Approaches: An Interdisciplinary Journal of Music Therapy | Approaches: Ένα Διεπιστημονικό Περιοδικό Μουσικοθεραπείας | ISSN: 2459-3338
On a very high level, web applications tend to have two major components: state and user interface (UI). An application starts with an initial state and presents that state to the user through UI. The user takes some action through a UI element that modifies the initial state. The new state is then presented back to the user for more actions. The figure below shows the interaction between a state and UI in a hypothetical application that allows logged in users to create blog posts.
At any given point in time, an application needs to store different types of information in memory. For example, it needs to know whether or not a user is logged in or how many blogs a user has posted. State is like a repository for storing all of this information. This state is then made available to various data structures in the application. Functions in the application perform different operations on it, resulting into a new state.
In the Pure Functions section, we defined a state as something that represents all the information stored at a given instant in time that a function has access to. Conceptually, an application state works the same way. It just contains a lot more information than a function level state.
In Elm we represent the state of an application with something called a model . A model is just a data structure that contains important information about the application. Imagine a simple app that allows us to increment or decrement a counter. For this app, the only state we need to track is the current value of the counter. Here is what the model definition for this app looks like:
It’s just an
Int
. A model doesn’t necessarily have to be complicated. It all depends on how complex the app is and how many different things it needs to track. For a simple counter app, all we need is a number that tells us what the current value is. A model is generally defined as a type alias.
Let’s add the above model definition into a file and start building a counter app in Elm. Create a new file called
Counter.elm
in the
beginning-elm/elm-examples
directory.
Now add the following code to
Counter.elm
.
The definition above doesn’t create a model. All it tells Elm is what our model looks like. Add the following code to the bottom of
Counter.elm
.
The constant above is what actually creates our initial model.
Next, we need to present our initial model to the user. Add the following code to the bottom of
Counter.elm
.
The
view
function takes a model and returns HTML code. Behind the scenes, the
div
function in Elm produces the
<div>
element in HTML, and the
button
function produces the
<button>
element. The
text
function doesn’t represent any HTML tag. It just displays a plain text by escaping special characters so that the text appears exactly as we specify in our code.
Depending on the specific analysis, we aggregated the results on the 50 or 100 data sets as explained in detail below and thus obtained more stable and reliable results.
Graphical models can be thought of as maps of dependence structures of a given probability distribution or a sample thereof (see for example [
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]). In order to illustrate the analogy, let us consider a road map. In order to be able to use a road map, one needs two given factors. Firstly, one needs the physical map with symbols such as dots and lines. Secondly, one needs a rule for interpreting the symbols. For instance, a railroad map and a map for electric circuits might look very much alike, but their interpretation differs a lot. In the same sense, a graphical model is a map. Firstly, a graphical model consists of a graph with dots, lines and potentially arrowheads. Secondly, a graphical model always comes with a rule for interpreting this graph. In general, nodes in the graph represent (random) variables and edges represent some kind of dependence.
An example of a graphical model is the Directed Acyclic Graph (DAG) model. The physical map here is a graph consisting of nodes and arrows (only one arrowhead per line) connecting the nodes. As a further restriction, the arrows must be directed in a way, so that it is not possible to trace a circle when following the arrowheads. The interpretation rule is d-separation, which is closely related to conditional independence. This rule is a bit more intricate and we refer the reader to [
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] for more details.
Another example of a graphical model is the so called "skeleton" (of a Directed Acyclic Graph, see [
16
]) model. The physical map in this model is a graph consisting of dots and lines (without arrowheads). Using this model, we will use the following rules for interpreting a graph: Two nodes are connected by an edge, if and only if the corresponding random variables are dependent if conditioning on any subset of the remaining random variables. Thus, an edge indicates a strong kind of dependence and it turns out that this is useful for estimating bounds on causal effects (also called intervention effects). See [
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] for a detailed discussion of this subject.
DAG models are particularly useful for estimating intervention effects. Imagine that a causal system is represented by a DAG: Nodes represent observable variables and arrows represent direct causes. Now assume that we gather data from the causal system by observing it many times in different states and recording the values of all involved variables. The observed data will entail some dependence information among the variables. Since every DAG on the same variables also entails dependence information via d-separation, we could find the DAG that fits the dependence information in the data best. It is a basic fact of DAG models, that we usually won't be able to identify a unique DAG that fits best. Rather, we will find several DAG models that fit all equally well. These DAG models are called "equivalent". The DAGs of the equivalent DAG models have a noteworthy property: When ignoring the arrowheads, they look the same. But some arrowheads point into different directions, i.e., the direction of some edges is ambiguous. It was shown in [
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], that under certain assumptions the unambiguous arrows in the estimated DAG models coincide with the true arrows in the underlying causal system. Thus, by estimating a DAG model and under some assumptions, we can get information about the underlying causal structure. This information is contained in the unambiguous arrows of the DAG model. However, the ambiguous arrows don't contain direct information on the underlying causal structure. Hence, estimating a DAG model from observational data gives insight into some aspects of the underlying causal structure, but other aspects will remain obscure. For this reason, it is in general only possible to estimate bounds (and not precise values) on causal effects from observational data.
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