Every time you speak into your smartphone, you
know, and it magically transcribes your voice
into a text message, it isn't actually hearing
your words. Right. Not the way a human does anyway.
Exactly. It's essentially blind to language.
Instead, what that phone is doing is calculating
this massive invisible web of probabilities in
real time. It's just guessing, really. Guessing
the most mathematically likely sequence of sounds
based on this totally chaotic ocean of audio
data. And today, we are taking you, the learner,
inside the architecture of that randomness. Welcome
to the Deep Dive. Glad to be here. So our mission
today is to decode a rather dense, highly technical
Wikipedia article we have. have in our sources.
It breaks down this secret cheat code that data
scientists and statisticians use to map out complex,
chaotic systems. Yeah, it's all about something
called probabilistic graphical models, or PGMs
for short. PGMs, right. And the goal here is
to give you a mental cheat sheet for how these
actually work. Because it is an incredibly powerful
framework. I mean, when you're trying to understand
any real world situation where dozens or even
hundreds of different random variables are interacting
all at once, the the math just becomes paralyzing.
Oh, I can imagine. Yeah, so a graphical model
is the tool that cuts through that noise. To
visualize what we're talking about today, I want
you to imagine stepping into a vast dimly lit
room. Okay, setting the scene, I like it. Right,
and floating all around you is this glowing,
highly complex, detective style string board.
Oh wow. You've got these push bins representing
different variables, and they are connected by
these pulsing threads of light, just weaving
this massive three -dimensional web. I love that
visual. So we're basically standing inside the
brain of a machine learning algorithm, just looking
at how it maps the world. Exactly. That's exactly
what it is. Alright, let's start with the fundamental
building blocks then. We have our floating push
pins and our strings of light. The source material
defines a graphical model as a graph that expresses
the, and I'm quoting here, the conditional dependence
structure between random variables. Right. It
also says it provides a compact, factorized representation
of a multi -dimensional space, which, you know,
is a lot of heavy terminology right out of the
gate. It is a mouthful. Let's break that down
because understanding that one sentence is really
the key to everything else we're going to talk
about. Please do. So. In statistics, a random
variable is really just something that can take
on different values. And we aren't completely
sure what value it's going to take. It could
be a coin. Yeah, or like the weather tomorrow,
the price of a stock, or, to use the detective
board analogy, the identity of a suspect in a
crime. Got it. Now, if you have 100 of these
variables, trying to calculate the probability
of every single possible combination of events
happening at the exact same time, computationally
impossible just too many variables way too many
even our most powerful supercomputers would choke
on it The math just scales exponentially into
infinity. So the graph, this PGM, is basically
a way of cheating that math. It's a way of isolating
what actually matters. What's fascinating here
is that it bridges graph theory and probability.
Instead of assuming everything in the universe
affects everything else equally, the graph maps
out specific relationships. It factors the massive,
incomprehensible problem into bite -sized, solvable
pieces. Oh, I see. the math along the strings
that are actually connected. By doing that, it
encodes a distribution over a multi -dimensional
space without having to calculate the whole space
at once. Okay, let's unpack this and ground it
in that detective string board analogy. Say we
have three suspects. Suspect A, Suspect B, and
Suspect C. Right. The classic trio. Right. And
you're the detective trying to figure out who
committed a robbery. Initially, you have strings
connecting all three of them because they all
run in the same circles. Makes sense. But then
you uncover airtight security footage proving
Suspect A was in a completely different country
at the time of the robbery. So now the value
of variable A is known. It's an absolute certainty.
Right. And because we know the absolute truth
about A, Let's say we realized that suspect B
and suspect C's possible motives had everything
to do with A's involvement. Okay, following you.
So without A, B, and C actually have nothing
to do with each other. If I were visualizing
this on a physical board, my instinct would be
to just take scissors and cut the string between
B and C forever. Is that conditional independence?
Well, let's refine that slightly. Because the
underlying relationship doesn't just cease to
exist in the real world. Oh, it doesn't. No,
think of the connection between them. Less like
a physical string that gets destroyed and more
like a road with a toll booth. Okay, a toll booth.
I'm tracking. In conditional independence, the
flow of new information is what gets blocked.
