Statistical inference is the process of drawing conclusions about populations or scientific truths from data. There are many modes of performing inference including statistical modeling, data oriented strategies and explicit use of designs and randomization in analyses. Furthermore, there are broad theories (frequentists, Bayesian, likelihood, design based, …) and numerous complexities (missing data, observed and unobserved confounding, biases) for performing inference. A practitioner can often be left in a debilitating maze of techniques, philosophies and nuance. This course presents the fundamentals of inference in a practical approach for getting things done. After taking this course, students will understand the broad directions of statistical inference and use this information for making informed choices in analyzing data.
02 02 Probability mass functions
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來自 Johns Hopkins University 的課程
Statistical Inference
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從本節課中
Week 1: Probability & Expected Values
This week, we'll focus on the fundamentals including probability, random variables, expectations and more.
與講師見面
Brian Caffo, PhDProfessor, Biostatistics
Bloomberg School of Public Health
Roger D. Peng, PhDAssociate Professor, Biostatistics
Bloomberg School of Public Health
Jeff Leek, PhDAssociate Professor, Biostatistics
Bloomberg School of Public Health
So, probability calculus is useful for understanding the rules that probability
must follow and forms the foundation for all thinking on probability.
However, we need something that's a bit easier to work with for numeric outcomes
of experiments, where here I'm using the word experiment in a broad sense.
So densities and mass functions for random variables are the best starting point for
this, and this is all we will need.
Probably the most famous example of a density is the so called bell curve.
So in this class you'll actually learn what it means to say that
the data follow a bell curve.
You'll actually learn what the bell curve represents.
Among other things, you'll know that when you talk about probabilities associated
with the bell curve or
the normal distribution, you're talking about population quantities.
You're not talking about statement about what occurs in the data.
How this is going to work and
where we're going with this is that we're going to collect data that's going to be
used to estimate properties of the population, okay.
And that's where we'd like to head.
But first before we start working with the data, we need to develop our intuition for
how the population quantities work.
A random variable is the numeric outcome of an experiment.
So the random variables that we will study come in two kinds, discrete or continuous.
Discrete are ones that you can count, like number of web hits, or the number,
the different outcomes that a die can take.
Or even things that aren't even numeric, like hair color.
And we can assign the numeric value, one for blonde, two for brown, three for
black, and so on.
Continuous random variables can take any value in a continuum.
The way we work with discrete random variables is,
we're going to assign a probability to every value that they can take.
The way that we're going to work with continuous random variables is we're
going to assign probabilities to ranges that they can take.
Let's just go over some simple examples of things that can be thought of
as random variables.
So the biggest ones for building up our intuition in this class will be
the flip of a coin, and we'll either say heads or tails, or 0 or 1.
This is discrete random variable, because it could only take two levels.
The outcome of the roll of a die is another discrete random variable,
because it could only take one of six possible values.
These are kind of silly random variables because they're,
the probability mechanics is so conceptually simple.
Some more complex random variables are given below.
For example, the amount of web, the website traffic on a given day,
the number of web hits, would be a count random variable.
We'll likely treat that as discrete.
However we don't really have an upper bound on it.
So it's an interesting kind of discrete random variable.
We'll use the Poisson distribution likely to model that.
The BMI of a subject four years after a baseline measurement is,
a somewhat random example I came up with.
And in this case we would likely model BMI body mass index as
a continuous random variable.
The hypertension status of a subject randomly drawn from a population could be
another random variable.
Here we might give a person a one if they have hypertension or
were diagnosed with hypertension, and a zero otherwise.
So, this random variable would likely be modeled as discrete, or
would be modeled as discrete.
The number of people who click on an ad.
Again, this is another discrete random variable, but a, but an unbounded one.
But still, we would still assign a problem, probability, for zero clicks,
one clicks, two clicks, three clicks and so on.
So intelligence quotients are often modeled as continuous.
When we talked about discrete random variables,
we said that the way we're going to work with probability for
them is to assign a probability to every value that they can take.
So why don't we just call that a function?
We'll call it the probability mass function.
And this is simply the function that takes any value that a discrete random variable
can take, and assigns the probability that it takes that specific value.
So a PMF for a die roll would assign one sixth for the value one,
one-sixth for the value two, one-sixth for the value three, and so on.
But you can come up with rules that a PMF must satisfy in order to then satisfy
the basic rules of probability that we outlined at the beginning of the class.
First, it must always be larger than zero because we, larger than or
equal to zero, because we've already seen that a probability has to
be a number between zero and one, inclusive.
But then also the sum of the possible values that the random variable can
take has to add up to one, just like if I add up the probability a die takes
the value one, plus the probability that it takes the value two,
plus the probability it takes the value three, plus the value it takes four, five,
six, that has to add up to one, otherwise.
The probability of something happening, right,
that the die takes one of the possible values, would not add up to one,
which would violate one of our basic tenets of probability.
So all a PMF does, has to satisfy, is these two rules.
We won't worry too much about these rules.
Instead, we will work with probability mass functions that
are particularly useful, like the binomial one, the canonical one for
flipping a coin, and the Poisson one, the canonical one for modelling counts.
Let's go over perhaps the most famous example of a probability mass function,
the result of a coin flip, the so-called Bernoulli distribution.
So let's let capital X be the result a coin flip,
where X equals 0 represents talks and X equal 1 represents heads.
Here we're using the notation where an upper case letter represents a potentially
unrealized value of the random variable.
So it makes sense to talk about the probability that x equals 0 and
the probability that capital X equals 1.
Where as a lower case x is just a placeholder that we're going to
plug a specific number into.
So in the PMF down here, we have p x equals one half to the x,
one half to the one minus x.
So if we plug in x equal to 0 we get one half, and
if we plug in x equal to 1 we get one half.
And this merely says is that the probability that the random
variable takes the value 0 is one half and
the probability that the random variable takes the value 1 is also one-half.
This is for a fair coin.
But what if we add an unfair coin?
Let's let theta be the probability of a head and 1 minus theta be
the probability of a tail, where theta's some number between 0 and 1.
Then we could write our probability math function like this.
P of x is theta to the x, 1 minus theta to the 1 minus x.
Now, notice if I plug in a lower case x of 1 we get theta, and if I plog in a,
plug in a lower case x of 0 I get 1 minus theta.
Thus for this population distribution the probability a random variable takes
the random 0 is 1 minus theta.
And the probability that it takes the value of 1 is theta.
This is incredibly useful, for example, for modeling the prevalence of something.
For example, if we wanted to model the prevalence of hypertension,
we might assume that the population or the sample that we're getting
is not unlike flips of biased coin with the success probability theta.
And just to connect it to what we'll be doing in the future.
The issue is that we don't know theta.
So, we're going to use our data to estimate this population proportion.
