Bayes Theorem – remember that mentioned way, way back in Lecture 2? No, it isn’t some new age way of predicting who you’ll be romantically involved with this winter, but there is a field of inference that comes from this theorem that plays into Bayesian statistics, and that is a subfield of statistics many scientists should be paying attention to.
Up until this point, we’ve basically been learning more frequentist statistics than Bayesian statistics (i.e., heavy on the linear regression, chi-squares, correlations, less so on multiple comparisons, etc.). This is evident by our HistoryStats projects: we’ve been looking at the lives and work of some of the founders of frequentists’ school of thought like Neyman, Pearson, and Wald. How do we best describe these frequentist statisticians? Well, let’s take a simple, intuitive analogy described by this StackExchange forum. According to “user28” having a frequentist frame of mind is like hearing the phone go off, referring to a model upon which helps you identify the area of your home that phone is going off to make the inference on where the phone is. Having a Bayesian frame of mind means you may have that model in mind, but you also take into account places where you’ve mistakenly left the phone in the past. Simply, frequentists believe that data is a frequency, or a repeatable random sample, while Bayesians believe that data is observed from a real sample. Furthermore, frequentists believe that parameters are fixed, whereas Bayesians believe the parameters to be unknown but can be described by probabilities. (So…that would make Fisher’s maximum likelihoods a closet Bayesian statistic, wouldn’t it?).
TJ gave us some great examples of Bayes Theorem applied to real life, like the probabilities in clinical trials with cancer treatments. However, we never really got to see how Bayesian inference affects the experiment’s statistics and experimental design.
To understand the experimental design, we need to understand exactly how experimental design is updated or modified by Bayes Theorem, generally. Let’s say you are going to flip a coin 10 times and you suspect a probability distribution to describe these coin flips. Therefore, h would represent the probability of heads, and p(h) would represent the distribution settled on prior to any coin flips. Then the coin is flipped and way more heads come up than usual, say 8 heads. By using Bayesian inference, we need to update our prior belief about the coin – it’s now unfair. So our new beliefs may be modeled like p(h|f) where f is the number of heads experienced in those 10 flips. This abstraction is read as “what is the probability distribution of heads given the number of heads resulting from 10 tosses [in this case 8]?” This seems like a reasonable update as we pare down our hypotheses to fit our experimental data. Mathematically, the update would look like p(h|f) = u(h, f) x p(h) where u(h,f) is an updating factor written out as u(h, f) = (l(f|h))/l(f) where l(f|h) is a likelihood function or the probability we observed 8 heads given the parameters we modeled in the beginning. The denominator of the updating factor is just the likelihood of the data under no conditions. Because Bayesian statistics doesn’t believe parameters are fixed, they can have conditions added to them. Therefore, the likelihood of the data can be written as an integral l(f) = ∫l(f|h)p(h)dh (this is similar to a general expectation value). The denominator turns out to be a weighted average of likelihoods across all possible parameters. Or simply, a ratio that is able to tell you what parameter values are most likely.
|Darth Vader: crafty with a lightsaber and some conditional probabilities.|
How does this play out in the lab? Let’s take a hypothetical animal trial where dose concentrations of many drugs are tested on large amounts of animals to test their potencies. The lab wants to apply regression analyses to the different drugs based on the specimen they inject the drug into. For experimentation of one drug, the experimental design included six equally spaced doses given to ten mice each; so, 60 animals to test a range of concentrations for one drug. The investigators measured the number of surviving mice one week after drug administration. It turns out that about 90% of mice died at high concentrations of the drug, while 10-20% died at low concentrations of the drug. After each of the experiments, maximum likelihood estimations were used to estimate an LD50 value (or the dose at which the probability of mice dying is 50%). As it turns out, the investigators used results from the first few sets of experiments to predict a distribution for following experiments, in anticipation of constructing an updating factor, as described above. In total, if 50 drugs are tested with similar experimental design, the investigators can use these 50 LD50 values as a sample from a distribution of LD50 values.
Overall, these Bayesian inferences and the statistics are mathematically rooted in Bayes Theorem. This theorem relies on conditional probability. These conditional probabilities make the system easy to update and a noteworthy design for scientists to consider -- because writing grant proposals on frequentist assumptions can be dangerous when we try to predict a model for data without any prior knowledge of the system.