Creencias y núcleos de sentido que configuran estereotipos en estudiantes universitarios

While all estimation problems in this thesis consider continuous parameter spaces, the concepts of Markov chain theory are simpler for discrete spaces. For this reason, definitions are given first for discrete spaces, followed by their continuous counterparts. For continuous parameter spaces, the probability of visiting any single value is 0. To be able to treat these chains similarly to discrete Markov chains, probabilities of single values have to be replaced by probabilities of measurable sets. For the purposes of this thesis, it suffices to note that a measurable set is a set that can be assigned a probability. For readers who desire a measure-theoretic treatment of probability, Rosenthal (2006) is recommended. A more thorough treatment of discrete spaces can be found in Roberts (1996). For comprehensive treatments of general spaces, see Robert and Casella (2004), Tierney (1994), Tierney (1996), Roberts and Rosenthal (2004) and Meyn and Tweedie (2009). In the definitions below, the parameter space

is denoted byΘ, while the set of all measurable subsets of Θis denoted B(Θ). A chain that is useful for MCMC should satisfy the following properties:

Irreducibility For discrete chains, let τy be the time index of the first time statey

is visited. A Markov chain is said to beirreducible if

P(τy <∞|θ(0) =x)>0, ∀x, y∈Θ. (3.57)

This means that it is possible to move from any parameter value to any other in a finite number of steps. If this property does not hold, only parts of the parameter space would be visited, with the initial value determining which parts. Such a chain cannot have a unique stationary distribution that is independent of the starting value.

A Markov chain on continuous spaces is said to beϕ-irreducible for a probability distributionϕif

P(τA<∞|θ(0)=x)>0, ∀x∈Θ, A∈ B(Θ) (3.58)

for any measurable setA such thatϕ(A)>0, whereτAis the first time that a

state inA is visited. As in the discrete case, this means that the probability of moving from x to a point in A in a finite number of steps is greater than zero. This definition is less strict than irreducibility, as it allows sets of zero probability to be unreachable. For the purposes of MCMC, we require that ϕ=π, so that all sets that are assigned positive probability by the stationary distribution can be reached from any starting state.

Aperiodicity A discrete Markov chain is said to aperiodic if the following holds:

greatest common divisorni >0 :κi(θ(i)=x|θ(0)=x)>0o= 1, ∀x∈Θ

(3.59) whereκi _{is thei-step transition kernel, κapplieditimes. If this condition does}

not hold, the chain will be split into subsets that are visited in a cyclic fashion, and the chain cannot have a stationary distribution.

In the continuous case, a Markov chain is periodic if it can be divided into disjoint sets such that each set is visited in a cyclic fashion. This definition can be made more rigorous by introducing the concept of small sets, something that is beyond the scope of this thesis. For more information, see Robert and Casella (2004).

x, and let τxx be the number of steps for the chain to return to state x,

τxx= min

i:θ(i) =x|θ(0) =x . A Markov chain is said to be recurrent if

E[ηx] =∞, ∀x∈Θ. (3.60)

The chain is positive recurrent if additionally

E[τxx]<∞, ∀x∈Θ. (3.61)

A chain that is not recurrent is calledtransient. If the chain is transient, there are states that are only visited finitely many times, so their proportion will be zero as the number of samples tends to infinity.

For continuous spaces, defineηAas the number of times the chain visits the set

A. A chain is recurrent if it isϕ-irreducible for someϕ, and

E[ηA] =∞, ∀A∈ B(Θ) :ϕ(A)>0. (3.62)

Often, a stronger form of recurrence is required, known asHarris recurrence. A chain is Harris recurrent if it is recurrent, and additionally

P(ηA=∞) = 1, ∀A∈ B(Θ) :ϕ(A)>0. (3.63)

Harris recurrence guarantees that any realisation will visit each set infinitely often, with probability1. Positive recurrence on the other hand only implies that each set willon average be visited infinitely often. For a particular realisation of the chain however, this may not hold, if the chain was started at an unfortunate parameter value.

Aϕ-irreducible, aperiodic chain withπas its stationary distribution is positive recurrent (Roberts and Rosenthal 2006). It can be shown that for a chain withπ as its stationary distribution, if it isπ-irreducible, aperiodic and Harris recurrent, then the following holds:

kκi(·|θ(0)=x)−πk →0, ∀x∈Θ (3.64) wherek · kdenotestotal variation distance (Tierney 1994). This means that the chain converges toπ independent of the starting state. While a proof will not be presented here, the following intuitive argument can be made for discrete chains: suppose we

start a chainC1 from any state x. Start another chain C2, initialised by drawing from the stationary distribution. SinceC2 is Harris recurrent, it will reach statex in a finite number of steps. Next, remove all samples fromC2 before the first time xis visited. Since the removed part has finite length, the statistics of the resulting chain will not be affected in the limit. But this modified chain starts in the same state asC1, and these chains necessarily have the same statistics due to the Markov property. A more detailed discussion can be found in Meyn and Tweedie (2009). For continuous chains, more care has to be taken, but the basic idea remains the same. Such a chain also obeys the SLLN.

Assuming that some mild technical conditions hold (Tierney 1996), which here will always be assumed to be the case, there is a variant of the CLT that holds forπ-irreducible, aperiodic, Harris-recurrent chains:

√

N¯hN −Eπ[h(θ)]

σ → N(0,1) asN → ∞ (3.65)

where now σ2 _{is given by:} σ2= Varhhθ(i) i + 2 ∞ X k=1 Covhhθ(i) , hθ(i+k) i . (3.66)

This result shows that strong correlations between samples will result in high variance. As expected, this expression reduces to the CLT (3.53) when the samples are uncorrelated. When constructing Markov chains, it is therefore important to strive to minimise correlations between samples.

In practice, Markov chains for MCMC are often constructed to satisfy the

detailed balance condition (Geyer 2011), namely:

π(y)κ(x|y) =π(x)κ(y|x), ∀x, y∈Θ. (3.67) This states that the probability of being in state x and moving to y is the same as the probability of being in statey and moving tox. A chain satisfying this condition is calledreversible. Detailed balance is a special case of the stationarity condition, which can be seen by integrating both sides with respect tox:

π(y) Z Θ κ(x|y) dx= Z Θ π(x)κ(y|x) dx⇐⇒π(y) = Z Θ π(x)κ(y|x) dx, (3.68)

sinceκ(x|y) is a probability distribution and therefore integrates to1 overΘ. While detailed balance is just a sufficient and not a necessary condition for a chain to have

π as its stationary distribution, it is often used in practice.

For more information on the general aspects of MCMC, see Gilks et al. (1996), Brooks et al. (2011), Chib and Greenberg (1995), Geyer (1992) and Kass et al. (1998).