assigned values. Factors are implemented in the probability module as the class Factor. They take as input variables and cpt.

Helper Functions

There are certain helper functions that help creating the cpt for the Factor given the evidence. Let us explore them one by one.

make_factor is used to create the cpt and variables that will be passed to the constructor of Factor. We use make_factor for each variable. It takes in the arguments var the particular variable, e the evidence we want to do inference on, bn the bayes network.

Here variables for each node refers to a list consisting of the variable itself and the parents minus any variables that are part of the evidence. This is created by finding the node.parents and filtering out those that are not part of the evidence.

The cpt created is the one similar to the original cpt of the node with only rows that agree with the evidence.

The all_events function is a recursive generator function which yields a key for the orignal cpt which is part of the node. This works by extending evidence related to the node, thus all the output from all_events only includes events that support the evidence. Given all_events is a generator function one such event is returned on every call.

We can try this out using the example on Page 524 of the book. We will make f₅(A) = P(m | A)

Here f5.cpt False key gives probability for P(MaryCalls=True | Alarm = False). Due to our representation where we only store probabilities for only in cases where the node variable is True this is the same as the cpt of the BayesNode. Let us try a somewhat different example from the book where evidence is that the Alarm = True

Here the cpt is for P(MaryCalls | Alarm = True). Therefore the probabilities for True and False sum up to one. Note the difference between both the cases. Again the only rows included are those consistent with the evidence.

Operations on Factors

We are interested in two kinds of operations on factors. Pointwise Product which is used to created joint distributions and Summing Out which is used for marginalization.

Factor.pointwise_product implements a method of creating a joint via combining two factors. We take the union of variables of both the factors and then generate the cpt for the new factor using all_events function. Note that the given we have eliminated rows that are not consistent with the evidence. Pointwise product assigns new probabilities by multiplying rows similar to that in a database join.

pointwise_product extends this operation to more than two operands where it is done sequentially in pairs of two.

Factor.sum_out makes a factor eliminating a variable by summing over its values. Again events_all is used to generate combinations for the rest of the variables.

sum_out uses both Factor.sum_out and pointwise_product to finally eliminate a particular variable from all factors by summing over its values.

Elimination Ask

The algorithm described in Figure 14.11 of the book is implemented by the function elimination_ask. We use this for inference. The key idea is that we eliminate the hidden variables by interleaving joining and marginalization. It takes in 3 arguments X the query variable, e the evidence variable and bn the Bayes network.

The algorithm creates factors out of Bayes Nodes in reverse order and eliminates hidden variables using sum_out. Finally it takes a point wise product of all factors and normalizes. Let us finally solve the problem of inferring

P(Burglary=True | JohnCalls=True, MaryCalls=True) using variable elimination.

Elimination Ask Optimizations

elimination_ask has some critical point to consider and some optimizations could be performed:

• Operation on factors:

  sum_out and pointwise_product function used in elimination_ask is where space and time complexity arise in the variable elimination algorithm (AIMA3e pg. 526).

The only trick is to notice that any factor that does not depend on the variable to be summed out can be moved outside the summation.

• Variable ordering:

  Elimination ordering is important, every choice of ordering yields a valid algorithm, but different orderings cause different intermediate factors to be generated during the calculation (AIMA3e pg. 527). In this case the algorithm applies a reversed order.

In general, the time and space requirements of variable elimination are dominated by the size of the largest factor constructed during the operation of the algorithm. This in turn is determined by the order of elimination of variables and by the structure of the network. It turns out to be intractable to determine the optimal ordering, but several good heuristics are available. One fairly effective method is a greedy one: eliminate whichever variable minimizes the size of the next factor to be constructed.

• Variable relevance

  Some variables could be irrelevant to resolve a query (i.e. sums to 1). A variable elimination algorithm can therefore remove all these variables before evaluating the query (AIMA3e pg. 528).

An optimization is to remove 'every variable that is not an ancestor of a query variable or evidence variable is irrelevant to the query'.

Runtime comparison

Let's see how the runtimes of these two algorithms compare. We expect variable elimination to outperform enumeration by a large margin as we reduce the number of repetitive calculations significantly.

In this test case we observe that variable elimination is slower than what we expected. It has something to do with number of threads, how Python tries to optimize things and this happens because the network is very small, with just 5 nodes. The elimination_ask has some critical point and some optimizations must be perfomed as seen above.
Of course, for more complicated networks, variable elimination will be significantly faster and runtime will drop not just by a constant factor, but by a polynomial factor proportional to the number of nodes, due to the reduction in repeated calculations.

