You know, whether you are asking Siri to set
a timer, or investing your life savings in the
stock market, or even having your DNA analyzed,
you are interacting with a ghost. A mathematical
ghost, to be precise. Right, yeah, not a literal
ghost, but there's this... this invisible architecture
operating right beneath the surface of our digital
lives. And it's constantly making these highly
educated guesses about things it can't directly
see. It is entirely behind the scenes. Exactly.
So today we're opening the hood on that invisible
architecture. We're doing a deep dive into the
hidden Markov model. This is the mathematical
engine that allows a machine to deduce the invisible
forces shaping our world purely by looking at,
well, the shadows they leave behind. We're using
a comprehensive breakdown from Wikipedia as our
source map today. It is a fantastic topic. The
mission today is really to demystify how these
algorithms actually work in practice. Yeah, so,
okay, let's unpack this. At its core, a hidden
Markov model or an HMM is basically a system
split into two distinct layers, right? That's
right, two layers. You have one layer. which
is just a sequence of events we can actually
observe in the real world. And then you have
the second layer, which is a sequence of hidden
states. We can't see them, but they are directly
causing those observable events. And the primary
objective of this model is to reverse engineer
reality. It tries to learn about that hidden
layer strictly by observing the visible layer.
Like playing detective. Exactly like that. And
the engine that makes this possible, the core
rule, is what we call the Markov. property. Which
is kind of a weird rule when you think about
it. It is. It's a strict mathematical rule stating
that the current hidden state is influenced only
by the state immediately preceding it. It operates
with zero long -term memory. Zero. None at all.
Right. What happened two steps ago or ten steps
ago doesn't mathematically matter to the current
state. Which, I mean, if you think about how
human memory works or even real -world physics,
that sounds totally counterintuitive at first.
Like, why force the system to have amnesia? Because
without that amnesia, the calculations would
literally break modern computers. By assuming
that only the immediate past matters, the Markov
property takes this chaotic, infinitely complex
universe and simplifies it into calculable probabilities.
Oh, I see. So it's a shortcut. A necessary one.
If a system had to remember the entire history
of everything that led up to a single moment,
just to guess what happens next, the math becomes
intractable. The Markov property trims the fat
so the algorithm can actually run. I mean, the
source material gives us two fantastic scenarios.
Let's start with the Alice and Bob weather game.
A classic example. Yeah, it's great. So imagine
two friends, Alice and Bob, who live far apart
and talk on the phone every day. Bob is a creature
of habit. He really only does three things. He
walks in the park, he goes shopping, or he cleans
his apartment. And his choice is determined exclusively
by the weather where he lives. Right, exclusively.
The catch is that Alice has no idea what the
weather is actually like where Bob lives. So
the weather is the hidden state. And the only
things Alice has access to are the observations.
So Bob telling her, you know, hey, I cleaned
my apartment today or I went for a walk. Exactly.
But Alice isn't totally in the dark here. She
knows the general rules of his city. Like, she
knows it rains a lot there. She knows that if
it's rainy today, there is maybe a 30 % chance
tomorrow will be sunny. And she knows his habits.
Right. She knows that if it's raining, there's
a 50 % chance he'll stay in and clean. But if
it's sunny, there's a 60 % chance he'll go for
a walk. So using just those observations and
those probabilities, Alice has to reconstruct
the hidden weather patterns day by day. The text
provides another, perhaps more mechanical example
to illustrate this architecture. Imagine a genie
in a hidden room. Okay, a genie. In this room,
there are several urns, and each urn contains
a specific known mix of uniquely labeled balls.
The genie randomly picks an urn, draws a ball,
and drops it onto a conveyor belt that leads
out of the room. So you, standing outside the
room, you just see the sequence of colored balls
rolling down the conveyor belt. You never see
the urns. Never. You only see the output. Watching
this model feels a lot like sitting in Plato's
cave. You know, like you're watching the shadows
on a cave wall to try and guess the shape of
the physical object casting them. You only ever
see the byproduct of the reality, never the reality
itself. Plato's cave is exactly the right way
to think about it. The sequence of balls rolling
out of the room is the shadow on the wall. and
the hidden math is trying to reverse -engineer
the true object casting it. To do that, the architecture
of a hidden Markov model relies on two specific
sets of rules. First you have transition probabilities.
Transition probabilities, okay. These are the
rules for moving from one hidden urn to the next
or, you know, one weather state to the next.