If suspect A is the toll booth and we don't know
anything about A yet, information about suspect
B's whereabouts might give us clues about C.
Because they're connected through A. Exactly.
The traffic is flowing freely between them. But
the moment we lock down the truth about suspect
A, the moment we close the toll booth, no new
information can travel from B to C. Wow. Knowing
more about B tells us absolutely nothing new
about C. The road is blocked. And that is the
essence of conditional independence. That makes
so much more sense. It's about the flow of information.
So once we establish that we are drawing this
map of connected variables, we have to look at
the strings themselves. Right, the connections.
And looking at the concepts in our sources, there's
a major fork in the road here. The entire field
basically splits into two main families of models
based on one really simple visual detail. Yep.
Do the strings have arrows or not? Exactly. Do
the strings connecting these variables have arrows
pointing in a specific direction, or are they
just plane lines? These are the two heavyweights
of the PGM world. Let's tackle the plane lines
first. Okay. These are called undirected graphical
models, or Markov random fields. In an undirected
graph, the presence of an edge, just a simple
line between two nodes, implies that there is
a mutual relationship or correlation. between
those variables. So just a general link, the
source actually gives a conceptual example of
four nodes, A, B, C, and D. Let's say node A
connects to B, C, and D, but there are no lines
connecting B, C, and D directly to each other.
Right. What is that structure actually doing
mathematically? It's establishing our tollbooth
rule on a larger scale. It tells us that B, C,
and D are conditionally independent given A.
So A is the tollbooth for all of them. Exactly.
If we lock down the value of A, The variables
b, c, and d are totally isolated from one another.
In a Markov random field, the math focuses on
pairs of connected nodes. You look at a and b,
and you evaluate how compatible their states
are. Yeah, the text mentions a specific formula
for this, where you calculate the joint probability
by multiplying functions of those connected pairs,
like function of A and B times function of A
and C. Yeah, the non -negative functions. But,
you know, reading math equations aloud is a surefire
way to lose anyone listening. How do we visualize
what that calculation is actually doing without
getting totally bogged down in the notation?
Think of it as a scoring system for a group project.
Oh. You don't need to evaluate how the entire
company works together all at once. You just
look at the pairs of people who who actually
sit next to each other. Oh, I like that. Yeah.
So you give a compatibility score to A and B
working together. Then you give a compatibility
score to A and C working together. And then you
just multiply them. Exactly. You multiply all
those local pairwise scores together, and it
gives you a global score for how well the whole
system is functioning. It's a way of calculating
the probability of the whole based purely on
the harmony of the local neighborhoods. a local
neighborhood scoring system. I really like that.
It feels very symmetrical and neat. It is. But
then we cross the aisle to the directed models.
These are called directed acyclic graphs or Bayesian
networks. And here, the strings are no longer
just plain lines. They have arrows. And those
arrows fundamentally change the interpretation
of the map. Because they enforce a direction.
Right. In a Bayesian network, we aren't just
looking at mutual correlation anymore. We are
looking at influence. The node where the arrow
starts is called a parent and the node the arrow
points to is the child. See, when I hear parent
and child, my mind immediately goes to a family
tree. It implies that this variable precedes
or physically creates that one. Is that the right
way to think about a directed graph? If we connect
this to the bigger picture, it's a very useful
starting point. But it's better to think of it
strictly as an influence or a generative process.
Okay, influence. The family tree analogy works
because time and genetics flow in one direction.
Right. Your grandfather influences your father
and your father influences you. Right. In a Bayesian
network, the probability of the entire system
is calculated by looking at the probability of
each child variable, given the state of its parents.
So the math is literally probability of X, given
the parents of X. Exactly. The conditionality
is baked directly into the direction of the arrow.
The text mentions naive Bayes classifiers, hidden
Markov models, neural networks. They all use
this directed flow. So if undirected graphs are
good for things sitting at the same table, directed
graphs are for a sequence of events. But the
source specifically calls them directed acyclic
graphs. Acyclic meaning what? No cycles. Right.
No cycles. You can't loop back. That seems like
a big rule. It is a strict rule for Bayesian
networks. You can never start at a node. Follow
the arrows and end up back where you started.