Approximate Inference in Bayesian Networks

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.

Before we consider the different algorithms in this section let us look at the BayesNode.sample method. It samples from the distribution for this variable conditioned on event's values for parent_variables. That is, return True/False at random according to with the conditional probability given the parents. The probability function is a simple helper from utils module which returns True with the probability passed to it.

Prior Sampling

The idea of Prior Sampling is to sample from the Bayesian Network in a topological order. We start at the top of the network and sample as per P(X_i | parents(X_i) i.e. the probability distribution from which the value is sampled is conditioned on the values already assigned to the variable's parents. This can be thought of as a simulation.

The function prior_sample implements the algorithm described in Figure 14.13 of the book. Nodes are sampled in the topological order. The old value of the event is passed as evidence for parent values. We will use the Bayesian Network in Figure 14.12 to try out the prior_sample

Traversing the graph in topological order is important. There are two possible topological orderings for this particular directed acyclic graph.

1. Cloudy -> Sprinkler -> Rain -> Wet Grass
2. Cloudy -> Rain -> Sprinkler -> Wet Grass
   
   We can follow any of the two orderings to sample from the network. Any ordering other than these two, however, cannot be used.
   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.
   We store the samples on the observations. Let us find P(Rain=True) by taking 1000 random samples from the network.

Now we filter to get the observations where Rain = True

Finally, we can find P(Rain=True)

Sampling this another time might give different results as we have no control over the distribution of the random samples

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)

Rejection Sampling

Rejection Sampling is based on an idea similar to what we did just now. First, it generates samples from the prior distribution specified by the network. Then, it rejects all those that do not match the evidence.
Rejection sampling is advantageous only when we know the query beforehand. While prior sampling generally works for any query, it might fail in some scenarios.
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. 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. In this situation, it might be possible that sampling never encounters a data-point where e is true. 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. We cannot find the required value using this sample.
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). 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.

Rejection sampling will be useful in this situation, as we already know the query.
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). We will only consider samples that do not violate any of the evidence variables. In this way, we will have enough data with the required evidence to infer queries involving a subset of that evidence.

The function rejection_sampling implements the algorithm described by Figure 14.14

The function keeps counts of each of the possible values of the Query variable and increases the count when we see an observation consistent with the evidence. It takes in input parameters X - The Query Variable, e - evidence, bn - Bayes net and N - number of prior samples to generate.

consistent_with is used to check consistency.

To answer P(Cloudy=True | Rain=True)

Likelihood Weighting

Rejection sampling takes a long time to run when the probability of finding consistent evidence is low. It is also slow for larger networks and more evidence variables. Rejection sampling tends to reject a lot of samples if our evidence consists of a large number of variables. Likelihood Weighting solves this by fixing the evidence (i.e. not sampling it) and then using weights to make sure that our overall sampling is still consistent.

The pseudocode in Figure 14.15 is implemented as likelihood_weighting and weighted_sample.

weighted_sample samples an event from Bayesian Network that's consistent with the evidence e and returns the event and its weight, the likelihood that the event accords to the evidence. It takes in two parameters bn the Bayesian Network and e the evidence.

The weight is obtained by multiplying P(x_i | parents(x_i)) for each node in evidence. We set the values of event = evidence at the start of the function.

likelihood_weighting implements the algorithm to solve our inference problem. The code is similar to rejection_sampling but instead of adding one for each sample we add the weight obtained from weighted_sampling.

Gibbs Sampling

In likelihood sampling, it is possible to obtain low weights in cases where the evidence variables reside at the bottom of the Bayesian Network. This can happen because influence only propagates downwards in likelihood sampling.

Gibbs Sampling solves this. The implementation of Figure 14.16 is provided in the function gibbs_ask

In gibbs_ask we initialize the non-evidence variables to random values. And then select non-evidence variables and sample it from P(Variable | value in the current state of all remaining vars) repeatedly sample. In practice, we speed this up by using markov_blanket_sample instead. This works because terms not involving the variable get canceled in the calculation. The arguments for gibbs_ask are similar to likelihood_weighting

Runtime analysis

Let's take a look at how much time each algorithm takes.

As expected, all algorithms have a very similar runtime. However, rejection sampling would be a lot faster and more accurate when the probabiliy of finding data-points consistent with the required evidence is small.
Likelihood weighting is the fastest out of all as it doesn't involve rejecting samples, but also has a quite high variance.

