So, in Bayesian Knowledge Tracing,

it's not designed to assess knowledge

for broad knowledge components such as ''additions.''

Rather, it's designed to assess knowledge of

very well-defined fine grained skills for example,

the addition of two digit numbers with no carry.

And so knowledge tracing has a couple of

very strong assumption that it makes on what the skills are.

So for example, a skill is going to

be atomic and it's either going to be known or not known.

So it's not possible to have a skill that is

partially known in Bayesian Knowledge Tracing.

And also it's possible for a student to learn the skill every time they have to apply it.

And once the skill is known,

the student can't forget.

So, this is probably not true.

When students learn something,

they can in fact forget it.

But in the context of Bayesian Knowledge Tracing,

to simplify the model,

we assume that there is no forgetting.

So, what exactly does a Bayesian Knowledge Tracing model look like?

Well first of all,

Bayesian Knowledge Tracing is a two states model.

It has two states that are latent,

either the student has learn the skill or the student has not learn the skill.

There's only two possibility.

Now, every time the student asked to practice applying the skill,

then there is a chance that the student will learn the skill,

which means that they will transition from not learned state to learned state.

And we say that there is a probability p(T),

that the student is going to make that transition.

Now, every practice opportunity,

that the student can either apply the skill correctly or incorrectly.

And because there are two states,

not learned and learned,

then there's different reasons why a student might incorrectly apply the skill,

or correctly apply the skill.

So first, if we look at a student that we would say

is in the not learned state, there's two possibility.

First of all, the student could still get the correct answer,

and this will happen when they guess what the answer should be.

So, even though they don't know,

how to apply the skill, they still succeed.

So, we're going to say that there is a probability p(G),

that the student is going to be able to guess,

even if they don't know the skill and they'll get the correct answer.

Now, the other outcome,

is that the student is getting

the incorrect answer which is probably the expected outcome,

if the student doesn't know the skill.

And so, we're going to say there's a probability 1 minus P(G),

that the student will get the incorrect answer if they don't know the skill.

Now, that's if they don't know the skill.

If they know skill, there's the same thing.

We can either succeed at applying the skill,

or fail at applying the skill.

So first of all, we're going to look at when the student

fails to apply the skill, even though they know it.

So, this is mostly caused by careless mistake that we call slip.

So, we're going to say, that in this model,

there is a probability p(S),

that the student will slip,

and incorrectly apply the skill even if they are in the learn state.

And then that means that,

in order to correctly apply the skill,

when the student isn't the learned state,

they need to not slip.

So, there's a probability 1 minus p(S)

that the student is going to be able to correctly apply the skill,

if they know the skill.

Right? So, in this model,

there's two states and there's observation of successes and failure.

So, the states themselve,

we can never observe them.

We can only infer whether the student

is in that state or not based on the observation that we make.

So, in addition, there's a last element to the Bayesian Knowledge Tracing model, is that,

before even the student even start practicing the skill,

it's possible that they already know it.

And so, we're going to say that there's probability p(L0) that the student has learned

the skill at the practice opportunity zero which is before they even start applying it.

So, as a quick summary of Bayesian Knowledge Tracing,

each bayesian knowledge tracing model is defined by a set of four parameters.

Two of those, are what we call learning parameters.

So L0 which is the probability that

the skill is known before the students start practicing it.

And T, which is the probability that the student will be able to

transition from the not learned state to the learned state.

There's also two performance operator parameters.

The first one is G,

It's the probability that the student is going to be able to guess

what the answer should be even when they don't know the skill,

and there's the problem with the S,

which is the problem that the student will slip and get

the incorrect answer on the skill even though they know it.

So, in order to build a Bayesian Knowledge Tracing model,

what we need to do is we need to find the best value for each of those four parameters.

And how this is done is we're going to go and collect data

about student trying to apply that skill in the learning environments.

And then we're going to use the computer to look

at the different possibility for those four parameters,

and find the ones that better predict

actual successes and failure of those students in the data that we collect.