You can't be your own ancestor. That makes sense.
Time and influence must march forward. And because
of this strict one -way flow, Bayesian networks
allow for a very specific type of analysis called
D separation. Wait. D separation. You totally
lost me there. The text mentioned it briefly
as a criterion to determine if sets of nodes
are conditionally independent. But how does looking
at the direction of arrows separate anything?
It's a fantastic concept, really. And it goes
back to our toll booth analogy, but this time
with directional traffic. Okay. Directional toll
booths. Right. D separation stands for directed
separation. Let's imagine a simple chain of three
events. Event A is your alarm clock failing to
go off. Oh no. B is you being late for work.
And event C is you getting fired. Wow. A very
stressful directed graph today. Very stressful,
yeah. Now the arrow points from alarm, which
is A, to late, which is B, and from late to fired,
which is C. Got it. A to B to C. If you don't
know anything about whether someone was late,
knowing their alarm failed gives you a pretty
strong clue that they might get fired. The information
flows all the way down the chain. Makes total
sense. But... What if you already know for an
absolute fact that the person was late for work?
You have observed variable B. Okay, so I know
B is true. In the rules of deseparation, observing
the middle node in a chain blocks the flow of
information. If I already know you were late,
discovering that your alarm failed doesn't change
my probability of you getting fired. Oh wow,
because being late is the thing that gets you
fired, not the alarm. Exactly. The late node
already contains all the predictive power. The
arrows dictate exactly how and when that information
gets blocked. So by tracing the arrows, an algorithm
can instantly know which variables to ignore
and which ones to actually pay attention to,
which must save massive amounts of competing
power. Huge amounts, yeah. That is brilliant.
But my detective brain is flashing a warning
light right now. Uh -oh. What's the issue? Well,
both of these heavyweights, the undirected mutual
models and the directed acyclic models, they
both assume the world is perfectly neat. Oh,
I see where you're going. Right. Like, either
everything is a two -way street or everything
flows forward in one direction. But what happens
when the real world is messy? What happens when
things do loop back on themselves? That is exactly
where we leave the neat, tidy world of the textbooks
and enter the Wild West of graphical models.
The Wild West? Yeah, because the researchers
who build these systems, they know reality is
messy. So they started breaking the foundational
rules to accommodate that reality. And the text
outlines several of these other types of models.
The first one that jumps out is the cyclic -directed
graphical model, which seems to directly violate
the acyclic rule we just spent all that time
establishing. It allows variables to depend on
parents in a closed loop. It does. And from a
purely mathematical standpoint, a cycle creates
an enormous headache. Think about it if you're
trying to calculate the probability of an event,
but the event is constantly feeding back into
its own cause. The standard rules of factorization
we discussed completely break down. I mean, how
do you calculate a score when the score keeps
changing itself? It's like a snake eating its
own tail. The example in the source is actually
wild. It describes a situation where variable
D depends on A, B, and C, but then C depends
on B and D. Exactly. It creates this infinite
feedback loop of influence. But think about real
life for a second. Ecosystems are feedback loops.
Predator and prey populations constantly influence
each other in cycles. Oh, that's true. Economics
is full of feedback loops. If inflation rises,
central banks raise interest rates, which slows
the economy, which then impacts inflation. You
can't model that accurately with a strict one
-way arrow. So the math just has to adapt. Right.
Statisticians developed complex discovery algorithms
specifically to handle these cyclic graphs, basically
forcing the math to reconcile with reality. There's
another hybrid in the source that I find absolutely
fascinating, the chain graph. Oh yeah, that's
a good one. It's described as a graph that allows
both directed arrows and undirected lines in
the same exact model. It's like the ultimate
diplomatic treaty between the two heavyweights.
Right, because it keeps the rule of no directed
cycles, so you still can't follow the arrows
back home. But along the way, you can have clusters
of variables that just mutually interact. It
gives you the best of both worlds, really. But
how does that actually look in practice? Like,
why would you need both in the same map. Okay.
Imagine modeling a complex disease. At the top
of the map, you have genetic predispositions.