If you have a specific number of hidden states,
say, n states, you have an n squared matrix of
transition probabilities mapping every possible
move. From any one state to any other. Correct.
Then you have the emission probabilities. These
dictate how likely a hidden state is to produce
a specific observation, like Bob's 50 % chance
of cleaning in the rain or the 80 % chance that
urn number three produces a red ball. Wait, I
have to push back here on behalf of the listener
for a second. Go ahead. If the genie is choosing
urns based only on the single previous urn...
or if the weather today only depends on yesterday,
because of that strict Markov property we talked
about, doesn't that make the system incredibly
short -sighted? It does seem that way. I mean,
how can this accurately mild complex reality
if it has zero long term memory? It feels like
trying to predict a novel by only looking at
the previous word. What's fascinating here is
that while it feels limiting, this is where the
math gets incredibly elegant. Even though the
model only looks one step back, these strict
interlocking probabilistic rules compound. Be
compound. Well, it is not just looking at one
transition in a vacuum. It is analyzing a long
chain of them. When you chain enough of these
short -term dependencies together through transition
and emission matrices, they become incredibly
powerful at mapping highly complex, seemingly
long -term patterns in the data. Oh, wow. A single
word doesn't tell you the plot of the novel,
but observing the transition probabilities of
10 ,000 words in a row absolutely maps out the
grammar and tone of the book. Okay, that makes
sense. So we've got the rules of the game. Yeah.
We know how the matrices are set up. Now how
do we actually win the game? Right. How do we
solve the puzzle? Exactly. How do we use this
model to solve the puzzle of the hidden truth?
The source lays out three main inference tasks
for these latent or hidden variables. There are
three major questions we can ask the model to
solve. The first is called filtering. Filtering,
right. This is when we want to find the distribution
over the hidden states at the very end of a sequence.
We use something called the forward algorithm
to figure out what state the process is in right
at this exact moment, based on accumulating the
probabilities of all the observations leading
up to it. OK, so that's the present moment. Then
there's the second task, which is smoothing.
Yes, smoothing. This is when you're looking backward.
You want to find the distribution of a hidden
state somewhere in the middle of a past sequence.
To do this, the model uses the forward -backward
algorithm. Which is very clever mathematically.
Yeah, it calculates the probabilities from the
beginning of the sequence up to that middle point
and also from the end of the sequence backward
to that middle point to basically pinch the exact
probability. And finally, the third task is finding
the most likely explanation. This doesn't just
look at one point in time. It asks, what is the
joint probability of the entire sequence of hidden
states that generated our observations? The whole
thing. The whole sequence. And for this, we use
the famous Viterbi algorithm. OK, here's where
it gets really interesting. Let's put this into
real -world terms for you. Think of these three
tools like tracking a friend on a cross -country
drive. I like this analogy Yeah, so filtering
is like using their current speed and direction
to guess exactly where they are right now smoothing
is trying to figure out which specific gas station
they stopped at three hours ago based on their
overall trajectory. Right. But the most likely
explanation, the Viterbi algorithm, is like looking
at a pile of crumpled gas receipts and reconstructing
the entire cross -country road trip map from
start to finish. The road trip analogy perfectly
captures the scale of the Viterbi algorithm,
and it brings up a crucial mechanical distinction.
Which is? You might wonder... Why can't we just
use the filtering tool over and over again? Why
not just find the most likely state for step
one, then step two, then step three, and stitch
them together to make our map? Right. I was actually
just thinking that. If I know the most likely
weather for Monday, Tuesday, and Wednesday individually.
Shouldn't that be the most likely sequence for
the week? Not necessarily. The most likely individual
states don't always form the most likely continuous
sequence. Wait, really? Why not? Because the
transitions between certain states might be mathematically
impossible or highly improbable. If you try to
guess a 10 word sentence, the number of possible
grammatical combinations is astronomical. If
a machine brute forced it by checking every single
path through the matrix, it would take years.
Oh, because of all the branches. Exactly. The
brilliance of the Viterbi algorithm is dynamic
programming. Instead of mapping every route,
it only remembers the single best path to reach
the current word, instantly discarding all the
suboptimal routes. It cuts a process that should
to take centuries down to milliseconds. So it's
ruthlessly efficient. It prunes the dead ends
immediately instead of following them to the
finish line. Exactly. The source gives a great
example regarding part of speech tagging and
language processing. If you are trying to understand
a sentence, you don't just want the probability
of what part of speech a single word is. Right.