HIDDEN MARKOV MODELS

Often, we need to carry out probabilistic inference on temporal data or a sequence of observations where the order of observations matter. We require a model similar to a Bayesian Network, but one that grows over time to keep up with the latest evidences. 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.
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. The first node, however, has a different distribution, called the prior distribution which models the initial state of the process. A state in a Markov model depends only on the previous state and the latest evidence and not on the states before it.
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. The possible values of the variable are the possible states of the world.
But what if we want to model a process with two or more state variables? In that case, we can still fit the process into the HMM framework by redefining our state variables as a single "megavariable". We do this because carrying out inference on HMMs have standard optimized algorithms. 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.
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.

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

$X_{t-1}$	$X_{t}$	P$(X_{t}	X_{t-1})$
${False}$	${False}$	0.7
${False}$	${True}$	0.3
${True}$	${False}$	0.3
${True}$	${True}$	0.7

And the the sensor model will be,

$X_{t}$	$U_{t}$	P$(U_{t}	X_{t})$
${False}$	${True}$	0.2
${False}$	${False}$	0.8
${True}$	${True}$	0.9
${True}$	${False}$	0.1

HMMs are implemented in the HiddenMarkovModel class. Let's have a look.

We instantiate the object hmm of the class using a list of lists for both the transition and the sensor model.

The sensor_dist() method returns a list with the conditional probabilities of the sensor model.

Now that we have defined an HMM object, our task here is to compute the belief $B_{t}(x)= P(X_{t}|U_{1:t})$ given evidence U at each time step t.
The basic inference tasks that must be solved are:

1. Filtering: Computing the posterior probability distribution over the most recent state, given all the evidence up to the current time step.
2. Prediction: Computing the posterior probability distribution over the future state.
3. Smoothing: Computing the posterior probability distribution over a past state. Smoothing provides a better estimation as it incorporates more evidence.
4. Most likely explanation: Finding the most likely sequence of states for a given observation
5. Learning: The transition and sensor models can be learnt, if not yet known, just like in an information gathering agent
   
   

There are three primary methods to carry out inference in Hidden Markov Models:

1. The Forward-Backward algorithm
2. Fixed lag smoothing
3. Particle filtering

Let's have a look at how we can carry out inference and answer queries based on our umbrella HMM using these algorithms.

FORWARD-BACKWARD

This is a general algorithm that works for all Markov models, not just HMMs. In the filtering task (inference) we are given evidence U in each time t and we want to compute the belief $B_{t}(x)= P(X_{t}|U_{1:t})$. We can think of it as a three step process:

1. In every step we start with the current belief $P(X_{t}|e_{1:t})$
2. We update it for time
3. We update it for evidence

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].

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.

In Day 2 our initial belief is the updated belief of Day 1. Again using the forward() function we can compute the probability of raining in Day 2

In the smoothing part we are interested in computing the distribution over past states given evidence up to the present. Assume that we want to compute the distribution for the time k, for $0\leq k<t $, the computation can be divided in two parts:

1. The forward message will be computed till and by filtering forward from 1 to k.
2. The backward message can be computed by a recusive process that runs from k to t.

Rather than starting at time 1, the algorithm starts at time t. In the umbrella example, we can compute the backward message from Day 2 to Day 1 by using the backward function. The backward function has as parameters the object created by the HiddenMarkovModel class, the evidence in Day 2 (in our case is True), and the initial probabilities of being in state in time t+1. Since no observation is available then it will be [1, 1]. The backward function will return a list with the conditional probabilities.

Some may notice that the result is not the same as in the book. The main reason is that in the book the normalization step is not used. If we want to normalize the result, one can use the normalize() helper function.

In order to find the smoothed estimate for raining in Day k, we will use the forward_backward() function. As in the example in the book, the umbrella is observed in both days and the prior distribution is [0.5, 0.5]

Since HMMs are represented as single variable systems, we can represent the transition model and sensor model as matrices. The forward_backward algorithm can be easily carried out on this representation (as we have done here) with a time complexity of $O({S}^{2} t)$ where t is the length of the sequence and each step multiplies a vector of size $S$ with a matrix of dimensions $SxS$.
Additionally, the forward pass stores $t$ vectors of size $S$ which makes the auxiliary space requirement equivalent to $O(St)$.

Is there any way we can improve the time or space complexity?
Fortunately, the matrix representation of HMM properties allows us to do so.
If $f$ and $b$ represent the forward and backward messages respectively, we can modify the smoothing algorithm by first running the standard forward pass to compute $f_{t:t}$ (forgetting all the intermediate results) and then running backward pass for both $b$ and $f$ together, using them to compute the smoothed estimate at each step. This optimization reduces auxlilary space requirement to constant (irrespective of the length of the sequence) provided the transition matrix is invertible and the sensor model has no zeros (which is sometimes hard to accomplish)

Let's look at another algorithm, that carries out smoothing in a more optimized way.