Those are directed arrows. Because you inherit
them. Right. The influence flows strictly downward
from your DNA to your cellular function. You
can't change your DNA based on a cold you caught.
Right. The arrow of biology only goes one way
there. Exactly. But once you get down to the
cellular level, you might have hundreds of different
proteins and enzymes interacting with each other
all at once. And they don't have a strict parent
-child thing going on. No, one protein doesn't
necessarily cause the other. They're just mutually
reacting in a chemical soup. Those relationships
are undirected. So a chain graph lets a data
scientist model the strict downward flow of genetics
while simultaneously capturing the mutual, undirected
correlation of the immediate cellular environment.
That is incredibly elegant. It really shows the
adaptability of the math. I mean, they've mutated
the basic graph concept to fit whatever messy
data they have. The text mentions a few honorable
engines too, like tree -augmented classifiers,
TAN models, and targeted Bayesian network learning,
or TBNL. Right, and ancestral graphs, which add
bi -directed edges, and restrictive Boltzmann
machines, which are these bipartite generative
models. Lots of specialized tools. But I want
to zero in on one specific concept it mentions
before we move on. Factor graphs and something
called belief propagation. Ah. belief propagation.
The text notes that factor graphs are undirected
by partite graphs used to implement this belief
propagation. But, as we established earlier,
we want to avoid just listing terms. We need
to know how this works, what is a factor graph
doing, and what exactly is a belief propagating.
Okay, to understand belief propagation, we need
a good analogy. Imagine you're at a massive crowded
cocktail party in a sprawling mansion. Sounds
fun. You're deep inside one of the interior rooms
and you want to know if it's raining outside.
You don't have a window and you don't have your
phone. Okay, so I'm a node in a graphical model
trying to figure out the state of a variable
I can't observe directly. Look at Q speaking
the language. Exactly. Now in a graphical model
you can only communicate with the nodes you're
directly connected to. So you ask the person
standing next to you. Hey, do you think it's
raining? And they don't know either. Right, they
don't know, so they ask the person next to them
who asks the person next to them. It's just a
giant game of telephone. But it's smarter than
telephone. Eventually, someone near the entryway
notices a guest walking in with a wet umbrella.
Ah, new data. Yes. That person updates their
belief about the weather to almost certainly
raining. They pass that updated belief back to
the person next to them. Hey, I just saw a wet
umbrella. I'm 90 % sure it's raining Okay, so
the message travels back and as it travels That
person combines that message with what they're
hearing from someone else who maybe heard thunder
They synthesize these messages, update their
own belief, and pass the new synthesis along
to you. And belief propagation is just that process
happening mathematically. Yes, exactly. It's
a message passing algorithm. all the variables
in the network, continuously pass mathematical
messages back and forth along the connected strengths.
Just constantly updating. Right, updating their
probabilities based on what their neighbors are
telling them until the entire room reaches a
consensus about the most likely state of the
world outside. And where does the factor graph
come in? A factor graph is just a specialized,
highly efficient map designed specifically to
organize how those messages are passed, making
sure the computation doesn't just get totally
overwhelmed by all the chatter. Okay, this brings
us to the ultimate so -what of this entire deep
dive. We've built these magnificent structures
in our minds. We have tollbooths blocking information,
we have local neighborhood scoring systems, we
have diplomatic treaties between arrows and lines,
and cocktail parties passing messages of belief.
It's a busy world. It is, but outside of theoretical
mathematics, what's this actually do? The source
has a whole section dedicated to applications,
and it claims this framework provides algorithms
for quoting, again, discovering and analyzing
structure in complex distributions. When you
look at the list of applications, it really reads
like a catalog of modern technological miracles.
Causal inference, information extraction, decoding
low -density parity check codes, modeling protein
structures. Gene regulatory networks, too. Let's
tie this all the way back to the hook from the
beginning of our deep dive. Speech recognition.
When I'm talking to my phone, how is it using
these maps to understand me? So historically,
speech recognition relied heavily on a specific
type of directed graphical model called a hidden
Markov model. When you speak, the phone receives
a chaotic wave of acoustic noise. It has to guess
the hidden state, which is the actual word you
intended to say based on the observed state,
which is the messy audio data. And it uses the
arrows to figure out the context. It uses the
conditional dependencies, yeah? The model knows
the statistical rules of the English language.