Now, once we've trained that model,

once we've built that model,

then we can apply it to new students and we can

apply it to try to compute the probability that

the student knows a skill after

each practice opportunity which allows us to track the students knowledge,

as they're using the learning environment,

and we can also use it to try to compute how likely it is that,

the student will succeed at applying the skill in future practice opportunity.

And so, in order to compute those probability,

we can use the following equations that I've shown here.

I'm not going to go into details about those equations,

but if you are interested in them,

then you can pause the video, look at them,

or you can look at papers on Bayesian Knowledge Tracing.

So, what I'm going to be doing now,

is I'm going to give you an example,

of what L can we apply Bayesian Knowledge Tracing model?

So, in this example,

we have a Bayesian Knowledge tracing model,

that's defined by four parameters.

The first one, is L0,

which is there's 40 percent chance,

that the student knows the skill before the first practice opportunity.

Then there's T which is,

10 percent chance that the student is going to

learn the skill every time they have to apply it.

There's S, which is,

there's 30 percent chance that the student will

slip and incorrectly apply the skill even if they know it.

And then there's G, which is,

20 percent chance that the student will guess the correct answer,

even if they don't know the skill.

So, if we want to apply the model,

the first thing we're going to do,

is for the first practice opportunity,

we're going to say well L0 is 40 percent.

So, there's 40 percent chance,

that the student already knows the skill before we know

anything about how the student is doing in the learning environment.

Then we can look at the outcome of

this first practice opportunity and

see that the students actually fail at applying the skill.

So, using that that information as evidence of knowledge,

we can adjust our estimate that the students knowledge and say well, now,

we're going to say that there's 28 percent chance,

that a student knows this skill,

because they only tried to apply it once and they failed.

Now, for a second practice opportunity, what we were going to do is,

we're going to take that 28 percent chance,

and we're going to say okay,

before the second practice opportunity,

we think that there's 28 percent chance that the student knows the skill.

And then we're going to look at okay,

did the student actually succeed?

Yes, they did.

So, because the students succeeded,

that's going to give us information about whether the student knows the skill or not.

So again, we're going to adapt our probability and now we're going to say, well,

we think that there's about 62 percent chance

that the student actually knows the skill now,

because they showed us that they were able to apply it.

For the third practice opportunity, we're going to say,

we start at 62 percent chance

that the student actually knows the skill and what we observe,

is that the student infact,

was able to successfully applied the skill,

which means that it's less likely that they're just guessing,

because they succeeded twice in a row,

which means that, now, we're going to say,

we think the student has about 86.5 percent chance that they know the skill.

And then finally, for the fourth practice opportunity,

we're going to start with this 86.5 Percent chance,

and then we're going to look at the outcome and say,

well, now the student failed to apply to skill.

What does that mean? So, because the student failed to apply to skill,

we're going to adjust our probability and say, well,

maybe 86 percent was a little bit too high,

so we're going to decrease to 0.73 Percent,

because, it's still possible

that a student knows the skill because they might have slipped.

There's a 30 percent chance that they slip.

So, it decreases a little bit,

but not by too much.

And so, this gives you kind of that very broad example of how we can apply

a Bayesian Knowledge Tracing model and

get an estimate of whether or not the student knows the skill.

And just to conclude quickly, in this video,

what we did is we looked at the concept of Latent Knowledge Estimation.

We looked at what it is,

and we've seen that it can be useful in order to

select for example optimal problem that a student should attempt next.

We can also use it to provide teachers with

an overview of the student's knowledge for example.

And we also looked at the classical Bayesian knowledge tracing model

that can be used to estimate student knowledge.

Even though classical Bayesian knowledge tracing model

is still very much used in the literature,

it's not the only model that can be used,

there are more advance Bayesian knowledge tracing models

that try to account for some of the limitation,

there's Performance Factor Analysis,

there's Item Response Theory.

I'm not going to talk about those approaches in this video,

but if you're interested in the Latent Knowledge Estimation,

I invite you to find papers and read about those.