FIXED LAG SMOOTHING

The matrix formulation allows to optimize online smoothing with a fixed lag.
Since smoothing can be done in constant, there should exist an algorithm whose time complexity is independent of the length of the lag. For smoothing a time slice $t - d$ where $d$ is the lag, we need to compute $\alpha f_{1:t-d}$ x $b_{t-d+1:t}$ incrementally.
As we already know, the forward equation is
$$f_{1:t+1} = \alpha O_{t+1}{T}^{T}f_{1:t}$$
and the backward equation is
$$b_{k+1:t} = TO_{k+1}b_{k+2:t}$$
where $T$ and $O$ are the transition and sensor models respectively.
For smoothing, the forward message is easy to compute but there exists no simple relation between the backward message of this time step and the one at the previous time step, hence we apply the backward equation $d$ times to get
$$b_{t-d+1:t} = \left ( \prod_{i=t-d+1}^{t}{TO_i} \right )b_{t+1:t} = B_{t-d+1:t}1$$
where $B_{t-d+1:t}$ is the product of the sequence of $T$ and $O$ matrices.
Here's how the probability module implements fixed_lag_smoothing.

This algorithm applies forward as usual and optimizes the smoothing step by using the equations above. This optimization could be achieved only because HMM properties can be represented as matrices.
vector_to_diagonal, matrix_multiplication and inverse_matrix are matrix manipulation functions to simplify the implementation.
normalize is used to normalize the output before returning it.

Here's how we can use fixed_lag_smoothing for inference on our umbrella HMM.

Given evidence T, F, T, F and T, we want to calculate the probability distribution for the fourth day with a fixed lag of 2 days.
Let e_t = False

We cannot calculate probability distributions when $t$ is less than $d$

As expected, the output is None

PARTICLE FILTERING

The filtering problem is too expensive to solve using the previous methods for problems with large or continuous state spaces. Particle filtering is a method that can solve the same problem but when the state space is a lot larger, where we wouldn't be able to do these computations in a reasonable amount of time as fast, as time goes by, and we want to keep track of things as they happen.
The downside is that it is a sampling method and hence isn't accurate, but the more samples we're willing to take, the more accurate we'd get.
In this method, instead of keping track of the probability distribution, we will drop particles in a similar proportion at the required regions. The internal representation of this distribution is usually a list of particles with coordinates in the state-space. A particle is just a new name for a sample.

Particle filtering can be divided into four steps:

1. Initialization: If we have some idea about the prior probability distribution, we drop the initial particles accordingly, or else we just drop them uniformly over the state space.

2. Forward pass: As time goes by and measurements come in, we are going to move the selected particles into the grid squares that makes the most sense in terms of representing the distribution that we are trying to track. When time goes by, we just loop through all our particles and try to simulate what could happen to each one of them by sampling its next position from the transition model. This is like prior sampling - samples' frequencies reflect the transition probabilities. If we have enough samples we are pretty close to exact values. We work through the list of particles, one particle at a time, all we do is stochastically simulate what the outcome might be. If we had no dimension of time, and we had no new measurements come in, this would be exactly the same as what we did in prior sampling.

3. Reweight: As observations come in, don't sample the observations, fix them and downweight the samples based on the evidence just like in likelihood weighting. $$w(x) = P(e/x)$$ $$B(X) \propto P(e/X)B'(X)$$
   As before, the probabilities don't sum to one, since most have been downweighted. They sum to an approximation of $P(e)$. To normalize the resulting distribution, we can divide by $P(e)$
   Likelihood weighting wasn't the best thing for Bayesian networks, because we were not accounting for the incoming evidence so we were getting samples from the prior distribution, in some sense not the right distribution, so we might end up with a lot of particles with low weights. These samples were very uninformative and the way we fixed it then was by using Gibbs sampling. Theoretically, Gibbs sampling can be run on a HMM, but as we iterated over the process infinitely many times in a Bayesian network, we cannot do that here as we have new incoming evidence and we also need computational cycles to propagate through time.
   A lot of samples with very low weight and they are not representative of the actual probability distribution. So if we keep running likelihood weighting, we keep propagating the samples with smaller weights and carry out computations for that even though these samples have no significant contribution to the actual probability distribution. Which is why we require this last step.