You need context. To actually make sense of the
grammar, you need the entire sequence of parts
of speech to align logically. You need the whole
road trip. Which means you require the viterbi
algorithm to find that most likely explanation
for the whole sentence Ensuring that a noun actually
follows an adjective in a way that makes linguistic
sense that makes total sense We use the terby
when the context of the whole sequence is what
actually matters but All of this puzzle solving
assumes we already know the rules, right? He
does assume that. Like, it assumes that we know
Alice's probabilities for Bob's weather, or the
exact mix of colored balls and the genies' urns.
How does a machine learn the rules of the hidden
world from scratch if no one inputs them? And
where does this actually touch your daily life?
Well, the history here is quite rich. The core
math was developed in the late 1960s by Leonard
E. Baum and his colleagues, but it really took
off in the mid -1970s when it was applied to
speech recognition. Speech recognition, okay.
And then it exploded in the 1980s with bioinformatics,
specifically analyzing DNA sequences. Today,
the applications are endless. We are talking
computational finance, protein folding, handwriting
recognition, even predicting solar irradiance
variability. And Siri. The source specifically
mentions Siri's speech recognition. Yes. Siri
is a perfect everyday example. But let's dig
into the mechanics of that. We're saying the
algorithm learns the transition and emission
probabilities, but how can it possibly learn
the rules of the hidden states if those states
are by definition, hidden. It is a paradox, isn't
it? And I really need to logic this out. If it's
guessing the rules based on the data, but it
needs the rules to understand the data, isn't
it just spinning its wheels? How does tweaking
the probabilities actually lock it into the correct
pattern instead of just, you know, a mathematically
convenient hallucination? It is the ultimate
chicken and egg problem. If we connect this to
the bigger picture to solve it, we rely on algorithms
like Baumwelsch, which is a type of expectation
maximization algorithm, or EM. For more complex
time series predictions, systems might use Markov
chain Monte Carlo or MCMC sampling. Okay, lots
of acronyms. True, but the core logic of expectation
maximization directly answers your question.
It uses iterative local maximum likelihood estimates.
Okay, let's translate that for everyone. Think
of it like trying to tune a blurry radio station
in the dark. Great way to visualize it. You don't
know the exact frequency. Those are the hidden
rules. But you can hear the static, which are
the errors in your output. Expectation maximization
is basically the process of turning the dial
just a millimeter to the left. Did the audio
get clearer? Yes. Keep going. Did it get worse?
Turn back. It iterates. Right. It iterates this
microtuning until the music comes through crystal
clear. That is precisely how it functions. The
algorithm splits the work into two distinct steps.
First is the expectation step. Given its current
fuzzy guess of the rules, it asks, what is the
most likely sequence of hidden states? It makes
a rough map. Okay, step one is the rough map.
Then comes the maximization step. Given that
rough map, it asks, how should we mathematically
update our transition and emission rules to make
this specific map even more likely to occur?
Oh, I see. It updates the rules, which changes
the map, which updates the rules again. It loops
this process over and over until the probabilities
stabilize, the static fades, and the pattern
locks in. So it literally pulls itself up by
its own mathematical bootstraps. It does. And
that brings it right back to you, the listener.
Every single time your phone's voice assistant
understands your spoken words, translating the
messy audio waves of your voice into a hidden
sequence of grammatical text, it is solving this
exact probabilistic puzzle. It's using expectation
maximization to tune the radio dial of your speech.
It is quite literally mapping the audio shadows
back to the linguistic urns in fractions of a
second. But wait, there is a fatal flaw in everything
we've talked about so far. A flaw? Yeah. Reality
doesn't happen in neat discrete steps like drawing
a single ball from an urn or flipping a coin
between rainy and sunny. The real world is continuous.
You know, it's messy. The stock market doesn't
just go up or down in rigid boxes. It fluctuates
wildly across a continuous spectrum. That is
very true. So what happens to our Markov model?
when the data doesn't fit into clean little boxes.
It evolves. The source outlines several major
extensions to handle exactly this kind of complexity.
When you were dealing with continuous state spaces
like tracking the continuous real -time trajectory
of a rocket through the atmosphere rather than
discrete weather states, the math shifts. Oh,
so. We use things like Kalman filters, which
adapt the Markov model to handle data that flows
without rigid breaks. But the truly significant
shift in modern AI has been the move toward discriminative
approaches. source heavily emphasizes this shift.