It knows that if the last hidden state it guessed
was the word peanut, the probability that the
next directed arrow points to the word butter
is incredibly high. Right, peanut butter. Exactly.
The probability that the next arrow points to
elephant is near zero. It uses the structure
of the graph to filter out the noise and lock
in on the most probable sequence of words. That's
amazing. The text also mentions computer vision.
I imagine that uses a different map entirely.
Usually computer vision relies on undirected
models, like the Markov random fields we discussed
earlier. Think about a digital photograph. Just
a bunch of pixels. Right, it's just a grid of
millions of pixels, each with a color value.
If a computer is trying to identify a stop sign
in a noisy image, it looks at the mutual correlation
between neighboring pixels. Oh, the local neighborhood
scoring system again? You got it. The algorithm
knows that if one pretzel is red, The pixels
immediately connected to it are statistically
highly likely to also be red. So it groups them?
Right. It uses those undirected connections to
smooth out the noise, group the red pixels together,
recognize that octagonal shape, and confidently
conclude that is a stop sign. It finds the structure
in the static. It is kind of blowing my mind
that the underlying logic used to read my texts
is the same underlying logic used by a self -driving
car to not run a red light. It's universal. And
the source also lists disease diagnosis. A medical
software program using a Bayesian network can
take dozens of disparate variables, a patient's
age, a slight fever, a specific genetic marker,
a recent travel history, and use those directed
arrows to calculate the actual probability of
a specific underlying disease. It is profound
when you step back and look at it. This raises
an important question about what it means to
discover structure. How so? Well, what do humans
speech? The pixels in a photograph and the symptoms
of a rare disease have in common. To the naked
eye, nothing. But to a probabilistic graphical
model, they're all just complex distributions
of random variables. They're all just pushpins
on the board. Yes. It means that hidden inside
the seemingly random chaos of speech patterns
or genetic mutations or digital noise, there's
an underlying structure of dependence. PGMs are
just the flashlights we use to go into that dark
room and find that structure. It's the invisible
architecture running beneath the surface of modern
science. It forces you to realize that very few
things are truly entirely independent. The world
is really just a web of conditional probabilities.
Let's take a second to recap the journey we've
been on because we have covered a massive amount
of intellectual ground today. We really have.
We started with a pile of random variables and
learned that to cheat the impossible math of
calculating everything at once, we have to build
a map, a graph. to map the flow of information.
Right. If the relationship is just a mutual correlation
sitting at the same table, we use Planeline's
undirected Markov models. And we evaluate them
using those local neighborhood scores. But if
there is a flow of influence, a parent -child
dynamic driving events forward, we use directed
arrows, Bayesian networks. We learned how observing
a middle event can act like a toll booth, shutting
down the flow of new information through deseparation.
And when the world refuses to fit into neat boxes,
we embrace the hybrids. We allow feedback loops
with cyclic graphs. We forge diplomatic treaties
with chain graphs. And we pass messages of belief
through factor graphs. the math to the messiness
of reality, and ultimately using it to decode
the world around us, from our DNA to human speech.
It is a dense, heavy topic, but the elegance
of how it all connects is just incredible. Thank
you for joining us on this exploration. We tackled
a highly technical subject today, but hopefully
you're walking away with a brand new lens through
which to view the technology you use every single
day. It certainly changes how you perceive randomness.
You start seeing the underlying strings connecting
everything. It really does. And actually, there
is one final lingering thought I want to leave
you with. Oh, I like these. The source notes
over and over that graphical are used for discovering
and analyzing structure in complex distributions.
It makes you wonder, if you were to sit down
right now and draw a probabilistic graphical
model of your own life. Wow. OK. Yeah, mapping
out all the seemingly random variables of your
daily choices, your habits, your social environment,
your stress levels, what hidden dependencies
would you discover? What unexpected parent nodes
are secretly acting as toll booths in your life,
controlling your outcomes and pulling the strings
without you even realizing it? That is a question
worth spending some time on. Something for you
to mull over until next time. Keep exploring.