4. Resample: Rather than tracking weighted samples, we resample. We choose from our weighted sample distribution as many times as the number of particles we initially had and we replace these particles too, so that we have a constant number of particles. This is equivalent to renormalizing the distribution. The samples with low weight are rarely chosen in the new distribution after resampling. This newer set of particles after resampling is in some sense more representative of the actual distribution and so we are better allocating our computational cycles. Now the update is complete for this time step, continue with the next one.

Let's see how this is implemented in the module.

Here, scalar_vector_product and vector_add are helper functions to help with vector math and weighted_sample_with_replacement resamples from a weighted sample and replaces the original sample, as is obvious from the name.
This implementation considers two state variables with generic names 'A' and 'B'.

Here's how we can use particle_filtering on our umbrella HMM, though it doesn't make much sense using particle filtering on a problem with such a small state space. It is just to get familiar with the syntax.

We got 5 samples from state A and 5 samples from state B

This time we got 2 samples from state A and 8 samples from state B

Comparing runtimes for these algorithms will not be useful, as each solves the filtering task efficiently for a different scenario.
forward_backward calculates the exact probability distribution.
fixed_lag_smoothing calculates an approximate distribution and its runtime will depend on the value of the lag chosen.
particle_filtering is an efficient method for approximating distributions for a very large or continuous state space.

MONTE CARLO LOCALIZATION

In the domain of robotics, particle filtering is used for robot localization. Localization is the problem of finding out where things are, in this case, we want to find the position of a robot in a continuous state space.
Monte Carlo Localization is an algorithm for robots to localize using a particle filter. Given a map of the environment, the algorithm estimates the position and orientation of a robot as it moves and senses the environment.
Initially, particles are distributed uniformly over the state space, ie the robot has no information of where it is and assumes it is equally likely to be at any point in space.
When the robot moves, it analyses the incoming evidence to shift and change the probability to better approximate the probability distribution of its position. The particles are then resampled based on their weights.
Gradually, as more evidence comes in, the robot gets better at approximating its location and the particles converge towards the actual position of the robot.
The pose of a robot is defined by its two Cartesian coordinates with values $x$ and $y$ and its direction with value $\theta$. We use the kinematic equations of motion to model a deterministic state prediction. This is our motion model (or transition model).
Next, we need a sensor model. There can be two kinds of sensor models, the first assumes that the sensors detect stable, recognizable features of the environment called landmarks. The robot senses the location and bearing of each landmark and updates its belief according to that. We can also assume the noise in measurements to be Gaussian, to simplify things.
Another kind of sensor model is used for an array of range sensors, each of which has a fixed bearing relative to the robot. These sensors provide a set of range values in each direction. This will also be corrupted by Gaussian noise, but we can assume that the errors for different beam directions are independent and identically distributed.
After evidence comes in, the robot updates its belief state and reweights the particle distribution to better aproximate the actual distribution.

Let's have a look at how this algorithm is implemented in the module

Our implementation of Monte Carlo Localization uses the range scan method. The ray_cast helper function casts rays in different directions and stores the range values.
a stores the v and w components of the robot's velocity.
z is a range scan.
P_motion_sample is the motion or transition model.
P_sensor is the range sensor noise model.
m is the 2D map of the environment
S is a vector of samples of size N

We'll now define a simple 2D map to run Monte Carlo Localization on.
Let's say this is the map we want

Let's define the motion model as a function P_motion_sample.

Define the sensor model as a function P_sensor.

Initializing variables.

Let's run monte_carlo_localization with these parameters to find a sample distribution S.

Let's plot the values in the sample distribution S.

The distribution is highly concentrated at (5, 3), but the robot is not very confident about its position as some other cells also have high probability values.

Let's look at another scenario.

In this case, the robot is 99.9% certain that it is at position (6, 7).

DECISION THEORETIC AGENT

We now move into the domain of probabilistic decision making.
To make choices between different possible plans in a certain situation in a given environment, an agent must have preference between the possible outcomes of the various plans.
Utility theory is used to represent and reason with preferences. The agent prefers states with a higher utility. While constructing multi-agent systems, one major element in the design is the mechanism the agents use for making decisions about which actions to adopt in order to achieve their goals. What is usually required is a mechanism which ensures that the actions adopted lead to benefits for both individual agents, and the community of which they are part. The utility of a state is relative to an agent.
Preferences, as expressed by utilities, are combined with probabilities in the general theory of rational decisions called decision theory.
An agent is said to be rational if and only if it chooses the action that yields the highest expected utility, averaged over all the possible outcomes of the action.