It details maximum R -SPAY Markov model's MEMS
and linear chain conditional random fields, or
CRFs. But let's not just name -drop the jargon.
How do these actually differ from the classic
hidden urns? The difference is in how they view
the world. Traditional HMMs are generative models.
They try to mathematically recreate exactly how
the hidden states generated the observations
from scratch. They model the entire joint distribution.
OK, recreating the whole world. Yes. A discriminative
model, like a conditional random field, doesn't
care about recreating the whole world. It just
models the conditional distribution. It only
cares about drawing the correct boundaries to
classify the observations. Think of a traditional
generative HMM. as looking at the words in a
sentence through a narrow straw, right? You only
see one word at a time, strictly limited by that
one -step Markov property we talked about. Yes,
the amnesia. Right, the amnesia. Conditional
random field takes the straw away. It allows
the model to look at the whole sequence simultaneously.
It recognizes context, like the fact that the
word bank... It means something entirely different
if the word river is nearby rather than if the
word money is nearby. Exactly. By taking the
straw away, CRFs solve what is known as the label
bias problem. They allow engineers to inject
domain -specific knowledge and look at combinations
of nearby observations all at once. Which is
huge for complex data. It is. We are also seeing
the integration of recurrent neural networks,
specifically reservoir networks, which feed temporal
dynamics into the model. This helps the HMM handle
non - stationary data. That's data where the
underlying rules of the universe are constantly
shifting over time. Which brings us to a massive
update from the source material. In 2023, two
breakthrough algorithms were introduced, the
discriminative forward -backward and discriminative
Viterbi algorithms. A very recent development.
Yeah, super recent. They bypassed the need to
model the joint distribution entirely using only
conditional distributions. But wait! Uh, doesn't
that rewrite the rulebook? We spent this whole
deep dive saying we needed the full generative
model that earns the specific emission probabilities
to figure out the hidden states. And now we don't.
It is a profound paradigm shift in the mathematics.
By skipping the joint law, by not forcing the
algorithm to learn the entire generative rulebook
of the universe, These 2023 algorithms make the
model vastly more computationally efficient.
Oh, because it's doing less math. Precisely.
You get the sequence -solving power of the Viterbi
algorithm without the immense computational baggage
of a traditional generative model. It makes the
HMM incredibly versatile for cutting -edge AI
applications where speed, scale and efficiency
are paramount. You are getting the road trip
map without having to mathematically simulate
the engine of the car. That is wild. I mean,
we've gone from a theoretical genie drawing balls
from hidden urns in the 1960s to teaching Siri
how to recognize the messy audio of your voice
to these 2023 algorithms that slice through the
underlying math faster and more efficiently than
ever before. It's an incredible evolution. So,
you know, the next time you are looking at a
sequence of events, whether it's the erratic
jumping in the stock market, the sequence of
nucleotides in a DNA test, or just trying to
guess if your friend is going to clean their
apartment based on the weather. Remember, there
is a hidden Markov chain operating right beneath
the surface, calculating the odds. And before
we wrap up, there is one final, almost philosophical
concept from the measure theory section of the
text that I think you should take with you. Ooh,
go for it. Let's hear it. We established at the
very beginning that the hidden part of a Markov
model strictly depends only on the immediate
past. It operates with mathematical amnesia.
Right, the one -step rule. However, the measure
theory proves that the observable sequence, the
part we actually see, can be non -Markovian.
Wait, meaning the shadows do have a memory, even
if the object casting them doesn't. That is exactly
what it means. For example, if you observe a
long enough sequence of a specific outcome on
the visible layer, let's call it outcome B, you
might become increasingly mathematically certain
that the underlying hidden state is A. This implies
that the visible, observable part of the system
can actually be affected by something infinitely
far back in the past, even if the hidden engine
driving it only looks one step back. Whoa! So
even if the hidden mechanics driving the universe
only care about what happened yesterday, our
visible world, the shadows we see on the cave
wall, might actually carry the mathematical fingerprints
of the infinite past. Exactly. The observables
remember what the hidden states forget. That
is so cool. It's a mathematical quirk with profound
philosophical weight. The ghost in the machine
might only be looking one step ahead, but the
shadows it casts stretch all the way back to
the beginning. A perfect way to summarize it.
Now that is a thought to leave you with. Until
next time, keep looking for the true shapes casting
the shadows.